Substitution for minimizing/maximizing a tropical linear (fractional) programming

Substitution for minimizing/maximizing a tropical linear (fractional) programming L. T ruffet IMT A tlan tique Departmen t Automation, Pro duction and Computer Sciences La Chan trerie, 4 rue A. Kastler 44300 Nan tes, F rance mail: lauren t.truffet@im t-atlan tique.fr Marc h 30, 2026 Abstract T ropical p olyhedra seem to pla y a cen tral role in static analysis of soft wares. These tropical geometrical ob jects play also a cen tral role in parity games esp ecially mean pa yoff games and energy games. And determining if an initial state of such game leads to win the game is kno wn to be equiv alent to solv e a tropical linear optimization problem. This paper mainly focus on the tropical linear minimization problem using a sp ecial substitution method on the tropical cone obtained by homogenization of the initial tropical p olyhedron. But due to a par- ticular case whic h can o ccur in the minimization pro cess based on substitution w e ha v e to switc h on a maximization problem. Nev er- theless, forw ard-backw ard substitution is known to b e strongly p oly- nomial. The sp ecial substitution developed in this pap er inherits the strong p olynomiality of the classical substitution for linear systems. This sp ecial substitution must not b e confused with the exp onential execution time of the tropical F ourier-Motzkin elimination. T ropical fractional minimization problem with linear ob jectiv e functions is also solv ed by tropicalizing the Charnes-Coop er’s transformation of a frac- tional linear program in to a linear program developed in the usual linear algebra. Let us also remark that no particular assumption is made on the p olyhedron of in terest. 1 Finally , the substitution metho d is illustrated on some examples b orro w ed from the litterature. Keyw ords : (max , +)-algebra, tropical linear programming, fractional pro- gramming, algorithmic complexity . MSC: 16Y60, 90C32, 03D15. 1 Main notations and definitions F or every k ≤ k ′ , k , k ′ ∈ Z w e define the discrete interv al [ | k , k ′ | ] := { k , k + 1 , . . . , k ′ − 1 , k ′ } . The inclusion of sets is denoted ⊆ and the strict inclusion of set is denoted ⊂ . Let us introduce the follo wing notations and definitions for the (max , +)- algebra. The reader is in vin ted to read eg. [3] to find more details on (max , +)-algebra and idemp otent semi-rings. The (max , +)-algebra denotes the set R O := R ∪ {−∞} equipped the “addition”: ( a, b ) 7→ max( a, b ) and the “multiplication” ( a, b ) 7→ a + b . In the sequel w e will use the follo wing notations: a ⊕ b := max( a, b ), a ⊗ b := a + b (sometimes a ⊗ b could b e denoted ab ). 1 := 0 is the neutral element for ⊗ , O := −∞ is the neutral element for ⊕ , O is also the absorbing element of ⊗ . W e will use the follo wing p o w er notation: a ⊗ ( b ) := a × b ( × : usual m ultiplication) . Note that in particular we ha ve: a ⊗ (0) = 1 . The pro duct a ⊗ b ⊗ ( − 1) will also b e denoted: a/b . Because division is required in this paper w e hav e: O ⊗ ( − 1) = + ∞ . Th us, w e work in the complete dioid by adding the top elemen t + ∞ : ( R O , ⊕ , ⊗ ) (1a) with: R O := R ∪ {−∞ , + ∞} , (1b) and the op erations ⊕ and ⊗ are extended as follows: ∀ a ∈ R O , ∀ b  = O , a ⊕ + ∞ = + ∞ , b ⊗ + ∞ = + ∞ , O ⊗ + ∞ = O . (1c) 2 R O is equipp ed with the natural order ≤ defined by: a ≤ b ⇔ a ⊕ b = b. F or any matrix A , a i,. denotes the i th row of A , a .,j denotes the j th column of A and a i,j denotes the entry ( i, j ) of A . The scalar op erations are generalized to matrices as follows: • addition of t wo matrices: ( a i,j ) ⊕ ( b i,j ) := ( a i,j ⊕ b i,j ) • pro duct of t w o matrices: the en try ( i, j ) of the matrix C = A ⊗ B is defined by: c i,j := ⊕ k a i,k ⊗ b k,j (=max k ( a i,k + b k,j )). • comparaison of t wo matrices: A ≤ B means ∀ i, j : a i,j ≤ b i,j . • all vectors are column vectors. And the transp ose op erator is denoted ( · ) ⊺ . • for ev ery m × n -matrix A w e define the submatrix of A denoted A I J b y: A I J := ( a i,j ) i ∈ I ,j ∈ J . (2) Where I ⊆ [ | 1 , m | ] and J ⊆ [ | 1 , n | ]. W e will use the b old sym b ols for the follo wing particular v ectors and matrices. • The b oldsymbol −∞ (resp. + ∞ ) denotes the column vector whic h all comp onen ts are −∞ (resp. + ∞ ). The n um b er of comp onents of the vector is determined by the context. • The matrix I n denotes the n × n -iden tit y matrix. Its diagonal entries are all 1 and its off-diagonal entries are all O . • O m,n will denote the m × n -null matrix (ie, all en tries are O ). When m = n the the square n ull matrix O n,n is denoted O n . 3 T o an y n × n -matrix A we can asso ciate a v alued graph G ( A ) with v alu- ation v suc h that the set of the v ertices of G ( A ) is { 1 , . . . , n } . The v aluation is defined by v i,j = a j,i for all vertices i, j . W e sa y that an arc i → j := ( i, j ) exists if v i,j  = O . A path p of lengh t k ≥ 1 is a series of k consecutiv e arcs whic h has the form: p = i 0 → i 1 → · · · i k − 1 → i k . And the weigh t of p is defined b y: w ( p ) := v i 0 ,i 1 ⊗ · · · ⊗ v i k − 1 ,i k = a i 1 ,i 0 ⊗ · · · ⊗ a i k ,i k − 1 . An elemen tary circuit of lengh t k ≥ 1 is a sp ecial path suc h that i k = i 0 and ∀ l = 1 , . . . , k − 1: i l app ears only once. Let q , s b e t w o n -dimensional v ectors and x = ( x j ) n j =1 . Then, to an y inequalit y of the form: q ⊺ ⊗ x ≥ s ⊺ ⊗ x (3a) Noticing that (3a) is expressed in usual notations as (max( q 1 + x 1 , . . . , q n + x n ) ≥ (max( s 1 + x 1 , . . . , s n + x n ) one can asso ciate the follo wing equiv alen t disjunction recalling that max( a, b ) ≥ c ⇔ ( a ≥ c ) or ( b ≥ c ): OR n j =1 { q j ⊗ x j ≥ s ⊺ ⊗ x } . (3b) And one can asso ciate the follo wing equiv alen t conjonction recalling that max( a, b ) ≤ c ⇔ ( a ≤ c ) and ( b ≤ c ): AND n j =1 { s j ⊗ x j ≤ q ⊺ ⊗ x } . (3c) 2 In tro duction 2.1 Problem description Let A and C b e tw o m × n -matrices and b and d b e tw o m -dimensional column v ectors. In this pap er we aim to solv e b y a sp ecial substitution metho d the follo wing max-plus linear programming problem called minimum pr oblem r ese ar ch ( mpr ): min { z = c ⊺ ⊗ x, x ∈ P ( A, b, C , d ) ∪ { ±∞ }} , (4a) where P ( A, b, C, d ) denotes the max-plus p olyhedron defined by: P ( A, b, C , d ) := { x ∈ R n O : A ⊗ x ⊕ b ≥ C ⊗ x ⊕ d } , (4b) 4 But instead of considering the p olyhedron w e will dev elop substitution metho d on the max-plus cone C ( A, b, C , d ) obtained by in tro ducing h as the homogenization v ariable, ie.: C ( A, b, C, d ) := { ( x, h ) ∈ R n +1 O : A ⊗ x ⊕ b ⊗ h ≥ C ⊗ x ⊕ d ⊗ h } ∪ { ±∞ } . (5a) The linear system of inequalities A ⊗ x ⊕ b ⊗ h ≥ C ⊗ x ⊕ d ⊗ h is supp osed to verify the following condition: ∀ i ∈ [ | 1 , m | ] { i ′ : i ′  = i and ∃ α i ′ ∈ R s.t. a i,. ⊗ x ⊕ b i ⊗ h ≥ α i ′ ⊗ ( c i ′ ,. ⊗ x ⊕ d i ′ ⊗ h ) } = ∅ . (5b) Remark 2.1 The c ondition (5b) c an b e assume d to b e always true b e c ause f ≥ g 1 and f ≥ g 2 is e quivalent to f ≥ g 1 ⊕ g 2 . T o form ulate the minimization problem mpr first we b orrow the F ourier’s tric k (see eg. [13], [20]) used in the usual linear algebra and applying it to the (max , +)-algebra. It comes that min( z = c ⊺ ⊗ x ⊕ c h ⊗ h ) can b e replaced b y: min( z ) and z ≥ c ⊺ ⊗ x ⊕ c h ⊗ h . And then, we will solve the following minimization problem mpr defined as: min( z ) (6a) suc h that: ( x, h ) ∈ R n +1 O , (6b) and z ≥ c ⊺ ⊗ x ⊕ c h ⊗ h A ⊗ x ⊕ b ⊗ h ≥ C ⊗ x ⊕ d ⊗ h. (6c) Where the homogenization v ariable h satisfies the follo wing condition: h > O . (6d) And z and h are not substituable . (6e) Finally , among the v ariables of the problem z , x 1 , . . . , x n w e ha ve: z has the highest priority b ecause b ounded by the cost function . (6f ) Sometimes it will b e useful to denote by mp r ( A, b, C , d, c, c h , z , x, h, ( m, n )) the mpr defined (6a)-(6b)-(6c)-(6d)-(6e)-(6f). 5 2.2 Motiv ations Motiv ations for studying tropical linear programms hav e b een widely ex- plained in several previous works by several authors. In [5] author motiv ate tropical linear optimization b y aiming to solve the problem of the sync hro- nization of m ultipro cessors interactiv e systems. In [10] and [11] authors in- dicate further motiv ations to solve not only tropical linear programms but also fractional tropical linear programms (see § 5 for the description of such problem). One can mention eg.: static analysis of softw ares, finding winning strategy in Mean Pa yoff Game. 2.3 Organization of the pap er In the Section 3 the necessary materials to solv e the minimization problem mp r . The substitution metho d is describ ed in Section 4. The reader m ust b e a w are that the substitution m ust not b e confused with the tropical F ourier-Motzkin elimination pro cedure whic h has an exp o- nen tial execution time [2]. The substitution is based on the prametrized research domain (22a)-(22b) for the x j ’s and the homogenization v ariable h . Dominating v ariable is de- fined in Definition 4.1. And the hierarc hy ⪯ v ar b et w een v ariables x j (33) c haracterizes the fact that the v alue domain of the dominating v ariables has non trivial upp er and low er b ounds. The partition of the ( x, h )-linear functions into h -bounded and h -un b ounded functions is defined in Defini- tion 4.2. The hierarc h y ⪯ f ct b et w een ( x, h )-linear functions (36) is based on the obvious prop ert y that v alue domain of a h -b ounded ( x, h )-linear func- tion α ⊺ ⊗ x ⊕ β ⊗ h only dep ends on β and is strictly included in the v alue domain of the function α ⊺ ⊗ x (see (34a)-(34b)-(34c)). The Theorem 4.3 of § 4.2.2 c haracterizes the relev ant v ariable x j ∗ whic h can be substituted. And the Theorem 4.2 of § 4.2.2 ensures the optimalit y of the result. The up date of the mp r is describ ed in § 4.2.3. Section 5 deals with the fractional linear programming. The method is illustrated on n umerical examples in Section 6. Finally , we conclude in Section 7 where the complexit y of our sub- stitution metho d (ab out O ( n 2 m )) is more discussed. W e add some remarks on this w ork vs other w orks. Finally , the maximization problem is presented in app endix A. 6 2.4 The switc h to a maximizing problem can o ccur The app endix A dealing with maximization problem is needed b ecause the follo wing situation can o ccur in the substitution pro cess for mpr . The reader has noticed that F ourier’s trick induces that the cost function is a low er b ound of the v ariable z . Th us, b ecause all linear functions are non-decreasing the researc h of low er b ounds on the x j ’s is the priorit y . But in the substitution pro cess we can ha ve the following situation: • the cost function only dep ends on the homogenization v ariable h , ie. z ≥ c h ⊗ h , • not all the v ariables hav e b een substituted and they ha v e no lo wer b ounds but upp er b ounds exist at least for one v ariable, • the optimalit y is not ensured, ie. Theorem 4.2 of § 4.2.2 do es not apply . Noticing that minimazing c h ⊗ h or maximazing c h ⊗ h o ver the set of the remaining v ariables do es not mak e differences one can switch to the maximization problem with z ≤ c h ⊗ h (F ourier’s trick for max) and compute upp er b ounds on the remaining x j ’s of the problem. Dually , the switch from maximization problem to minimization problem can o ccur. 3 Preliminary results Let us consider tw o m × n -matrices L and W . And let us consider the cone C ( L, W ) := { x : L ⊗ x ≥ W ⊗ x } . T o this cone w e asso ciate the function setro wtozero ( L, W ) defined by: setro wtozero ( L, W ) := F or i = 1 to m do if l i,. ≥ w i,. then l i,. := O , w i,. := O (7) 3.1 Elemen tary results from interv al arithmetic in dioids In this subsection we presen t elemen tary materials dealing with in terv al arith- metic in the (max , +)-algebra. The in terv al arithmetic on dioids has b een already used in the context of automatic-control problem (see eg. [15], [14]). An interv al is a set denoted I := [ u, v ] where u, v ∈ R O and u ≤ v . W e define hereafter the elementary needed in this pap er: 7 • Addition of tw o in terv als. Let I i = [ u i , v i ], i = 1 , 2 be t wo in terv als, then the addition of I 1 and I 2 is denoted I 1 ⊕ I 2 and defined by: [ u 1 , v 1 ] ⊕ [ u 2 , v 2 ] := [ u 1 ⊕ u 2 , v 1 ⊕ v 2 ] . (8) • Scalar m ultiplication. Let I = [ u, v ] be an in terv al and let α ∈ R O . Then, the scalar multiplication of I b y the scalar α is denoted α ⊗ I and defined by: α ⊗ [ u, v ] := [ α ⊗ u, α ⊗ v ] . (9) • And the linear com bination of k in terv als is the follo wing interv al de- fined as: ⊕ k i =1 α i ⊗ [ u i , v i ] := [ ⊕ k i =1 α i ⊗ u i , ⊕ k i =1 α i ⊗ v i ] . (10) Where α i ∈ R O , i = 1 , . . . , k . W e will ha v e to use results dealing with particular in terv als. These par- ticular interv als are defined as follo ws: I := { [ u, + ∞ ] , u ∈ R O } . (11) Let us p oin t out that the familly of interv als I is ob viously stable b y (max , +)-linear combinations. Prop osition 3.1 L et [ u l , + ∞ ] , l = 1 , . . . , k , k ≥ 2 b e k intervals of I with u 1 ≤ . . . ≤ u k . Then, we have the fol lowing series of intervals inclusion: [ u k , + ∞ ] ⊆ [ u k − 1 , + ∞ ] ⊆ · · · ⊆ [ u 1 , + ∞ ] . (12) If Min denotes the minim um in the sense of interv al inclusion then, from Prop osition 3.1 we ha ve the following noticeable interv al equalit y: ∩ k l =1 [ u l , + ∞ ] = Min { [ u l , + ∞ ] , l = 1 , . . . , k } . (13) Let A b e a n × n -matrix, b a n -dimensional v ector and x = ( x i ) n i =1 . W e ha v e the follo wing well-kno wn result. 8 Result 3.1 (Sub and sup er fix p oint) L et us c onsider the fol lowing sets define d by: S ≤ ( A, b ) := { x : x ≤ A ⊗ x ⊕ b } . (14a) and S ≥ ( A, b ) := { x : x ≥ A ⊗ x ⊕ b } . (14b) If the fol lowing c ondition holds: lim k → + ∞ A ⊗ ( k ) = O n ( O n : n × n -nul l matrix) (15) then the gr e atest element of the set S ≤ ( A, b ) (r esp. the smal lest element of the set S ≥ ( A, b ) ) is: x = A ∗ ⊗ b, wher e A ∗ := I n ⊕ A ⊕ A ⊗ (2) ⊕ A ⊗ (3) ⊕ · · · is the infinite “sum” of the p owers of the matrix A known as the Kle ene star of the matrix A . A nd under c ondition (15) we have: A ∗ = I n ⊕ A ⊕ A ⊗ (2) ⊕ · · · ⊕ A ⊗ ( n − 1) . This gr e atest element is the solution of the (max , +) -line ar e quation: x = A ⊗ x ⊕ b which is obtaine d by the satur ation of the ine quality: x ≤ A ⊗ x ⊕ b . There exist differen t conditions suc h that (15) is true. Notably the one whic h states that all elementary circuits of the v alued graph G ( A ) asso ciated with the matrix A hav e weigh t < 1 . W e sp ecify the previous Result 3.1 in the following case. Prop osition 3.2 (V alid inequalit y) L et a b e a sc alar, a ∈ R O . L et v b e a n -dimensional ve ctor. And let y = ( y i ) n i =1 b e a n -dimensional c olumn ve ctor of variables in R O . Then, an ine quality of the form: a ⊗ y j ⋚ v ⊺ ⊗ y , (16) is said to b e v alid iff the fol lowing c ondition is fulfil le d: a  = O and a > v j . (17) A nd in this c ase the ine quality (16) is e quivalent to: y j ⋚ a ⊗ ( − 1) ⊗ v ⊺ ⊗ y , (18a) 9 wher e v is the n -dimensional ve ctor define d by: ∀ j ′ , v j ′ :=  O if j ′ = j v j ′ if j ′  = j . (18b) Pro of. W.l.o.g we can assume j = n in the inequalit y (16). And the inequalit y (16) is equiv alent to y ⋚ V ⊗ y , where V is the n × n -matrix defined b y: V :=  I n − 1 O n − 1 , 1 D C  , D =  v 1 · · · v n − 1  , C = ( a ⊗ ( − 1) ⊗ v n ) . No w w e express V ⊗ y as follows: V ⊗ y = A ⊗ y ⊕ B ⊗ u, with : A :=  O n − 1 ,n − 1 O n − 1 , 1 O 1 ,n − 1 C  , B :=  I n − 1 D  , u = ( y i ) n − 1 i =1 . And clearly by Result 3.1 the set { y : y ⋚ A ⊗ y ⊕ B ⊗ u } admits a smallest (or greatest) elemen t iff condition (15) is verified. Here, the condition (15) is equiv alent to lim k → + ∞ C ⊗ ( k ) = O 1 , 1 whic h is equiv alent to a ⊗ ( − 1) ⊗ v n < 1 b ecause the matrix C has only one elemen tary circuit n → n of w eigh t a ⊗ ( − 1) ⊗ v n . T o conclude w e just ha v e to note that by defining v ⊺ := ( u ⊺ , O ) the inequality (18a) is verified. 2 4 The minimization problem mp r A t the step 0 of the substitution the cone associated with the constrain ts system (6c) of the mpr problem is denoted C ( A + , A − ) [0] and is defined by: C ( A + , A − ) [0] := { w ∈ R n +2 O : A +[0] ⊗ w ≥ A − [0] ⊗ w } (19a) where: 10 ( A + , A − , w ) [0] :=  1 −∞ ⊺ O −∞ A b  ,  O c ⊺ c h −∞ C d  ,   z x h   . (19b) W e use the following con v entions. Numerotation con v en tion 4.1 The r ows of the matric es A + , A − ar e num- b er e d fr om 0 to m . The c olumns of the matric es A + , A − ar e indexe d by j , j varying fr om j = 0 to j = n + 1 . But sometimes it wil l b e useful to use the variables z , x 1 , . . . , x n , h . The c omp onents of the ve ctor w ar e numb er e d fr om 0 to n + 1 . The cost function at step 0 of the substitution is denoted and defined by: cost [0] ( x, h ) := c ⊺ ⊗ x ⊕ c h ⊗ h , with c h = O . The set of the stored linear equalities is denoted L [ k ] at each step k of the substitution. And for k = 0 we hav e L [0] = ∅ . The v ector x is called the vector of the remaining v ariables of the mpr . Of course at step 0 of the metho d: x = ( x j ) n j =1 . T o x w e asso ciate the set of the remaining v ariables denoted x . And of course at step 0 of the metho d: x [0] = { x 1 , . . . , x n } . W e ha ve the following equiv alence: x j / ∈ x ⇔ x j = O in the v ector of remaining v ariables x. (20) Because x = + ∞ ∈ P ( A, b, C , d ) ∪ { ±∞ } and b ecause min = −∞ is an acceptable answer for the minimization problem mp r one make the following assumption for the parametrized reseach of a minim um reac hed by x . Assumption 4.1 (P arametrized reseac h domain for the v ariables x j ) F or the r ese ar ch of a minimum for z we c an assume that ∀ x j ∈ x : x j ∈ [ λ, + ∞ ] for some arbitr ary O ≤ λ ≤ + ∞ . The homogenization v ariable h is not substituable (recall). And the gen- eral scheme of the substitution for the mpr problem is to arriv e after the n substitutions of the v ariables x j at the following situation: 11 z ≥ c [ n ] h ⊗ h b [ n ] ⊗ h ≥ d [ n ] ⊗ h. (21) Where b [ n ] , d [ n ] are t wo m -dimensional column v ectors. W e then hav e the follo wing cases. • Case 1: b [ n ] ≥ d [ n ] . In this case h can b e set to 1 and min( z ) = c [ n ] h . • Case 2: b [ n ] ≱ d [ n ] . In this case h = + ∞ is the only p ossible v alue for h b ecause h > O and min( z ) = + ∞ . This discussion leads to assume: Assumption 4.2 (P arametrized domain for h ) F or the mp r pr oblem we assume that the homo genization variable h has a p ar ametrize d interval [ µ, + ∞ ] for some arbitr ary O < µ ≤ + ∞ . In conclusion, the parametrized domain of research of a minim um is de noted R λµ ( x, h ) and defined by: R λµ ( x, h ) := × x j ∈ x [ λ, + ∞ ] × [ µ, + ∞ ] . (22a) Where because h must b e > O w e can assume that the following condition holds: ∀ θ , ∀ β  = O , ∃ λ, µ : θ ⊗ λ < β ⊗ µ. (22b) 4.1 Study of the cone C ( i ) := { z ≥ c ⊺ ⊗ x ⊕ c h ⊗ h, a + i,. ⊗ w ≥ a − i,. ⊗ w } Based on the result of Prop osition 3.2 ∀ i = 1 , . . . , m , ∀ j = 1 , . . . , n such that the following condition is satisfied: a + i,j  = O and a + i,j > a − i,j (23) w e define the following v alid inequality: I ≥ ( x j , i ) := { a + i,j ⊗ x j ≥ a − i,. ⊗ w } . (24) Under condition (23) the inequality I ≥ ( x j , i ) can b e rewritten as: 12 I ≥ ( x j , i ) = { x j ≥ f ≥ ij ( x, h ) } . (25) Where the ( x, h )-linear function f ≥ ij ( x, h ) is defined by: f ≥ ij ( x, h ) := ℓ ij ( x ) ⊕ r ij ⊗ h, (26a) with ℓ ij is the following x -linear function whic h do es not dep end on x j : ℓ ij ( x ) := v ⊺ ij ⊗ x, (26b) where v ij denotes the n -dimensional vector such that: ∀ j ′ = 1 , . . . , n : v ij,j ′ =  ( a + i,j ) ⊗ ( − 1) ⊗ a − i,j ′ if j ′  = j O if j ′ = j (26c) and the scalar r ij is defined by: r ij := ( a + i,j ) ⊗ ( − 1) ⊗ d i . (26d) Assuming that I ≥ ( x j , i ) is v alid we define the following function: f z ,ij ( x, h ) := ⊕ j ′  = j c j ′ ⊗ x j ′ ⊕ c j ⊗ f ≥ ij ( x, h ) ⊕ c h ⊗ h. (27) By replacing f ≥ ij ( x, h ) in (27) we obtain the new expression for f z ,ij ( x, h ): f z ,ij ( x, h ) := ⊕ j ′  = j ( c j ′ ⊕ c j ⊗ v ij,j ′ ) ⊗ x j ′ ⊕ ( c h ⊕ c j ⊗ r ij ) ⊗ h. (28) The next Theorem will b e useful to justify the substitution method de- v elop ed in this pap er. Theorem 4.1 (Saturation of an inequalit y) F or al l x j such that I ≥ ( x j , i ) is valid in the sense of Pr op osition 3.2 we have: inf { x : x j ≥ f ≥ ij ( x,h ) }  c ⊺ ⊗ x ⊕ c h ⊗ h x j  =  f z ,ij ( x, h ) f ≥ ij ( x, h )  . (29) A nd the infimum is r e ache d at x such that we have: x j = f ≥ ij ( x, h ) . (30) 13 Pro of. W e hav e the follo wing implication: x j ≥ f ≥ ij ( x, h ) ⇒ ∀ α ∈ R O , α ⊗ x j ≥ α ⊗ f ≥ ij ( x, h ) . In particular, the ab ov e implication is true for α = c j . All the ( x, h )-linear functions are non-decreasing. Noticing that the cost function: ( x, h ) 7→ c ⊺ ⊗ x ⊕ c h ⊗ h is a ( x, h )-linear function the result is pro v ed. 2 4.2 Ho w to c ho ose the next v ariable for substitution in mp r 4.2.1 Classification of the v ariables and the linear functions of mp r Based on the result of Prop osition 3.2, ∀ i = 1 , . . . , m , ∀ j = 1 , . . . , n suc h that the following condition is satisfied: a − i,j  = O and a + i,j < a − i,j (31) w e define the following v alid inequality: I ≤ ( x j , i ) := { a − i,j ⊗ x j ≤ a + i,. ⊗ w } . (32) Under condition (31) the result of Prop osition 3.2 allo ws us to remov e x j from the expression a + i,. ⊗ w . This new ( x, h )-linear function which do es not dep end on x j is denoted f ≤ ij ( x, h ). The expression is not given here b ecause the only information to know in the mpr problem is that this function is well defined. W e use the con ven tion that I ≥ ( x j , i )  = ∅ (resp. I ≤ ( x j , i )  = ∅ ) equiv a- len tly means that I ≥ ( x j , i ) (resp. I ≤ ( x j , i )) is v alid in the sense of Prop osi- tion 3.2. Definition 4.1 A variable x j is said to b e • min -b ounde d if I ≥ ( x j ) := ∪ m i =1 I ≥ ( x j , i )  = ∅ . Otherwise x j is said to b e min -unb ounde d. • max -b ounde d if I ≤ ( x j ) := ∪ m i =1 I ≤ ( x j , i )  = ∅ . Otherwise x j is said to b e max -unb ounde d. 14 • dominating if x j is min -and- max -b ounde d. And clearly , the dominating v ariables ha v e a v ariation domain smaller (in the sense of the set inclusion) than the others v ariables. So that there exists an order b et ween v ariables denoted ⪯ v ar based on their v ariation domain suc h that: { x j ∈ x : I ≥ ( x j )  = ∅ and I ≤ ( x j )  = ∅} ⪯ v ar { x j ∈ x : I ≥ ( x j )  = ∅ and I ≤ ( x j ) = ∅} . (33) And of course we also men tioned that: { x j ∈ x : I ≥ ( x j )  = ∅ and I ≤ ( x j ) = ∅} ⪯ v ar { x j ∈ x : I ≥ ( x j ) = ∅ and I ≤ ( x j ) = ∅} . Definition 4.2 A non nul l ( x, h ) -line ar function f : ( x, h ) 7→ α ⊺ ⊗ x ⊕ β ⊗ h is said to b e h -b ounde d if β  = O . Otherwise the function f is said to b e h -unb ounde d. W e denote h B the set of h -b ounded non n ull linear functions and w e denote h U the set of non n ull h -unbounded linear functions. And ( h B , h U ) is a partition of the set of all non n ull ( x, h )-linear functions. W e ha v e the follo wing in terv al calculus results. • ∀ f = α ⊺ ⊗ x ⊕ β ⊗ h ∈ h B , using interv al calculus form ulae w e ha ve: f ( R λµ ( x, h )) = [ ⊕ i α i ⊗ λ ⊕ β ⊗ λ, + ∞ ] , (34a) and by assumption ((22b) with θ = ⊕ i α i ) on λ and µ we ha ve: ∀ α ∈ R n O , ∀ β  = O : f ( R λµ ( x, h )) = [ β ⊗ µ, + ∞ ] (34b) Let us stress that this in terv al do es not dep end on the n -dimensional ro w v ector α and by assumption (22b): [ β ⊗ µ, + ∞ ] ⊂ [ ⊕ i α i ⊗ λ, + ∞ ] . (34c) • And ∀ f = α ⊺ ⊗ x ∈ h U : f ( R λµ ( x, h )) = [ ⊕ i α i ⊗ λ, + ∞ ] . (35) F rom this ab o v e results w e ha v e the following ordering b et ween the non n ull linear functions denoted ⪯ f ct whic h is based on their v alue domain: { f : f ∈ h B } ⪯ f ct { f : f ∈ h U } . (36) 15 4.2.2 Cho osing a v ariable for a new substitution after k substitu- tions in mp r In this section w e describ e the pro cedure for choosing a remaining v ariable to be substituted. Assuming that k substitutions ha ve b een done the cone asso ciated with the mp r is denoted C ( A + , A − ) [ k ] . The set of linear equalities at step k , sa y L [ k ] , contains k equalities of the form { x j = f } where f is a ( x, h )-linear function. T o ligh ten notations the sup erscript [ k ] will not b e applied on the vector of the remaining v ariables x and the set of the remaining v ariables x in the sequel. Recall that we use n umerotation conv ention 4.1 p. 11. In the next theorem w e c haracterize a normal end of the substitution pro cess. Theorem 4.2 (Minimalit y and reacha bility at −∞ of h 7→ c h ⊗ h ) The function h 7→ c h ⊗ h is the lower b ound of the c ost function ( x, h ) 7→ c ⊺ x ⊗ x ⊕ c h ⊗ h which is attaine d at x = −∞ iff the fol lowing c ondition holds: A +[ k ] [ | 1 ,m | ] h ≥ A − [ k ] [ | 1 ,m | ] h . (37) Pro of. Considering the system of inequalities A +[ k ] ⊗ w ≥ A − [ k ] ⊗ w . The [ | 1 , m | ] × x ∪ { h } part of the system is expressed as follows: A +[ k ] [ | 1 ,m | ] x ⊗ x ⊕ A +[ k ] [ | 1 ,m | ] h ⊗ h ≥ A − [ k ] [ | 1 ,m | ] x ⊗ x ⊕ A − [ k ] [ | 1 ,m | ] h ⊗ h. W e hav e z = c h ⊗ h at x = −∞ iff ( z , −∞ , h ) ∈ C ( A + , A − ) [ k ] . And based on the ab ov e inequalit y it means that A +[ k ] [ | 1 ,m | ] h ⊗ h ≥ A − [ k ] [ | 1 ,m | ] h ⊗ h . And this last inequalit y do es not imply that h = + ∞ is the only solution iff A +[ k ] [ | 1 ,m | ] h ≥ A − [ k ] [ | 1 ,m | ] h . And the equiv alence is pro ved. 2 In the next Theorem w e presen t the pro of of the optimal c hoice for one or sev eral v ariables whic h can b e substituted. Each v ariable is asso ciated with an inequality which b elongs to a given set of inequalities. Theorem 4.3 L et I b e a subset of valid ine qualities I ≥ ( x j , i ) , ie. a set such that: I ⊆ I ≥ ( x ) . (38) 16 Wher e I ≥ ( x ) := { ( i, j ) : x j ∈ x and I ≥ ( x j , i ) is valid } . (39) We have the fol lowing implic ation: I ≥ ( x j , i )  = ∅ or e quivalently I ≥ ( x j , i ) is valid ⇒ z ≥ f z ,ij ( x, h ) . (40) R e c al ling that f z ,ij ( x, h ) is define d by (28). L et T and T ′ denoting either B or U , r esp e ctively. The 2 -tuple ( T , T ′ ) is define d as a function of the set I ac c or ding to ⪯ f ct as fol lows (priority to the c ost function): ( T , T ′ ) :=        ( B , B ) if ∃ ( i, j ) ∈ I s.t. f z ,ij ∈ h B and f ≥ ij ∈ h B ( B , U ) if ∃ ( i, j ) ∈ I s.t. f z ,ij ∈ h B and f ≥ ij ∈ h U ( U , B ) if ∃ ( i, j ) ∈ I s.t. f z ,ij ∈ h U and f ≥ ij ∈ h B ( U , U ) if ∃ ( i, j ) ∈ I s.t. f z ,ij ∈ h U and f ≥ ij ∈ h U . (41) Then, define the fol lowing set of indexes: T ( T , T ′ , I ) := { ( i, j ) ∈ I : ( f z ,ij , f ≥ ij ) ∈ ( h T , h T ′ ) } . (42) L et us define the fol lowing assertion which de als with the c ost z and an interval τ : Z( τ ) : ∀ ( i, j ) ∈ T ( T , T ′ , I ) ∀ z ( z ∈ τ ⇒ ∃ ( x, h ) ∈ R λµ ( x, h ) z ≥ f z ,ij ( x, h )) . (43) A nd let us define the fol lowing assertion de aling with variables of the pr oblem and intervals τ , τ ′ : X( τ , τ ′ ) : ∀ ( i, j ) ∈ T ( T , T ′ , I ) ∩ a rgMin ( τ ) ∀ x j ( x j ∈ τ ′ ⇒ ∃ ( x, h ) ∈ R λµ ( x, h ) x j ≥ f ≥ ij ( x, h )) . (44) Then, we have the fol lowing e quivalenc es. Z( τ ) true ⇔ τ = Min { f z ,ij ( R λµ ( x, h )) , ( i, j ) ∈ T ( T , T ′ , I ) } . (45) 17 A nd X( τ , τ ′ ) true ⇔ τ ′ = Min { f ≥ ij ( R λµ ( x, h )) , ( i, j ) ∈ argMin ( τ ) ∩ T ( T , T ′ , I ) } . (46) Pro of. The pro of of the implication (40) is a straigh tforw ard consequence of Theorem 4.1. The equiv alence (45) comes from the prop erties of the familly of interv als we are considering in this pap er. W e ha ve a total order (see Prop osition 3.1) and the Min of a set of interv als coincides with the in tersection of the same set of in terv als (see (13)). The pro of is achiev ed by noticing that ∀ ( i, j ) ∈ a rgMin ( τ ) ∩ T ( T , T ′ , I ) the v ariables x j can b e exc hanged. And thus one ha ve to take the Min . 2 W e are no w in p osition to indicate ho w to choose a remaining v ariable to b e substituted. The c hoice also dep ends on the follo wing differen t cases whic h are listed hereafter. Case 0: min ( z ) = + ∞ or no minim um. Case 0 . 1: x = ∅ and the condition (37) of Theorem 4.2 do es not holds. x = ∅ means that k = n (no more v ariable to b e substituted) and cost [ n ] ( x, h ) = c [ n ] h ⊗ h . The condition (37) of Theorem 4.2 do es not holds means A +[ n ] [ | 1 ,m | ] h ≱ A − [ n ] [ | 1 ,m | ] h then h = + ∞ is the only p ossible v alue for h and b y backw ard substitution we ha v e z = + ∞ , x = + ∞ . Or Case 0 . 2: x  = ∅ and I ≤ ( x ) = ∅ and I ≥ ( x ) = ∅ and the condition (37) of Theorem 4.2 do es not holds. The substitution pro cess stops b ecause no v alid inequalities can b e generated. No minimum reached. Case 1: switc hing case , cf. § 2.4. x + = ∅ and cost [ k ] ( x, h ) = c [ k ] h ⊗ h and the condition (37) of Theorem 4.2 do es not holds and x O  = ∅ and I ≥ ( x O ) = ∅ and I ≤ ( x O )  = ∅ . The substitution must switc h to the follo wing maximizing problem: PMR ( A, b, C, d, c, c h , z , x, h, ( m, n )) , (47a) where the matrices A and C are defined by: A := A − [ k ] [ | 1 ,m | ][ | 1 ,n | ] , C := A +[ k ] [ | 1 ,m | ][ | 1 ,n | ] . (47b) 18 The vectors b and d are defined b y: b := A − [ k ] [ | 1 ,m | ] h , d := A +[ k ] [ | 1 ,m | ] h . (47c) The initial cost function of the MPR is characterized by the vector c and the constan t c h suc h that: c := −∞ , c h := c [ k ] h . (47d) And the v ariables of the problem are: z := z , x ← x O , h := h. (47e) Where the notation x ← x O means that w e store in the n -dimensional vec- tor x the remaining v ariables of the set x O . The other comp onen ts (already substituted v ariables) are set to O . And the dimensions ( m, n ) of the MPR are ob viously the same as in the mp r . Case 2: Not Case 0 and not Case 1. The first case is Case 2 . 1: x = ∅ and the condition (37) of Theorem 4.2 holds. This means that k = n (no more v ariable to b e substituted) and cost [ n ] ( x, h ) = c [ n ] h ⊗ h and A +[ n ] [ | 1 ,m | ] h ≥ A − [ n ] [ | 1 ,m | ] h then h can take all p ossible v alues in particular we can take h = 1 and the mp r succeeds with finite minimum z = c [ n ] h . The case is Case 2 . 2: x  = ∅ and ∃ x j ∈ x suc h that: I ≥ ( x j )  = ∅ . Let us distinguish the follo wing sub cases whic h are based on the form of the cost function: cost [ k ] ( x, h ) = c ⊺ x + ⊗ x + ⊕ c ⊺ x O ⊗ x O ⊕ c h ⊗ h, (48a) where: x + := { x j ∈ x : c j > O } and x O := { x j ∈ x : c j = O } . (48b) And the vectors x + and x O are defined by x + := ( x j ) { j : x j ∈ x + } and x O := ( x j ) { j : x j ∈ x O } . (48c) Case 2 . 2 . 1: the condition (37) of Theorem 4.2 holds. Then, z = c h ⊗ h is the minimum for the mpr . And set x to −∞ . Case 2 . 2 . 2: otherwise, the condition (37) of Theorem 4.2 does not hold. Based on the hierarch y ⪯ v ar (see (33)) b et ween the remaining v ariables let 19 us define the set of the dominating v ariables D + O := { x j ∈ x : I ≤ ( x j )  = ∅ and I ≥ ( x j )  = ∅} . And the set of the v alid inequalities: I + O ( D ) :=  I ≥ ( x ∩ D + O ) if D + O  = ∅ I ≥ ( x ) , cf. (39) otherwise. (49) W e apply Theorem 4.3 recalling that the 2-tuple ( T , T ′ ) is defined ac- cording with ⪯ f ct b y (41) and the subset of v alid inequalities is I := I + O ( D ). Th us, the “w orking space” of indexes of rows and columns (or v ariables) of the system of inequalities is: T ( T , T ′ , I + O ( D )). Th us, we compute the interv al τ suc h that Z( τ ) is true (see (43) and (45)). If argMin ( τ ) is a singleton { ( i ∗ , j ∗ ) } then take the unique 2-tuple ( i ∗ , j ∗ ) for the substitution of x j ∗ = f ≥ i ∗ j ∗ ( x, h ). Otherwise, tak e an y 2-tuple ( i ∗ , j ∗ ) elemen t of a rgMin ( τ ′ ) recalling that the interv al τ ′ is the in terv al suc h that X( τ , τ ′ ) is true (see (44) and (46)). The new set of equalities L [ k +1] , the new cost function cost [ k +1] ( x, h ) and the new cone C ( A + , A − ) [ k +1] are defined b y the pro cedure describ ed in sec- tion 4.2.3. 4.2.3 The new characteristic elemen ts of the mp r after substitu- tion The new set of equalities is defined as follo ws. If conditions of Theorem 4.2 are verified then set k + 1 = n and L [ n ] := L [ k ] ∪ x j ∈ x { x j = O } ∪ { z = c h ⊗ h } . Then, pmrp stops and we solve the linear system L [ n ] and express all the x j ’s as function of the homogenization v ariable h . Otherwise, w e define L [ k +1] as: L [ k +1] := L [ k ] ∪ { x j ∗ = f i ∗ j ∗ ( x, h ) } and w e apply Theorem 4.1 of § 4.1. And based on the previous cases the new cost function is defined as: cost [ k +1] ( x, h ) := f z ,i ∗ j ∗ ( x, h ) . (50) And in all cases the new cone C ( A + , A − ) [ k +1] asso ciated with pmrp is deduced from the following set of inequalities: z ≥ cost [ k +1] ( x, h ) , ∀ i ∈ [ | 1 , m | ] : a +[ k ] i,. ⊗ ℓ ≥ a − [ k ] i,. ⊗ ℓ. (51a) 20 and ℓ is the n + 2-dimensional column v ector which has the same comp onen ts as the vector w [ k ] except its j ∗ th comp onent wic h is: ℓ j ∗ := f i ∗ j ∗ ( x, h ) . (51b) Let us define the following ( n + 2) × ( n + 2) tr ansition matrix T k → k +1 b y: ∀ j : t k → k +1 j,. :=  e j if j  = j ∗ ( O , v ⊺ i ∗ j ∗ , r i ∗ j ∗ ) if j = j ∗ . (52) Where e j denotes the j th n + 2-dimensional ro w v ector of the ( n + 2) × ( n + 2)- iden tit y matrix I n +2 . The new matrices ( A + , A − ) [ k +1] are defined by: ( A + , A − ) [ k +1] := setro wtozero ( A +[ k ] ⊗ T k → k +1 , A − [ k ] ⊗ T k → k +1 ) (53) recalling that function setrowtozero is defined b y (7). And the new vector w [ k +1] is defined by: w [ k +1] :=        z . . . 0 . . . h        ← j ∗ (54) And the new set x of the remaining v ariables is defined by: x := x \ { x j ∗ } . (55) 5 Linear fractional programming In this section we consider the following (max , +)-analogue of the linear fractional programming (see [6]) defined as follows. min { z = p ⊺ ⊗ x/r ⊺ ⊗ x, x ∈ P ( A, b, C , d ) ∪ { ±∞ }} . (56a) Where p , r are tw o n -dimensional vectors satisfying the follo wing assumption: Assumption 1 f : p  = −∞ and r  = −∞ . (56b) 21 This kind of problem has b een studied in eg. [10] and [11] where pseudo- p olynomial algorithms hav e b een pro vided. Hereafter, we follow the (max , +)-analogue of the Charnes-Co op er trans- formation [6] which has b een used in [11]. Using also the F ourier’s trick the problem (56a) is unfolded as follows: min( z ) z ≥ p ⊺ ⊗ y : F ourier’s tric k A ⊗ y ⊕ b ⊗ t ≥ C ⊗ y ⊕ d ⊗ t : ( y , t )-cone part r ⊺ ⊗ y ≥ 1 : ( y )-p olyhedron part t ≥ O : t added v ariable suc h that y = t ⊗ x. (57) Remark 5.1 In [11] the authors fol low exactly the Charnes-Co op er tr ans- formation. This me ans that the ( y ) -p olyhe dr on p art is not an ine quality but the e quality: r ⊺ ⊗ y = 1 . This latter e quality is e quivalent to: (1). r ⊺ ⊗ y ≥ 1 and (2). 1 ≥ r ⊺ ⊗ y . But substitution al lows us to dr op the ine quality (2) b e c ause when an ine quality is chosen using our criteria develop e d in § 4.2.2 then we use the r esult of The or em 4.1 of § 4.1 to satur ate this ine quality, ie. ≥ b e c omes = . No w, w e just hav e to homogenize the ( y )-p olyhedron part b y adding the homogeneous v ariable h and we obtain the following mp r problem: min( z ) z ≥ p ⊺ ⊗ y : F ourier’s tric k . (58a) The research of a minimum is developed on the following cone: A ⊗ y ⊕ b ⊗ t ≥ C ⊗ y ⊕ d ⊗ t : ( y , t )-cone part r ⊺ ⊗ y ≥ h : ( y )-p olyhedron part homogenized . (58b) The v ariable t used in the Charnes-Co op er transformation is such that: t ≥ O , y = t ⊗ x. (58c) And the homogenization v ariable h satifies: h > O . (58d) 22 6 Numerical examples In ths section w e study p edagogical examples. In § 6.1 w e illustrate the coherency of our strongly p olynomial metho d on a toy example. In § 6.2 (problem of minimization) and in § 6.3 (problem of fractional minimization) we illustrate our strongly p olynomial metho d on some numer- ical examples b orro wed from the litterature which are solv ed using pseudo- p olynomial metho d in [10]. 6.1 A minim um whic h is equal to + ∞ In this subsection w e consider the following minimization problem: the ho- mogenization v ariable h : min { z = x 1 , x = ( x 1 ) ∈ P ( A, b, C , d ) ∪ { ±∞ }} , where the p olyhedron P ( A, b, C, d ) is characterized b y the follo wing vectors and matrices: A :=  1 2 ⊗ ( − 1)  , b :=  1 1  , C :=  1 ⊗ ( − 1) 1  , d :=  2 O  . Because the example is v ery small we do not need to describe the cone as- so ciated with mpr . Using the homogenization v ariable h the mpr is unfolded as follows: min( z ) l 0 : z ≥ x 1 l 1 : x 1 ⊕ h ≥ 1 ⊗ ( − 1) ⊗ x 1 ⊕ 2 ⊗ h l 2 : 2 ⊗ ( − 1) ⊗ x 1 ⊕ 1 ⊗ h ≥ x 1 . (59) Only l 1 generates the v alid inequalit y I ≥ ( x 1 , 1) = { x 1 ≥ 2 ⊗ h } . Thus, the substitution of x 1 b y 2 ⊗ h (ie. w e ha ve L [1] = { x 1 = 2 ⊗ h } ) pro vides the new system: min( z ) l 0 : z ≥ 2 ⊗ h l 1 : 2 ⊗ h ≥ 2 ⊗ h l 2 : 1 ⊗ h ≥ 2 ⊗ h (60) Constrain t l 2 implies that h = + ∞ is the only possible v alue for h , where h is assumed to b e > O . Then, by bac k substitution we ha v e: z = 2 ⊗ + ∞ = + ∞ and x 1 = 2 ⊗ + ∞ = + ∞ . And + ∞ is elemen t of the set P ( A, b, C , d ) ∪{ ±∞ } . 23 This illustrates the coherency of the metho d b ecause min( z ) = + ∞ is actually reached at x = (+ ∞ ) whic h is a p oin t of the p olyhedron ∪{ ±∞ } . 6.2 Minimization Example In this subsection we consider the Example 2 p. 1463 of [10] which is: min { z = 2 ⊗ x 1 ⊕ 4 ⊗ ( − 1) ⊗ x 2 , x = ( x 1 , x 2 ) ⊺ ∈ P ( A, b, C , d ) ∪ { ±∞ }} , Where: A :=           2 ⊗ ( − 1) 1 1 1 ⊗ ( − 1) 1 2 ⊗ ( − 1) 2 O 1 O 2 ⊗ ( − 1) O 4 ⊗ ( − 1) O           , b :=           O O O O 1 1 1           , C :=           O O O O O O O 3 ⊗ ( − 1) O 4 ⊗ ( − 1) O 5 ⊗ ( − 1) O 6 ⊗ ( − 1)           , d :=           1 1 1 1 O O O           . with the following change of notations A ↔ B , b ↔ d , C ↔ A and d ↔ c . The mp r is defined on the cone C ( A, b, C , d ) obtained from the p olyhedron P ( A, b, C , d ) b y adding the homogenization v ariable h as already explained previuously . A t step 0 of the substitution the cone associated with the ab o v e con- strain ts of the mp r problem is denoted C ( A + , A − ) [0] and is defined by the follo wing system of inequalities: A +[0] ⊗ w ≥ A +[0] ⊗ w , (61a) with w = ( z x 1 x 2 h ) ∈ R 4 O and the matrices ( A + , A − ) [0] are defined by: A +[0] :=             1 O O O O 2 ⊗ ( − 1) 1 O O 1 1 ⊗ ( − 1) O O 1 2 ⊗ ( − 1) O O 2 O O O 1 O 1 O 2 ⊗ ( − 1) O 1 O 4 ⊗ ( − 1) O 1             , A − [0] :=             O 2 4 ⊗ ( − 1) O O O O 1 O O O 1 O O O 1 O O 3 ⊗ ( − 1) 1 O O 4 ⊗ ( − 1) O O O 5 ⊗ ( − 1) O O O 6 ⊗ ( − 1) O             . (61b) 24 And the F ourier’s trick for this example is: min( z ) (62a) z ≥ cost [0] ( x, h ) (62b) cost [0] ( x, h ) = 2 ⊗ x 1 ⊕ 4 ⊗ ( − 1) ⊗ x 2 ⊕ O ⊗ h. (62c) x = { x 1 , x 2 }  = ∅ and the mpr problem is b ounded b ecause one can chec k that: ∀ x j ∈ x : I ≥ ( x j ) ∪ I ≤ ( x j )  = ∅ . W e hav e x + = { x 1 , x 2 } and x O = ∅ . The set of dominating v ariables is D + O = { x 2 } . Indeed, it is easy to see here that I ≤ ( x 1 ) = ∅ b ecause A − [0] [ | 1 ,m | ] x 1 = −∞ . Clearly , the condition (37) of Theorem 4.2 do es not hold and Case 2 . 2 . 2 applies. W e hav e (see 49): I + O ( D ) = I ≥ ( x 2 ) = { (1 , 2) , (2 , 2) , (3 , 2) } . And we ha ve the following three possibilities for the substitution of the dominating v ariable x 2 in the system of inequalities detailed hereafter:  z x 2  0 , 1 ≥  2 ⊗ x 1 ⊕ 4 ⊗ ( − 1) ⊗ h h  ::  [4 ⊗ ( − 1) ⊗ µ, + ∞ ] [ 1 ⊗ µ, + ∞ ]  ::  h B h B   z x 2  0 , 2 ≥  2 ⊗ x 1 ⊕ 3 ⊗ ( − 1) ⊗ h 1 ⊗ h  ::  [3 ⊗ ( − 1) ⊗ µ, + ∞ ] [1 ⊗ µ, + ∞ ]  ::  h B h B   z x 2  0 , 3 ≥  2 ⊗ x 1 ⊕ 2 ⊗ ( − 1) ⊗ h 2 ⊗ h  ::  [2 ⊗ ( − 1) ⊗ µ, + ∞ ] [2 ⊗ µ, + ∞ ]  ::  h B h B  . Applying Theorem 4.3 where from the abov e array w e ha v e ( T , T ′ ) = ( B , B ) the minim um interv al τ is τ = [2 ⊗ ( − 1) ⊗ µ, + ∞ ]. And argMin ( τ ) = { (3 , 2) } whic h is a singleton. Thus, ( i ∗ , j ∗ ) = (3 , 2) whic h means that x 2 = 2 ⊗ h in row 3 of the linear system of inequalities (61a). W e apply the results of § 4.2.3. Then, the set of linear equalities is now: L [1] = { x 2 = 2 ⊗ h } . (63) The new cost function is: cost [1] ( x, h ) := f z , 32 ( x, h ) = 2 ⊗ x 1 ⊕ 2 ⊗ ( − 1) ⊗ h. (64) The 4 × 4-transition matrix T 0 → 1 is defined by: 25 T 0 → 1 :=     1 O O O O 1 O O O O O 2 O O O 1     . (65) W e calculate the following matrices A + := A +[0] ⊗ T 0 → 1 and A − := A − [0] ⊗ T 0 → 1 and we obtain: A + :=             1 O O O O 2 ⊗ ( − 1) O 2 O 1 O 1 O 1 O 1 O 2 O O O 1 O 1 O 2 ⊗ ( − 1) O 1 O 4 ⊗ ( − 1) O 1             , A − :=             O 2 O 2 ⊗ ( − 1) O O O 1 O O O 1 O O O 1 O O O 1 O O O 2 ⊗ ( − 1) O O O 3 ⊗ ( − 1) O O O 4 ⊗ ( − 1)             . W e remark that ∀ i  = 0 , 4 w e hav e a + i,. ≥ a − i,. th us the new cone C ( A + , A − ) [1] is defined by: A +[1] ⊗ w ≥ A +[1] ⊗ w , (66a) with w = ( z , x 1 , O , h ) ⊺ ∈ R 4 O and the matrices ( A + , A − ) [1] are defined as the result of setrowtozero ( A + , A − ): A +[1] :=             1 O O O O O O O O O O O O O O O O 2 O O O O O O O O O O O O O O             , A − [1] :=             O 2 O 2 ⊗ ( − 1) O O O O O O O O O O O O O O O 1 O O O O O O O O O O O O             . (66b) x = { x 1 }  = ∅ and the mp r problem is bounded because one can c heck eg. that: I ≥ ( x 1 )  = ∅ . W e hav e x + = { x 1 } and of course x O = ∅ . The set of dominating v ariables is D + O = ∅ . The condition (37) of Theorem 4.2 do es not hold and Case 2 . 2 . 2 applies. W e hav e (see 49): I + O ( D ) = I ≥ ( x 2 ) = { (4 , 1) } . 26 Th us, ( i ∗ , j ∗ ) = (4 , 1) and the 4 × 4-transition matrix T 0 → 1 is defined b y: T 1 → 2 :=     1 O O O O O O 2 ⊗ ( − 1) O O O O O O O 1     . (67) W e calculate the following matrices A + := A +[1] ⊗ T 1 → 2 and A − := A − [1] ⊗ T 1 → 2 and we obtain: A + :=             1 O O O O O O O O O O O O O O O O O O 1 O O O O O O O O O O O O             , A − :=             O O O 1 O O O O O O O O O O O O O O O 1 O O O O O O O O O O O O             . with w = ( z , O , O , h ) ⊺ ∈ R 4 O and the matrices ( A + , A − ) [2] are defined as a result of setrowtozero ( A + , A − ): A +[2] :=             1 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O             , A − [2] :=             O O O 1 O O O O O O O O O O O O O O O O O O O O O O O O O O O O             . (68) And here we hav e: −∞ = A +[2] [ | 1 , 7 | ] h ≥ A − [2] [ | 1 , 7 | ] h = −∞ . Thus, the reachabilit y condition (37) of Theorem 4.2 is v erified. So, mpr has the following solution: z = h, x 1 = 2 ⊗ ( − 1) ⊗ h, x 2 = 2 ⊗ h. And we retrieve the solution of the Example 2 p. 1463 of [10]. 27 6.3 F ractional Example In this subsection w e illustrate and compare our strongly p olynomial metho d with the pseudo-p olynomial metho d applied on Example 3 p. 1469 of [10]. The starting p oin t of the pseudo-p olynomial metho d is not a p olyhedron but the homogenized cone of the following set P ( A, b, C , d ) ∪ { ±∞ } defined by: A ⊗ x ⊕ b ≥ C ⊗ x ⊕ d, (69) with A :=     O O O O 1 O 1 O O 1 O O     , b :=     1 O O 3     , C :=     3 ⊗ ( − 1) 4 ⊗ ( − 1) O 1 ⊗ ( − 1) 1 O O O O 1 O 1     , d :=     O 1 1 O     . (70) With the following change of notations ( A b ) ↔ D and ( C d ) ↔ C . The fractional minimization problem fmpr we solve here is defined as: min { z = x 2 / 3 ⊗ x 1 , x = ( x 1 , x 2 , x 3 ) ⊺ ∈ P ( A, b, C , d ) } . (71) Applying the results of § 5, the previous fmpr is transformed into the mpr problem defined by (58a)-(58b)-(58c)-(58d) whic h is sp ecified hereafter. min( z ) z ≥ cost [0] ( y , t, h ) := y 2 (72) w := ( z , y 1 , y 2 , y 3 , t, h ) ⊺ ∈ C ( A + , A − ) [0] (73) ( A + , A − ) [0] :=         1 O O O O O O O O O 1 O O O 1 O O O O 1 O O O O O 1 O O 3 O O 3 O O O O         ,         O O 1 O O O O 3 ⊗ ( − 1) 4 ⊗ ( − 1) O O O O 1 ⊗ ( − 1) O O 1 O O O O O 1 O O 1 O 1 O O O O O O O 1         . (74) The set { y 1 , y 2 , y 3 , t } is abbreviated by ( y , t ). The mpr is b ounded with I ≥ ( y 3 ) = ∅ . ( y , t ) + = { y 2 } and ( y , t ) O = { y 1 , y 3 , t } . The dominating v ariables are in D + O = { y 1 , y 2 , t } . The condition 28 (37) of Theorem 4.2 do es not hold and Case 2 . 2 . 2 applies. W e ha ve (see 49): I + O ( D ) = { (1 , t ) , (2 , 2) , (3 , 1) , (4 , t ) , (5 , 1) } . W e enumerate all the p ossibilities in the next array .  z t  0 , 1 ≥  y 2 3 ⊗ ( − 1) ⊗ y 1 ⊕ 4 ⊗ ( − 1) ⊗ y 2  ::  [ 1 ⊗ λ, + ∞ ] [3 ⊗ ( − 1) ⊗ λ, + ∞ ]  ::  h U h U   z y 2  0 , 2 ≥  1 ⊗ ( − 1) ⊗ y 1 ⊕ 1 ⊗ t 1 ⊗ ( − 1) ⊗ y 1 ⊕ 1 ⊗ t  ::  [1 ⊗ λ, + ∞ ] [1 ⊗ λ, + ∞ ]  ::  h U h U   z y 1  0 , 3 ≥  y 2 t  ::  [ 1 ⊗ λ, + ∞ ] [ 1 ⊗ λ, + ∞ ]  ::  h U h U   z t  0 , 4 ≥  y 2 2 ⊗ ( − 1) ⊗ y 1 ⊕ 3 ⊗ ( − 1) ⊗ y 3  ::  [ 1 ⊗ λ, + ∞ ] [2 ⊗ ( − 1) ⊗ λ, + ∞ ]  ::  h U h U   z y 1  0 , 5 ≥  y 2 3 ⊗ ( − 1) ⊗ h  ::  [ 1 ⊗ λ, + ∞ ] [3 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h U h B  By Theorem 4.3 w e ha ve: ( T , T ′ ) = ( U , B ), th us: ( i ∗ , j ∗ ) = (5 , 1). This corresp onds to  z y 1  0 , 5 in the array just ab o ve. The new cost function is: cost [1] ( y , t, h ) = y 2 . And we define the 6 × 6- transition matrix T 0 → 1 as follows: T 0 → 1 :=         1 O O O O O O O O O O 3 ⊗ ( − 1) O O 1 O O O O O O 1 O O O O O O 1 O O O O O O 1         . The set of linear equalities L [1] is: L [1] = L [0] ∪ { y 1 = 3 ⊗ ( − 1) ⊗ h } = { y 1 = 3 ⊗ ( − 1) ⊗ h } . (75) W e calculate the matrices A + = A +[0] ⊗ T 0 → 1 and A − = A − [0] ⊗ T 0 → 1 and it comes: 29 A + =         1 O O O O O O O O O 1 O O O 1 O O O O O O O O 3 ⊗ ( − 1) O O O O 3 3 ⊗ ( − 1) O O O O O 1         , A − =         O O 1 O O O O O 4 ⊗ ( − 1) O O 6 ⊗ ( − 1) O O O O 1 4 ⊗ ( − 1) O O O O 1 O O O O 1 O 2 ⊗ ( − 1) O O O O O 1         . W e hav e w = ( z , O , y 2 , y 3 , t, h ) ⊺ and the matrices ( A + , A − ) [1] are defined as the result of setrowtozero ( A + , A − ): A +[1] =         1 O O O O O O O O O 1 O O O 1 O O O O O O O O 3 ⊗ ( − 1) O O O O 3 3 ⊗ ( − 1) O O O O O O         , A − [1] =         O O 1 O O O O O 4 ⊗ ( − 1) O O 6 ⊗ ( − 1) O O O O 1 4 ⊗ ( − 1) O O O O 1 O O O O 1 O 2 ⊗ ( − 1) O O O O O O         . (76) W e hav e ( y , t ) = { y 2 , y 3 , t }  = ∅ , ( y , t ) + = { y 2 } and ( y , t ) O = { y 3 , t } . The mp r is b ounded. Indeed, it is easy to chec k that eg. I ≥ ( y 2 )  = ∅ , I ≤ ( y 3 )  = ∅ and I ≥ ( t )  = ∅ . The set of dominating v ariables is D + O = { y 2 , t } . The condition (37) of Theorem 4.2 do es not hold and Case 2 . 2 . 2 applies. W e ha v e (see 49): I + O ( D ) = { (1 , t ) , (2 , 2) , (4 , t ) } . And the three p ossibilities are studied in the next array .  z t  0 , 1 ≥  y 2 4 ⊗ ( − 1) ⊗ y 2 ⊕ 6 ⊗ ( − 1) ⊗ h  ::  [ 1 ⊗ λ, + ∞ ] [6 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h U h B   z y 2  0 , 2 ≥  1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h  ::  [4 ⊗ ( − 1) ⊗ µ, + ∞ ] [4 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h B h B   z t  0 , 4 ≥  y 2 3 ⊗ ( − 1) ⊗ y 3 ⊕ 5 ⊗ ( − 1) ⊗ h  ::  [ 1 ⊗ λ, + ∞ ] [5 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h U h B  . By Theorem 4.3: ( T , T ′ ) = ( B , B ). So, we ha ve: ( i ∗ , j ∗ ) = (2 , 2) from the previous array . The new cost function is cost [2] ( y , t, h ) = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h . And we define the 6 × 6-transition matrix T 1 → 2 whic h corresponds to y 2 = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h as follows: 30 T 1 → 2 :=         1 O O O O O O O O O O O O O O O 1 4 ⊗ ( − 1) O O O 1 O O O O O O 1 O O O O O O 1         . The set of linear equalities L [1] is: L [2] = L [1] ∪ { y 2 = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h } = { y 1 = 3 ⊗ ( − 1) ⊗ h, y 2 = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h } . (77) W e calculate the matrices A + = A +[1] ⊗ T 1 → 2 and A − = A − [1] ⊗ T 1 → 2 and it comes: A + =         1 O O O O O O O O O 1 O O O O O 1 4 ⊗ ( − 1) O O O O O 3 ⊗ ( − 1) O O O O 3 3 ⊗ ( − 1) O O O O O O         , A − =         O O 1 O 1 4 ⊗ ( − 1) O O O O 3 ⊗ ( − 1) 6 ⊗ ( − 1) O O O O 1 4 ⊗ ( − 1) O O O O 1 O O O O 1 O 2 ⊗ ( − 1) O O O O O O         . W e ha ve w = ( z , O , O , y 3 , t, h ) ⊺ and the matrices ( A + , A − ) [2] are defined as the result of setrowtozero ( A + , A − ): A +[2] =         1 O O O O O O O O O 1 O O O O O O O O O O O O 3 ⊗ ( − 1) O O O O 3 3 ⊗ ( − 1) O O O O O O         , A − [2] =         O O 1 O 1 4 ⊗ ( − 1) O O O O 3 ⊗ ( − 1) 6 ⊗ ( − 1) O O O O O O O O O O 1 O O O O 1 O 2 ⊗ ( − 1) O O O O O O         . (78) ( y , t ) = { y 3 , t }  = ∅ and the mpr problem is b ounded because one can c hec k eg. that: I ≤ ( y 3 )  = ∅ and I ≥ ( t )  = ∅ . W e ha v e ( y , t ) + = { t } and ( y , t ) O = { y 3 } . The set of dominating v ariables is D + O = { t } . Indeed, it is easy to see here that I ≥ ( y 3 ) = ∅ b ecause A +[0] [ | 1 ,m | ] y 3 = −∞ . Clearly , the 31 condition (37) of Theorem 4.2 do es not hold and Case 2 . 2 . 2 applies. W e ha v e (see 49): I + O ( D ) = I ≥ ( t ) = { (1 , t ) , (4 , t ) } . And we hav e the follo wing t wo p ossibilities for the substitution of the dominating v ariable t which en umerated hereafter:  z t  0 , 1 ≥  4 ⊗ ( − 1) ⊗ h 6 ⊗ ( − 1) ⊗ h  ::  [4 ⊗ ( − 1) ⊗ µ, + ∞ ] [6 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h B h B   z t  0 , 4 ≥  3 ⊗ ( − 1) ⊗ y 3 ⊕ 4 ⊗ ( − 1) ⊗ h 3 ⊗ ( − 1) ⊗ y 3 ⊕ 5 ⊗ ( − 1) ⊗ h  ::  [4 ⊗ ( − 1) ⊗ µ, + ∞ ] [5 ⊗ ( − 1) ⊗ µ, + ∞ ]  ::  h B h B  . Applying Theorem 4.3: ( T , T ′ ) = ( B , B ). W e compute the interv al τ and we ha v e: τ = [4 ⊗ ( − 1) ⊗ µ, + ∞ ]. And argMin ( τ ) = { (1 , t ) , (4 , t ) } is not a singleton. So, we hav e to compute τ ′ , Theorem 4.3. And τ ′ = [5 ⊗ ( − 1) ⊗ µ, + ∞ ]. Then, w e take ( i ∗ , j ∗ ) ∈ argMin ( τ ′ ) = { (4 , t ) } . Thus, cost [3] ( y , t, h ) = 3 ⊗ ( − 1) ⊗ y 3 ⊕ 4 ⊗ ( − 1) ⊗ h . And w e apply the results of § 4.2.3. Then, the set of linear equalities is now: L [3] = { y 1 = 3 ⊗ ( − 1) ⊗ h, y 2 = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h } ∪ { t = 3 ⊗ ( − 1) ⊗ y 3 ⊕ 5 ⊗ ( − 1) ⊗ h } . (79) And we define the 6 × 6-transition matrix T 1 → 2 as follows: T 2 → 3 :=         1 O O O O O O 1 O O O O O O 1 O O O O O O 1 O O O O O 3 ⊗ ( − 1) O 5 ⊗ ( − 1) O O O O O 1         . W e calculate the matrices A + = A +[2] ⊗ T 2 → 3 and A − = A − [2] ⊗ T 2 → 3 and w e ha ve: A + =         1 O O O O O O O O 3 ⊗ ( − 1) O 5 ⊗ ( − 1) O O O O O O O O O O O 3 ⊗ ( − 1) O O O 1 O 2 ⊗ ( − 1) O O O O O O         , A − =         O O O 3 ⊗ ( − 1) O 4 ⊗ ( − 1) O O O 6 ⊗ ( − 1) O 6 ⊗ ( − 1) O O O O O O O O O 3 ⊗ ( − 1) O 5 ⊗ ( − 1) O O O 1 O 2 ⊗ ( − 1) O O O O O O         . 32 W e ha v e w = ( z , O , O , y 3 , O , h ) ⊺ and the matrices ( A + , A − ) [3] are defined as the result of setrowtozero ( A + , A − ): A +[3] =         1 O O O O O O O O O O O O O O O O O O O O O O 3 ⊗ ( − 1) O O O O O O O O O O O O         , A − [3] =         O O O 3 ⊗ ( − 1) O 4 ⊗ ( − 1) O O O O O O O O O O O O O O O 3 ⊗ ( − 1) O 5 ⊗ ( − 1) O O O O O O O O O O O O         . (80) W e hav e: A +[3] [ | 1 , 5 | ] h =       O O 3 ⊗ ( − 1) O O       ≥ A − [3] [ | 1 , 5 | ] h =       O O 5 ⊗ ( − 1) O O       , and the condition (37) of Theorem 4.2 is v erified. Then, the function h 7→ 4 ⊗ ( − 1) of the cost function cost [3] ( y , t, h ) = 3 ⊗ ( − 1) ⊗ y 3 ⊕ 4 ⊗ ( − 1) ⊗ h is reached at y 3 = O . Thus, w e ha ve to solve the follo wing triangular linear system of equalities L [4] : L [4] : z = 3 ⊗ ( − 1) ⊗ y 3 ⊕ 4 ⊗ ( − 1) ⊗ h y 1 = 3 ⊗ ( − 1) ⊗ h y 2 = 1 ⊗ t ⊕ 4 ⊗ ( − 1) ⊗ h t = 3 ⊗ ( − 1) ⊗ y 3 ⊕ 5 ⊗ ( − 1) ⊗ h y 3 = O . (81) And we ha ve the solution of the mp r obtained b y obvious bac k substitution in L [4] : min( z ) = 4 ⊗ ( − 1) ⊗ h, y =   3 ⊗ ( − 1) ⊗ h 4 ⊗ ( − 1) ⊗ h O   , t = 5 ⊗ ( − 1) ⊗ h. (82) The solution x to the initial fmpr is then: x = t ⊗ ( − 1) ⊗ y =   2 1 O   and x 2 / 3 ⊗ x 1 = 1 ⊗ (3 ⊗ 2) ⊗ ( − 1) = 1 ⊗ 5 ⊗ ( − 1) = 4 ⊗ ( − 1) . W e retrieve the minimum 33 (( − 4) with the usual notations for reals) found b y the pseudo-p olynomial metho d developed in [10] and applied on Example 3 p. 1469 of [10]. 7 Conclusion In this conclusion w e try to pro vide a more precise analysis of the complexity of the substitution metho d. W e also compare our result with other known results. F or the forw ard substitution we hav e to establish the hierarc hy b et w een the v ariables of the problem we need to study m inequalities of the form (3a) whic h generate at most n AND -inequalities (3c) and at most n OR -inequalities (3b): O (2 nm ). F rom the OR -inequalities the computation complexity of the functions f ≥ ij (26a)-(26b)-(26c)-(26d) is O ( nm ). F rom the f ≥ ij the computa- tion complexit y of the f z ,ij functions (the p ossible next cost function (28)) is also O ( nm ). T o decide whic h v ariable can b e substituted we need The- orem 4.3 and compute the interv als f z ,ij ( R λµ ( x, h )) and f ij ( R λµ ( x, h )) in O (2 nm ). This pro cedure is rep eated n times and the o verall complexit y of the forward susbtitution is thus: O (6 n 2 m ) . No w we ha ve to add the time complexit y of the resolution of the triangular system of equalities L [ n ] whic h is O ( n ) b ecause we do not need to complute a pseudo inv erse of a matrix by residuation theory (see eg. [3]). Th us, the whole complexity is: O (6 n 2 m ) + O ( n ) . (83) Let us recall that to the b est of our knowledge this substitution metho d is the first strongly p olynomial metho d for such problem. All other prop osed sc hemes of resolution are pseudo-p olynomial (see [5], [10], [11]). Th us, b e- cause (max , +)-linear programming is now prov ed to b e strongly p olynomial then the mean pay off games problem is also strongly p olynomial (see eg. [10]). Let us remark that this result was already prov ed in [19] when solving the max-atom problem (MAP). The imp ortan t consequence of this result is that other six PTIME equiv alen t problems to MAP which were kno wn to b e in NP ∩ co-NP (see eg [1], [4], [7], [8], [9], [12], [16], [17], [18], [21]) are also 34 strongly p olynonial. W e list them hereafter. P1: Lo oking for non trivial solu- tions of a tropical cone, P2: Computation of a tropical rank of a matrix, P3: Computation of optimal strategies in parity games (t ypically: Mean Pa yoff Games), P4: Sc heduling with and/or precedence constrain ts, P5: Shortest path problem in hypergraph, P6: Model chec king and µ -calculus. And we rep eat that the b est pseudo-p olynomial metho d known till now can b e replaced by our strongly p olynomial approach. References [1] M Akian, S Gaubert, and A Guterman. The correspondence b etw een tropical con vexit y and mean pay off game. In Int. Pr o c. 19th Int. Symp. MTNS 2010 , 2010. [2] X. 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L. Litvinov and A. N. Sob olevski. Idemp oten t interv al analysis and optimization problems. R eliable Computing , 7(5), 2001. (353–377). [16] A. Min ´ e. A new numerical abstract domain based on difference-b ound matrices. In LNCS 2053 , pages 155–172, 2001. [17] R. H. Mohring, M. Skutella, and F. Stork. Scheduling with and/or precedence constraints. SIAM Journ. on Comp. , 33(2):393–415, 2004. [18] R. Nieuw enhuis, A. Oliv eras, and C. Tinelli. Solving sat and sat mo dulo theories from an abstract da vis-putman-logemann-lo v eland procedure to dpll(t). Journal of the A CM , 53(6):937–977, 2006. [19] Lauren t T ruffet. Lo oking F or All Solutions of a Set of Max-Atoms Solv es the Max Atom Problem in Strongly Polynom ial Time. In Mo d ´ elisation des Syst` emes R ´ eactifs (MSR’25) , Reims, F rance, Nov em- b er 2025. CReSTIC, LICI IS. (hal-05491586). [20] H. P . Williams. F ourier’s Metho d of Linear Programming and Its Dual. A mer. Math. Month. , 93(9), 1986. (681-695). [21] U. Zwick and M. P aterson. The complexit y of mean pa yoff games on graphs. The or. Comp. Scienc e , 158:343–359, 1996. 36 A Maximizing a tropical linear problem In this app endix we follow the same organization of the main part of the pa- p er dealing with the minimization problem. All the results b elo w are listed without pro ofs. T o form ulate the maximization problem MPR first we b orrow the F ourier’s tric k (see eg. [13], [20]) used in the usual linear algebra and applying it to the (max , +)-algebra. It comes that max( z = c ⊺ ⊗ x ⊕ c h ⊗ h ) is replaced b y: max( z ) and z ≤ c ⊺ ⊗ x ⊕ c h ⊗ h . And then, w e will solv e the following minimization problem MPR defined as: max( z ) (84a) suc h that: ( x, h ) ∈ R n +1 O , (84b) and z ≤ c ⊺ ⊗ x ⊕ c h ⊗ h A ⊗ x ⊕ b ⊗ h ≤ C ⊗ x ⊕ d ⊗ h. (84c) Where the homogenization v ariable h satisfies the follo wing condition: h < + ∞ . (84d) And z and h are not substituable . (84e) Finally , among the v ariables of the problem z , x 1 , . . . , x n w e ha ve: z has the highest priority b ecause b ounded by the cost function . (84f ) When necessary such problem will b e denoted: MPR ( A, b, C, d, c, c h , z , x, h, ( m, n )). T o the cone C ( L, W ) := { x : L ⊗ x ≤ W ⊗ x } we asso ciate the function setro wtozero ( L, W ) defined by: setro wtozero ( L, W ) := F or i = 1 to m do if l i,. ≤ w i,. then l i,. := O , w i,. := O (85) F or the MPR we will consider the following familly of in terv als: J := { [ O , u ] , u ∈ R O } . (86) 37 Prop osition A.1 L et [ O , u l ] , l = 1 , . . . , k , k ≥ 2 b e k intervals of J with F or al l u k ≤ . . . ≤ u 1 , we have the fol lowing intervals inclusion: [ O , u k ] ⊆ [ O , u k − 1 ] ⊆ · · · ⊆ [ O , u 1 ] . (87) And if Min denotes the minimum in the sense of interv al inclusion then: ∩ k l =1 [ O , u l ] = Min { [ O , u l ] , l = 1 , . . . , k } . (88) A t the step 0 of the substitution the cone asso ciated with the constraints system (6c) of the MPR problem is denoted C ( A + , A − ) [0] and is defined by: C ( A + , A − ) [0] := { w ∈ R n +2 O : A +[0] ⊗ w ≤ A − [0] ⊗ w } (89a) where: ( A + , A − , w ) [0] :=  1 −∞ ⊺ O −∞ A b  ,  O c ⊺ c h −∞ C d  ,   z x h   . (89b) W e also use the same n umerotation conv ention 4.1 p. 11 for matrices and v ectors in volv ed in this problem. The cost function at step 0 of the substitution is denoted and defined by: cost [0] ( x, h ) := c ⊺ ⊗ x ⊕ c h ⊗ h , with c h = O . The set of the stored linear equalities is denoted L [ k ] at each step k of the substitution. And for k = 0 we hav e L [0] = ∅ . The vector x is called the v ector of the remaining v ariables of the MPR . Of course at step 0 of the metho d: x = ( x j ) n j =1 . T o x w e asso ciate the set of the remaining v ariables denoted x . The parametrized domain of research of a maximum is denoted ˜ R ˜ λ ˜ µ ( x, h ) and defined by: ˜ R ˜ λ ˜ µ ( x, h ) := × x j ∈ x [ O , ˜ λ ] × [ O , ˜ µ ] . (90a) Where b ecause h m ust b e < + ∞ we can assume that the following condition holds: ∀ θ  = O , ∀ β , ∃ ˜ λ, ˜ µ : β ⊗ ˜ µ < θ ⊗ ˜ λ. (90b) 38 Based on the result of Prop osition 3.2 ∀ i = 1 , . . . , m , ∀ j = 1 , . . . , n suc h that the following condition is satisfied: a + i,j  = O and a + i,j > a − i,j (91) w e define the following v alid inequality: I ≤ ( x j , i ) := { a + i,j ⊗ x j ≤ a − i,. ⊗ w } . (92) Under condition (91) the inequality I ≤ ( x j , i ) can b e rewritten as: I ≤ ( x j , i ) = { x j ≤ f ≤ ij ( x, h ) } . (93) Where the ( x, h )-linear function f ≤ ij ( x, h ) is defined by: f ≤ ij ( x, h ) := ℓ ij ( x ) ⊕ r ij ⊗ h, (94a) with ℓ ij is the following x -linear function whic h do es not dep end on x j : ℓ ij ( x ) := v ⊺ ij ⊗ x, (94b) where v ij denotes the n -dimensional vector such that: ∀ j ′ = 1 , . . . , n : v ij,j ′ =  ( a + i,j ) ⊗ ( − 1) ⊗ a − i,j ′ if j ′  = j O if j ′ = j (94c) and the scalar r ij is defined by: r ij := ( a + i,j ) ⊗ ( − 1) ⊗ d i . (94d) Assuming that I ≤ ( x j , i )  = ∅ , or equiv alen tly be a v alid inequalit y , we define the following function: f z ,ij ( x, h ) := ⊕ j ′  = j c j ′ ⊗ x j ′ ⊕ c j ⊗ f ≤ ij ( x, h ) ⊕ c h ⊗ h. (95) By replacing f ≤ ij ( x, h ) in (27) we obtain the new expression for f z ,ij ( x, h ): f z ,ij ( x, h ) := ⊕ j ′  = j ( c j ′ ⊕ c j ⊗ v ij,j ′ ) ⊗ x j ′ ⊕ ( c h ⊕ c j ⊗ r ij ) ⊗ h. (96) The next Theorem will b e useful to justify the substitution method de- v elop ed in this pap er. 39 Theorem A.1 (Saturation of an inequlit y) F or al l x j such that I ≤ ( x j , i ) is valid in the sense of Pr op osition 3.2 we have: sup { x : x j ≤ f ≤ ij ( x,h ) }  c ⊺ ⊗ x ⊕ c h ⊗ h x j  =  f z ,ij ( x, h ) f ≤ ij ( x, h )  . (97) A nd the supr emum is r e ache d at x such that we have: x j = f ≤ ij ( x, h ) . (98) A.1 Classification of the v ariables and the linear func- tions of MPR Based on the result of Prop osition 3.2, ∀ i = 1 , . . . , m , ∀ j = 1 , . . . , n suc h that the following condition is satisfied: a − i,j  = O and a + i,j < a − i,j (99) w e define the following v alid inequality: I ≥ ( x j , i ) := { a − i,j ⊗ x j ≥ a + i,. ⊗ w } . (100) Under condition (31) the result of Prop osition 3.2 I ≥ ( x j , i ) = { x j ≥ f ≥ ij ( x, h ) } (see (26a)-(26d)). The ordering b et w een the v ariables of the problem is denoted ⪯ v ar and based on their v ariation domain such that: { x j ∈ x : I ≤ ( x j )  = ∅ and I ≥ ( x j )  = ∅} ⪯ v ar { x j ∈ x : I ≤ ( x j )  = ∅ and I ≥ ( x j ) = ∅} . (101) W e also mentioned that: { x j ∈ x : I ≤ ( x j )  = ∅ and I ≥ ( x j ) = ∅} ⪯ v ar { x j ∈ x : I ≤ ( x j ) = ∅ and I ≥ ( x j ) = ∅} . W e now define the follo wing partition of non n ull functions in volv ed in the MPR . Definition A.1 A non nul l ( x, h ) -line ar function f : ( x, h ) 7→ α ⊺ ⊗ x ⊕ β ⊗ h is said to b e h -b ounde d if α = −∞ . Otherwise the function f is said to b e h -unb ounde d. 40 W e denote h ˜ B the set of h -b ounded non n ull linear functions and we denote h ˜ U the set of non null h -unbounded linear functions. And ( h ˜ B , h ˜ U ) is a partition of the set of all non n ull ( x, h )-linear functions. W e ha v e the follo wing in terv al calculus results. • ∀ f = α ⊺ ⊗ x ⊕ β ⊗ h ∈ h ˜ U , using interv al calculus formulae w e ha ve: f ( ˜ R ˜ λ ˜ µ ( x, h )) = [ O , ⊕ i α i ⊗ ˜ λ ⊕ β ⊗ ˜ µ ] , (102a) and by assumption ((90b) with θ = ⊕ i α i ) on ˜ λ and ˜ µ w e ha ve: ∀ α ∈ R n O \ { −∞ } , ∀ β : f ( ˜ R ˜ λ ˜ µ ( x, h )) = [ O , ⊕ i α i ⊗ ˜ λ ] (102b) Let us stress that this in terv al does not dep end on β and b y assumption (90b): [ O , β ⊗ ˜ µ ] ⊂ [ O , ⊕ i α i ⊗ ˜ λ ] . (102c) • And ∀ f = β ⊗ h ∈ h ˜ B : f ( ˜ R ˜ λ ˜ µ ( x, h )) = [ O , β ⊗ ˜ µ ] . (103) F rom this ab o v e results w e ha v e the following ordering b et ween the non n ull linear functions denoted ⪯ f ct whic h is based on their v alue domain: { f : f ∈ h ˜ B } ⪯ f ct { f : f ∈ h ˜ U } . (104) A.2 Cho osing a v ariable for a new substitution after k substitutions in MPR The pro cedure for choosing a remaining v ariable to b e substituted is based on the theorems listed below. Recall that w e use numerotation con v ention 4.1 p. 11. Theorem A.2 (Maximalit y and reachabilit y at −∞ of h 7→ c h ⊗ h ) The function h 7→ c h ⊗ h is the lower b ound of the c ost function ( x, h ) 7→ c ⊺ x ⊗ x ⊕ c h ⊗ h which is attaine d at x = −∞ iff the fol lowing c ondition holds: A +[ k ] [ | 1 ,m | ] h ≤ A − [ k ] [ | 1 ,m | ] h . (105) 41 Theorem A.3 L et I b e a subset of valid ine qualities I ≤ ( x j , i ) , ie. a set such that: I ⊆ I ≤ ( x ) . (106) Wher e I ≤ ( x ) := { ( i, j ) : x j ∈ x and I ≤ ( x j , i ) is valid } . (107) I ≤ ( x j , i )  = ∅ or e quivalently I ≤ ( x j , i ) is valid ⇒ z ≤ f z ,ij ( x, h ) . (108) R e c al ling that f z ,ij ( x, h ) is define d by (96). L et T and T ′ denoting either ˜ B or ˜ U , r esp e ctively. The 2 -tuple ( T , T ′ ) is a function of I and is define d ac c or ding to ⪯ f ct as fol lows with priority to the c ost function: ( T , T ′ ) :=          ( ˜ B , ˜ B ) if ∃ ( i, j ) s.t. f z ,ij ∈ h ˜ B and f ≤ ij ∈ h ˜ B ( ˜ B , ˜ U ) if ∃ ( i, j ) s.t. f z ,ij ∈ h ˜ B and f ≤ ij ∈ h ˜ U ( ˜ U , ˜ B ) if ∃ ( i, j ) s.t. f z ,ij ∈ h ˜ U and f ≤ ij ∈ h ˜ B ( ˜ U , ˜ U ) if ∃ ( i, j ) s.t. f z ,ij ∈ h ˜ U and f ≤ ij ∈ h ˜ U . (109) Then, define the fol lowing set of indexes: T ( T , T ′ , I ) := { ( i, j ) ∈ I : ( f z ,ij , f ≤ ij ) ∈ ( h T , h T ′ ) } . (110) L et us define the fol lowing assertion which de als with the c ost z and an interval τ : Z( τ ) : ∀ ( i, j ) ∈ T ( T , T ′ , I ) ∀ z ( z ∈ τ ⇒ ∃ ( x, h ) ∈ ˜ R ˜ λ ˜ µ ( x, h ) z ≤ f z ,ij ( x, h )) . (111) A nd let us define the fol lowing assertion de aling with variables of the pr oblem and intervals τ , τ ′ : X( τ , τ ′ ) : ∀ ( i, j ) ∈ T ( T , T ′ , I ) ∩ a rgMin ( τ ) ∀ x j ( x j ∈ τ ′ ⇒ ∃ ( x, h ) ∈ ˜ R ˜ λ ˜ µ ( x, h ) x j ≤ f ≤ ij ( x, h )) . (112) 42 Then, we have the fol lowing e quivalenc es. Z( τ ) true ⇔ τ = Min { f z ,ij ( ˜ R ˜ λ ˜ µ ( x, h )) , ( i, j ) ∈ T ( T , T ′ , I ) } . (113) A nd X( τ , τ ′ ) true ⇔ τ ′ = Min { f ≤ ij ( ˜ R ˜ λ ˜ µ ( x, h )) , ( i, j ) ∈ argMin ( τ ) ∩ T ( T , T ′ , I ) } . (114) W e are no w in p osition to indicate ho w to choose a remaining v ariable to b e substituted. The c hoice also dep ends on the follo wing differen t cases whic h are listed hereafter. Case 0: max( z ) = O or no maximum. Case 0 . 1: x = ∅ and the condition (105) of Theorem A.2 does not holds. x = ∅ means that k = n (no more v ariable to b e substituted) and cost [ n ] ( x, h ) = c [ n ] h ⊗ h . The condition (105) of Theorem A.2 do es not holds means A +[ n ] [ | 1 ,m | ] h ≰ A − [ n ] [ | 1 ,m | ] h then h = O is the only p ossible v alue for h and by backw ard substitution w e ha ve z = O , x = −∞ . Or Case 0 . 2: x  = ∅ and I ≤ ( x ) = ∅ and I ≥ ( x ) = ∅ and the condition (105) of Theorem A.2 does not holds. The substitution pro cess stops because no v alid inequalities can b e generated. No maximum. Case 1: switching case , cf. § 2.4. x + = ∅ and cost [ k ] ( x, h ) = c [ k ] h ⊗ h and the condition (105) of Theorem A.2 do es not holds and x O  = ∅ and I ≤ ( x O ) = ∅ and I ≥ ( x O )  = ∅ . The substitution must switch to the following minimizing problem: pmr ( A, b, C, d, c, c h , z , x, h, ( m, n )) , (115a) where the matrices A and C are defined by: A := A − [ k ] [ | 1 ,m | ][ | 1 ,n | ] , C := A +[ k ] [ | 1 ,m | ][ | 1 ,n | ] . (115b) The vectors b and d are defined b y: b := A − [ k ] [ | 1 ,m | ] h , d := A +[ k ] [ | 1 ,m | ] h . (115c) The initial cost function of the MPR is characterized by the vector c and the constan t c h suc h that: c := −∞ , c h := c [ k ] h . (115d) 43 And the v ariables of the problem are: z := z , x ← x O , h := h. (115e) Recalling that the notation x ← x O means that w e store in the n -dimensional v ector x the remaining v ariables of the set x O . The other comp onents (al- ready substituted v ariables) are set to O . And the dimensions ( m, n ) of the mp r are ob viously the same as the ones of the MPR . Case 2: Not Case 0 and not Case 1. The first case is Case 2 . 1: x = ∅ . This means that k = n (no more v ariable to b e substituted) and cost [ n ] ( x, h ) = c [ n ] h ⊗ h . And if A +[ n ] [ | 1 ,m | ] h ≤ A − [ n ] [ | 1 ,m | ] h then h can tak e all p ossible v alues in par- ticular w e can tak e h = 1 and the MPR succeeds with maxim um z = c [ n ] h . Otherwise h = O is the only p ossible v alue for h and z = O . The case is Case 2 . 2: x  = ∅ and ∃ x j ∈ x : I ≤ ( x j )  = ∅ . Let us distinguish the following sub cases whic h are based on the form of the cost function: cost [ k ] ( x, h ) = c ⊺ x + ⊗ x + ⊕ c ⊺ x O ⊗ x O ⊕ c h ⊗ h, (116a) where: x + := { x j ∈ x : c j > O } and x O := { x j ∈ x : c j = O } . (116b) And the vectors of remaining x + and x O are defined by: x + := ( x j ) { j : x j ∈ x + } and x O := ( x j ) { j : x j ∈ x O } . (116c) Case 2 . 2 . 1: the condition (105) of Theorem A.2 holds. Then, z = c h ⊗ h is the maxim um for the MPR . And the vector of the remaining v ariables x is set to −∞ . Case 2 . 2 . 2: otherwise, the condition (105) of Theorem A.2 do es not hold. Based on the hierarch y ⪯ v ar (see (101)) b etw een the remaining v ariables let us define the set of the dominating v ariables D + O := { x j ∈ x : I ≥ ( x j )  = ∅ and I ≤ ( x j )  = ∅} . And the set of inequalities: I + O ( D ) :=  I ≤ ( x ∩ D + O ) if D + O  = ∅ I ≤ ( x ) , defined by (107) otherwise. (117) 44 W e apply Theorem A.3 recalling that ( T , T ′ ) is defined according to ⪯ f ct b y (109) and the subset of v alid inequalities is I := I + O ( D ). Thus, w e are “w orking” with the indexes of rows and columns (or v ariables) of the system of inequalities which is: T ( T , T ′ , I + O ( D )). Then, we ha v e to compute the interv al τ such that Z( τ ) is true (see (111) and (113)). If argMin ( τ ) is a singleton { ( i ∗ , j ∗ ) } then take the unique 2-tuple ( i ∗ , j ∗ ) for the substitution of x j ∗ = f ≤ i ∗ j ∗ ( x, h ). Otherwise, tak e an y 2-tuple ( i ∗ , j ∗ ) elemen t of a rgMin ( τ ′ ) recalling that the interv al τ ′ is the in terv al suc h that X( τ , τ ′ ) is true (see (112) and (114)). A.3 The new c haracteristic elemen ts of the MPR after substitution The new set of equalities is defined as follows. If conditions of Theorem A.2 are verified then set k + 1 = n and L [ n ] := L [ k ] ∪ x j ∈ x { x j = O } ∪ { z = c h ⊗ h } . Then, MPR stops and w e solv e the linear system L [ n ] and express all the x j ’s as function of the homogenization v ariable h . Otherwise, w e define L [ k +1] as: L [ k +1] := L [ k ] ∪ { x j ∗ = f ≤ i ∗ j ∗ ( x, h ) } and we apply Theorem A.1. And based on the previous cases the new cost function is defined as: cost [ k +1] ( x, h ) := f z ,i ∗ j ∗ ( x, h ) . (118) And in all cases the new cone C ( A + , A − ) [ k +1] asso ciated with MPR is deduced from the following set of inequalities: z ≤ cost [ k +1] ( x, h ) , ∀ i ∈ [ | 1 , m | ] : a +[ k ] i,. ⊗ ℓ ≤ a − [ k ] i,. ⊗ ℓ. (119a) and ℓ is the n + 2-dimensional column v ector which has the same comp onen ts as the vector w [ k ] except its j ∗ th comp onent wic h is: ℓ j ∗ := f ≤ i ∗ j ∗ ( x, h ) . (119b) Let us define the following ( n + 2) × ( n + 2) tr ansition matrix T k → k +1 b y: ∀ j : t k → k +1 j,. :=  e j if j  = j ∗ ( O , v ⊺ i ∗ j ∗ , r i ∗ j ∗ ) if j = j ∗ . (120) 45 Where e j denotes the j th n + 2-dimensional ro w v ector of the ( n + 2) × ( n + 2)- iden tit y matrix I n +2 . The new matrices ( A + , A − ) [ k +1] are defined by: ( A + , A − ) [ k +1] := setro wtozero ( A +[ k ] ⊗ T k → k +1 , A − [ k ] ⊗ T k → k +1 ) (121) recalling that function setrowtozero is defined b y (85). And the new vector w [ k +1] is defined by: w [ k +1] :=        z . . . 0 . . . h        ← j ∗ (122) Finally , the new set of the remaining v ariables at step k + 1 is defined by: x := x \ { x j ∗ } . (123) A.4 A n umerical example In this section we illustrate our strongly p olynomial metho d on a numeri- cal example b orro w ed from the litterature which are solv ed using pseudo- p olynomial metho d in [10]. And w e apply substitution on [10], Example 1 p. 1458 which is: max { z = 1 ⊗ x 1 ⊕ 3 ⊗ x 2 , x = ( x 1 , x 2 ) ⊺ ∈ P ( A, b, C , d ) ∪ { ±∞ }} , where: A :=     O 1 ⊗ ( − 1) 2 ⊗ ( − 1) 2 ⊗ ( − 1) 1 ⊗ ( − 1) O 1 O     , b := −∞ , C :=     1 O O O O 1 O 2     , d :=     1 1 1 1     . With the following change of notations c ↔ b and B ↔ C . The MPR is defined on the homogenized cone C ( A, b, C , d ) by adding the homogenization v ariable h . 46 A t step 0 of the substitution the cone associated with the ab o v e con- strain ts is denoted C ( A + , A − ) [0] and is characterized by the following system of inequalities: A +[0] ⊗ w ≤ A − [0] ⊗ w , (124a) with w = ( z , x 1 , x 2 , h ) ⊺ ∈ R 4 O and the matrices ( A + , A − ) [0] are defined by: A +[0] :=       1 O O O O O 1 ⊗ ( − 1) O O 2 ⊗ ( − 1) 2 ⊗ ( − 1) O O 1 ⊗ ( − 1) O O O 1 O O       , A − [0] :=       O 1 3 O O 1 O 1 O O O 1 O O 1 1 O O 2 1       . (124b) And the F ourier’s trick for this example is: max( z ) (125a) z ≤ cost [0] ( x, h ) (125b) cost [0] ( x, h ) = 1 ⊗ x 1 ⊕ 3 ⊗ x 2 ⊕ O ⊗ h. (125c) x = { x 1 , x 2 } . It is easy to c heck that ∀ x j ∈ x : I ≤ ( x j ) ∪ I ≥ ( x j )  = ∅ . So that the MPR is b ounded. W e hav e x + = { x 1 , x 2 } and x O = ∅ . The set of dominating v ariables is D + O = { x 1 , x 2 } . The condition (37) of Theorem 4.2 is not v erified. Th us, Case 2 . 2 . 2 applies. W e ha v e (see (117)): I + O ( D ) = I ≤ ( { x 1 , x 2 } ) = { (1 , 2) , (2 , 1) , (2 , 2) , (3 , 1) , (4 , 1) } . And we enumerate all the p ossibilities in the following arra y: 47  z x 2  0 , 1 ≤  4 ⊗ x 1 ⊕ 4 ⊗ h 1 ⊗ x 1 ⊕ 1 ⊗ h  ::  [ O , 4 ⊗ ˜ λ ] [ O , 1 ⊗ ˜ λ ]  ::  h ˜ U h ˜ U   z x 1  0 , 2 ≤  3 ⊗ x 2 ⊕ 3 ⊗ h 2 ⊗ h  ::  [ O , 3 ⊗ ˜ λ ] [ O , 2 ⊗ ˜ µ ]  ::  h ˜ U h ˜ B   z x 2  0 , 2 ≤  1 ⊗ x 2 ⊕ 5 ⊗ h 2 ⊗ h  ::  [ O , 1 ⊗ ˜ λ ] [ O , 2 ⊗ ˜ µ ]  ::  h ˜ U h ˜ B   z x 1  0 , 3 ≤  3 ⊗ x 2 ⊕ 2 ⊗ h 1 ⊗ x 2 ⊕ 1 ⊗ h  ::  [ O , 3 ⊗ ˜ λ ] [ O , 1 ⊗ ˜ λ ]  ::  h ˜ U h ˜ U   z x 1  0 , 4 ≤  3 ⊗ x 2 ⊕ 1 ⊗ h 2 ⊗ x 2 ⊕ h  ::  [ O , 3 ⊗ ˜ λ ] [ O , 2 ⊗ ˜ λ ]  ::  h ˜ U h ˜ U  . By Theorem A.3 w e ha ve ( T , T ′ ) = ( ˜ U , ˜ B ). Thus, w e only ha ve to study  z x 1  0 , 2 and  z x 2  0 , 2 . The set τ , Theorem A.3, is suc h that τ = [ O , 1 ⊗ ˜ λ ]. And a rgMin ( τ ) = { (2 , 2) } whic h is a singleton. Th us, ( i ∗ , j ∗ ) = (2 , 2). So, we hav e: x 2 = 2 ⊗ h in row 2 of the linear system of inequalities (124a)-(124b). The new characteristic elements are listed b elow. L [1] = { x 2 = 2 ⊗ h } . (126) The new cost function is: cost [1] ( x, h ) := f z , 22 ( x, h ) = 1 ⊗ x 1 ⊕ 5 ⊗ h. (127) The 4 × 4-transition matrix is defined by: T 0 → 1 :=     1 O O O O 1 O O O O O 2 O O O 1     . (128) Then, w e compute the following matrices A + := A +[0] ⊗ T 0 → 1 and A − := A − [0] ⊗ T 0 → 1 and we obtain: 48 A + :=       1 O O O O O O 1 O 2 ⊗ ( − 1) O 1 O 1 ⊗ ( − 1) O O O 1 O 1       , A − :=       O 1 O 5 O 1 O 1 O O O 1 O O O 2 O O O 4       . The new cone C ( A + , A − ) [1] is defined by: A +[1] ⊗ w ≤ A − [1] ⊗ w , (129a) with w = ( z , x 1 , O , h ) ⊺ ∈ R 4 O and the matrices ( A + , A − ) [1] are defined as a result of setrowtozero ( A + , A − ): A +[1] :=       1 O O O O O O 1 O 2 ⊗ ( − 1) O 1 O 1 ⊗ ( − 1) O O O 1 O 1       , A − [1] :=       O 1 O 5 O 1 O 1 O O O 1 O O O 2 O O O 4       . (129b) x = { x 1 }  = ∅ . It is easy to see that the MPR problem is b ounded. W e ha v e x + = { x 1 } and x O = ∅ . The set of dominating v ariables is D + O := { x 1 } . The condition (105) of Theorem A.2 is not v erified. Thus, Case 2 . 2 . 2 applies. W e hav e (see (117)): I + O ( D ) = I ≤ ( { x 1 } ) = { (2 , 1) , (3 , 1) , (4 , 1) } . And we enumerate all the p ossibilities in the following arra y:  z x 1  0 , 2 ≤  5 ⊗ h 2 ⊗ h  ::  [ O , 5 ⊗ ˜ µ ] [ O , 2 ⊗ ˜ µ ]  ::  h ˜ B h ˜ B   z x 1  0 , 3 ≤  5 ⊗ h 3 ⊗ h  ::  [ O , 5 ⊗ ˜ µ ] [ O , 3 ⊗ ˜ µ ]  ::  h ˜ B h ˜ B   z x 1  0 , 4 ≤  5 ⊗ h 4 ⊗ h  ::  [ O , 5 ⊗ ˜ µ ] [ O , 4 ⊗ ˜ µ ]  ::  h ˜ B h ˜ B  . Applying Theorem A.3 we hav e: ( T , T ′ ) = ( ˜ B , ˜ B ). And the set τ is τ = [ O , 5 ⊗ ˜ µ ]. W e remark that a rgMin ( τ ) = { (2 , 1) , (3 , 1) , (4 , 1) } is not a singleton. So, we hav e to compute τ ′ , Theorem A.3, τ ′ = [ O , 2 ⊗ ˜ µ ]. And we tak e ( i ∗ , j ∗ ) ∈ a rgMin ( τ ′ ) = { (2 , 1) } . 49 Th us, ( i ∗ , j ∗ ) = (2 , 1). So, w e ha v e: x 1 = 2 ⊗ h in row 2 of the linear system of inequalities (129a)-(129b). The new characteristic elments of the problem are listed b elo w. L [2] = L [1] ∪ { x 1 = 2 ⊗ h } = { x 2 = 2 ⊗ h, x 1 = 2 ⊗ h } . (130) The new cost function is: cost [2] ( x, h ) := f z , 21 ( x, h ) = 5 ⊗ h. (131) The 4 × 4-transition matrix is defined by: T 1 → 2 :=     1 O O O O O O 2 O O 1 O O O O 1     . (132) W e compute the follo wing matrices A + := A +[1] ⊗ T 1 → 2 and A − := A − [1] ⊗ T 1 → 2 and we obtain: A + :=       1 O O O O O O 1 O O O 1 O O O 1 O O O 2       , A − :=       O O O 5 O O O 2 O O O 1 O O O 2 O O O 4       . The new cone C ( A + , A − ) [2] is defined by: A +[2] ⊗ w ≤ A − [2] ⊗ w , (133a) with w = ( z , x 1 , O , h ) ⊺ ∈ R 4 O and the matrices ( A + , A − ) [2] are defined as a result of setrowtozero ( A + , A − ): A +[2] :=       1 O O O O O O O O O O O O O O O O O O O       , A − [2] :=       O O O 5 O O O O O O O O O O O O O O O O       . (133b) Clearly , A +[2] [ | 1 , 4 | ] h = −∞ ≤ A − [2] [ | 1 , 4 | ] h = −∞ . And the condition (105) of Theorem A.2 is verified and the cost function h 7→ 5 ⊗ h is reached. 50 The solution of the MPR is obtained b y a trivial bac k substitution: z = 5 ⊗ h, x 2 = 2 ⊗ h, x 1 = 2 ⊗ h. And w e retriev e by our strongly p olynomial metyhod the same maximum as in Example 1 p. 1458 of [10]. But the corner (2 , 2) is different from the one obtained b y the pseudo-p olynomial metho d developed by the authors of [10], and mentioned p. 1474: x 1 = 1 , x 2 = 2. 51