N-fold integer programming in cubic time

N-fold in teger programming in cubic time Ra y m ond Hemmeck e ∗ Shm u el Onn † Lyub ov Romanc huk ‡ Abstract N-fold in teger programming is a fundamen tal problem with a v ariet y of natural applications in op erations researc h and statistic s. Moreo ver, it is uni- v ersal and pro vid es a new, v a r iable-dimension, parametrization of all of inte ger programming. The fastest algo r ithm for n -fold in teger programming p r edat- ing the pr esen t article ru n s in time O  n g ( A ) L  with L the binary length of the numerical part of the in p ut and g ( A ) the s o-cal led Gra ver complexit y of the bimatrix A defining the system. In this article w e p r o vide a d r astic impro vemen t a n d establish an algorithm which r u ns in time O  n 3 L  ha ving cubic dep endency on n regardless of the bim atrix A . Our algorithm can b e extended to separable conv ex piecewise affine ob jectiv es as w ell, an d also to systems defin ed b y b im atrices w ith v ariable en tries. Moreo v er, it can b e used to defin e a hierarch y of appro ximations for any integ er p rogramming problem. 1 In tro duct i o n N-fold integer pro gramming is the following problem in v ariable dimension nt , min  wx : A ( n ) x = b , l ≤ x ≤ u , x ∈ Z nt  , (1) where A ( n ) :=        A 1 A 1 · · · A 1 A 2 0 · · · 0 0 A 2 · · · 0 . . . . . . . . . . . . 0 0 · · · A 2        (2) is a n ( r + ns ) × nt matrix whic h is the n -f o ld pr o duct of a fixed ( r, s ) × t b imatrix A , that is, of a ma t rix A consisting of tw o blo c ks A 1 , A 2 , with A 1 its r × t submatrix consisting of the first r rows a nd A 2 its s × t submatrix consisting o f the last s ro ws. ∗ T echnisc he Universit¨ at Munich, Germany . † T echnion – Isra el Institute of T echnology . Suppor ted in part by the Israel Science F oundation. ‡ T echnion – Isr ael Institute o f T ech nology . Supp orted in part by the T echnion Graduate Scho ol. 1 2 This is a f undamental problem with a v ariety of natural applications in op eratio ns researc h and statistics whic h, along with extensions and v ariatio ns, include m ultiin- dex and m ulticommo dity transp ortatio n problems, priv acy and disclosure con trol in statistical dat a bases, and sto c ha stic in teger programming. W e briefly discuss some of these applications in Section 5 (Corollaries 5.1 and 5.2 ). F or more inf ormation see e.g. [4, 5, 7, 9, 10, 11, 12, 14, 15, 20], [1 7, Chapters 4,5], and the references therein. Moreo ve r , n -fold integer programming is univ ersal [6] and prov ides a new, v ariable- dimension, parametrization of all of in teger programming: ev ery program is an n -fold program for some m o ver the bimatrix A := A ( m ) with first blo c k A 1 the 3 m × 3 m iden tity matrix and second blo c k A 2 the (3 + m ) × 3 m incidence matr ix o f the complete bipartite graph K 3 ,m . W e make further discussion of this in Section 7. The fa stest algorithm for n -fold in teger prog r amming predating the presen t a rti- cle is in [10] and runs in time O  n g ( A ) L  with L = h w , b , l , u i the binary length of t he n umerical part of the input, and g ( A ) the so-called Gr aver c omplex i ty of the bimatrix A . Unfor t una t ely , the Gra v er complexit y is typic ally ve r y la rge [2, 18]: for instance, the bimatrices A ( m ) men tio ned a b o ve ha v e Grav er complexit y g ( A ( m )) = Ω(2 m ), yielding p olynomial but very large n Ω(2 m ) dep endency of the running time on n . In this art icle w e provide a drastic improv emen t and establish an algo r ithm whic h runs in time O ( n 3 L ) hav ing the cubic dep endency on n which is alluded to in the title, regardless of the fixed bimatrix A . So the Grav er complexit y g ( A ) no w dro ps do wn from t he exp onent of n to the constan t m ultiplying n 3 . This is established in Section 3 (Theorem 3.9). Moreo v er, our construction can b e used to define a natural hierarc hy of appro ximatio ns fo r (1 ) for the bimatr ices A ( m ) with m v a riable, and therefore, b y the univ ersalit y theorem of [6], for any in t eger programming problem. These appro ximatio ns are currently under study , implemen tation and testing, and will b e discusse d briefly in Section 7 and in more detail elsewhere. Our algor it hm extends, moreo ver, for certain nonlinear ob jectiv e functions: using results o f [16] on certain optimalit y criteria, we provide in Section 4 an o ptimalit y certification pro cedure fo r separable conv ex ob jectiv es whose time complexit y is lin- ear in n (Theorem 4.1) and an a lgorithm for solving problems with separable con vex piecewise affine ob jectiv es whose time complexit y is a g ain cubic in n (Theorem 4.2). F urthermore, the algor it hm also leads to the first p olynomial time solution of n -fold in teger programming problems ov er bimatrices with v ariable entries (Theorem 6.1). 2 Notation and preliminaries W e start with some notation and review of some preliminaries on Gra ver bases and n -fold integer programming that w e need later on. See the b o ok [17 ] for more details. Gra v er bases w ere in tro duced in [8] as optimality certificates for in teger progra m- ming. Define a pa r t ial order ⊑ on R n b y x ⊑ y if x i y i ≥ 0 and | x i | ≤ | y i | for all i . 3 So ⊑ extends the co ordinat e-wise partial order ≤ on the nonnega t ive o r t han t R n + to all of R n . By a classical lemma of Gordan, ev ery subset Z ⊆ Z n has finitely-man y ⊑ -minimal elemen ts, that is, x ∈ Z suc h that no other y ∈ Z satisfies y ⊑ x . W e ha ve the following fundamen tal definition from [8]. Definition 2.1 The Gr aver b asis of an in teger m × n matrix A is defined to b e the finite set G ( A ) ⊂ Z n of ⊑ -minimal elemen ts in { x ∈ Z n : A x = 0 , x 6 = 0 } . F or instance, the Grav er basis of the matrix A := (1 2 1) consists of 8 ve ctors, G ( A ) = ± { (2 − 1 0) , (0 − 1 2) , (1 0 − 1) , (1 − 1 1) } . Consider the general integer programming problem in standard fo r m, min { w x : A x = b , l ≤ x ≤ u , x ∈ Z n } . (3) A fe a s ible step for feasible p oint x in (3) is any v ector v suc h that x + v is also feasible, that is, A v = 0 so A ( x + v ) = b , and l ≤ x + v ≤ u . An augmenting step for x is a feasible step v suc h tha t x + v is b etter, that is, w v < 0 so w ( x + v ) < w x . Gra v er has sho wn that a feasible p o in t x in (3) is optimal if and only if there is no elemen t g ∈ G ( A ) in the Gra ver basis of A whic h is an augmenting step fo r x . This suggests the follo wing simple a ug men tation sche me: start fro m a n y feasible p oin t in (3) and iterativ ely augment it to an optimal solutio n using Gra v er aug- men ting steps g ∈ G ( A ) as long as p ossible. While the num b er of iterations in this simple sc heme as is may b e expo nen tial, it was recen t ly sho wn in [10] that if in eac h iteration the b est p ossible augmen ting step of the for m γ g with γ p ositiv e in teger and g ∈ G ( A ) is tak en, then the n umber of iterations do es b ecome p olynomial. In what fo llo ws, w e call an augmen ting step whic h is at least as g o o d a s the b est p ossible augmen t ing step γ g with g ∈ G ( A ), a Gr aver-b est augmenting step . It w as shown in [5] that for fixed bimatrix A , the G ra ve r ba sis G  A ( n )  of the n -fold pro duct of A can b e computed in time p olynomial in n . Th us, to find a Gra v er-b est augmenting step of the f orm γ g for an n - fold in teger program (1), it is p ossible, as sho wn in [10], to c hec k eac h elemen t g ∈ G  A ( n )  , and for eac h, find the b est p ossible step size γ . How ev er, a s w e explain b elow, the Gra ver ba sis G  A ( n )  is v ery large. Therefore, in this article, we do it the other w ay around. F or each of O ( n ) critical p ositiv e in teger p oten tial step sizes γ , w e determine an aug men ting step γ h whic h is at least as go o d a s the b est p ossible augmen ting step γ g with g ∈ G  A ( n )  . W e then show that the b est among these steps ov er all suc h γ is a Gra v er-b est step. In prepara t io n for this, w e need to review some material on Gra v er bases of n - fold pro ducts. Let A b e a fixed in teger ( r , s ) × t bimatrix. F or any n w e write eac h v ector x ∈ Z nt as a tuple x = ( x 1 , . . . , x n ) o f n bricks x i ∈ Z t . It has b een shown in [1], [18], and [13], in increasing generality , that for ev ery bimatrix A , the n umber of nonzero bricks app earing in a n y elemen t in the Grav er basis G  A ( n )  for an y n is b ounded by a constan t indep enden t of n . So w e can mak e the fo llowing definition. 4 Definition 2.2 The Gr aver c omplexity of an integer bimatrix A is defined to b e the largest n umber g ( A ) of nonzero brick s g i in any elemen t g ∈ G  A ( n )  for any n . This w as used in [5] to show t ha t the Gra ver basis G ( A ( n ) ) ha s a p olynomial n umber O ( n g ( A ) ) of elemen ts and is computable in time O ( n g ( A ) ) p olynomial in n . Th us, the computat io n of a G ra ver-best augmenting step in [10] was done b y finding the b est step size γ for eac h o f these O ( n g ( A ) ) elemen t s of G ( A ( n ) ), resulting in p olynomial but v ery large O ( n g ( A ) ) dep endency of the running time on n . In Section 3 w e sho w ho w to find a Grav er- b est step without constructing G ( A ( n ) ) explicitly in quadratic time O ( n 2 ) regardless of the bimat r ix A and its Grav er complexit y . W e conclude this subsection with some remarks ab out complexit y and finiteness. The binary length of an in t eger num b er z is the num b er of bits in its binary enco ding, whic h is O (log | z | ), and is denoted b y h z i . The binary length h z i of an in teger ve ctor z is the sum of binary lengths of its entries . W e denote by L the binary length of all n umerical part of the input. In particular L = h w , b , l , u i for problem (1) . All n um- b ers manipulated b y our algorithms remain p olynomial in the binary length o f the input and our algorithms are p olynomial time in the T uring machine mo del. But w e are mostly in terested in the n umber of a rithmetic op erations p erformed ( a dditions, m ultiplications, divisions, comparisons), so time in our complexit y statemen ts is the n umber of suc h op erations as in the real arit hmetic mo del o f computation. F o r simplicit y of presen tation w e assume througho ut that all entries of the b ounds l , u in prog ram (1) a re finite and hence the set of feasible p oin ts in (1) is finite. This is no loss of g enerality since, as is w ell kno wn, it is alw ays p ossible to add suitable p olynomial upp er and lo wer b ounds without excluding some optimal solution if an y . 3 The algorithm W e now sho w how to decide if a give n feasible p oint x = ( x 1 , . . . , x n ) in (1) is optimal in linear time O ( n ), and if not, determine a Grav er-b est step γ g for x in quadratic time O ( n 2 ). This is then incorp orated into an iterative algorithm fo r solving (1). W e b egin with a lemma ab out elemen ts o f Gr av er bases o f n -fo ld pro ducts. Lemma 3.1 L et A b e inte ge r ( r , s ) × t bimatrix with Gr aver c omplexity g ( A ) . L et Z ( A ) :=  z ∈ Z t : z is the sum of at mo s t g ( A ) elements of G ( A 2 )  . (4) Then for any n , any g ∈ G  A ( n )  and any I ⊆ { 1 , . . . , n } , we have P i ∈ I g i ∈ Z ( A ) . Pro of. Consider an y Grav er basis elemen t g ∈ G  A ( n )  for some n . Then A ( n ) g = 0 and hence P n i =1 A 1 g i = 0 and A 2 g i = 0 for all i . Therefore (see [17 , Chapter 4]) 5 eac h g i can b e written as the sum g i = P k i j = 1 h i,j of some elemen ts h i,j ∈ G ( A 2 ) for all i, j . Let m := k 1 + · · · + k n and let h b e the v ector h := ( h 1 , 1 , . . . , h 1 ,k 1 , . . . , h n, 1 , . . . , h n,k n ) ∈ Z mt . Then P i,j A 1 h i,j = 0 and A 2 h i,j = 0 fo r all i, j a nd hence A ( m ) h = 0 . W e claim that mor eo v er, h ∈ G  A ( m )  . Supp ose indirectly this is not the case. Then there is an ¯ h ∈ G  A ( m )  with ¯ h ⊏ h . But then the ve cto r ¯ g ∈ Z nt defined b y ¯ g i := P k i j = 1 ¯ h i,j for all i satisfies ¯ g ⊏ ¯ g con t r adicting g ∈ G  A ( n )  . This pro v es the claim. Therefore, b y Definition 2.2 of Gra v er complexit y , the num b er of no nzero bric ks h i,j of h is at most g ( A ). So for ev ery I ⊆ { 1 , . . . , n } , we ha v e that P i ∈ I g i = P i ∈ I P k i j = 1 h i,j is a sum of at most g ( A ) nonzero elemen ts h i,j ∈ G ( A 2 ) and hence P i ∈ I g i ∈ Z ( A ).  Example 3.2 Let A := A (3) b e the (9 , 6) × 9 bimat r ix men tioned in the in tro duc- tion, whic h arises in the univ ersalit y of n -fold in teger programming discussed further in Section 7 , ha ving first blo c k A 1 = I 9 the 9 × 9 iden tity matrix and second blo c k the following 6 × 9 incidence matrix of the complete bipartite graph K 3 , 3 , A 2 =         1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1         . Since A 2 is t o tally unimo dular, its Grav er basis G ( A 2 ) consists of the 30 v ectors in { 0 , ± 1 } 9 supp orted on circuits of K 3 , 3 with alternating ± 1, see [17]. Also, it is know n that the Grav er complexit y of this bimatrix is g ( A ) = 9, see [2, 18]. Therefore, the set Z ( A ) in (4) whic h correspo nds to A , consists of all sums of a t most 9 suc h circuit v ectors, and turns out to b e comprised of 429 31 v ectors in Z 9 , suc h as  9 − 2 − 7 − 4 5 − 1 − 5 − 3 8  . W e no w define a dynamic prog ram, that is, a w eighted digraph, whic h will enable to find a G ra ve r-b est step γ g for a feasible p oint x of (1) or detect that no ne exists. Definition 3.3 (t he dynamic program) Let A b e a fixed ( r, s ) × t bimatrix and let g ( A ) b e its G r av er complexit y . Giv en n , w , b , l , u , feasible p o int x in (1), and p ositiv e integer γ , define a w eigh t ed digraph a s fo llows. Its v ertices a re partitioned in to n + 1 stages defined in terms of the fixed finite set Z ( A ) ⊂ Z t in (4), b y S 0 := { 0 } , S 1 := S 2 := · · · := S n − 1 := Z ( A ) , S n := { z ∈ Z ( A ) : A 1 z = 0 } . Denote the v ertices of S i b y h i ∈ Z t . In tro duce an arc ( h i − 1 , h i ) from h i − 1 ∈ S i − 1 to h i ∈ S i if g i := h i − h i − 1 ∈ Z ( A ) and l i ≤ x i + γ g i ≤ u i , and giv e it w eigh t w i g i . 6 T o eac h dipath h = ( h 0 , h 1 , . . . , h n ) from S 0 to S n in t his digraph we asso ciate a v ector g ( h ) := ( h 1 − h 0 , . . . , h n − h n − 1 ) ∈ Z nt . Not e that 0 ∈ Z ( A ) and hence the trivial path h = ( 0 , . . . , 0 ) with weigh t 0 and ve ctor g ( h ) = ( 0 , . . . , 0 ) alw a ys exists. Note also that wg ( h ) = P n i =1 w i ( h i − h i − 1 ) is precisely the we ig h t of the dipath h . The followin g lemma relates this dynamic program to Grav er augmentations. Lemma 3.4 A fe a sible step γ g for x wh ich satisfies w ( x + γ g ) ≤ w ( x + γ ¯ g ) for any fe asible step γ ¯ g w ith ¯ g ∈ G  A ( n )  c an b e c onstructe d in line ar time O ( n ) . Pro of. Let h b e a minim um we igh t dipath fro m S 0 to S n and let g := g ( h ) b e the v ector asso ciated with h . W e claim t hat γ g is the desired feasible step for x . W e b egin with the complexit y statemen t. Since A is fixed, so is g ( A ), and hence so is eac h S i . As the digra ph is acyclic, the minim um w eight dipath from S 0 to h i ∈ S i decomp oses in to a minim um w eigh t dipath from S 0 to some h i − 1 ∈ S i − 1 plus the arc from h i − 1 to h i . Th us, we ha ve to c heck at most a constant n umber | S i − 1 | · | S i | of suc h pairs ( h i − 1 , h i ) to find the minim um weigh t dipaths from S 0 to ev ery h i ∈ S i giv en the minim um w eigh t dipaths from S 0 to ev ery h i − 1 ∈ S i − 1 . Rep eating this for eac h of the sets S 1 , . . . , S n one after the other tak es O ( n ) time. W e next sho w that γ g is a go o d feasible step. Since g i = h i − h i − 1 and ( h i − 1 , h i ) is an arc, w e hav e l i ≤ x i + γ g i ≤ u i , and g i ∈ Z ( A ) and hence A 2 g i = 0 , for all i . Also, P n i =1 g i = h n ∈ S n and hence P n i =1 A 1 g i = A 1 h n = 0 . So x + γ g is feasible in ( 1). Moreov er, wg = P n i =1 w i g i is the w eigh t of the minim um w eigh t dipath h . No w consider an y feasible step γ ¯ g for x with ¯ g ∈ G  A ( n )  . D efine ¯ h i := P j ≤ i ¯ g j for all i . Then ¯ h i ∈ Z ( A ) for all i by Lemma 3.1. Moreov er, A 1 ¯ h n = A 1 P n j = 1 ¯ g j = 0 . Therefore ¯ h i ∈ S i for a ll i . F urthermore, ¯ h i − ¯ h i − 1 = ¯ g i ∈ Z ( A ) and l i ≤ x i + γ ¯ g i ≤ u i and therefore ( ¯ h i − 1 , ¯ h i ) is a n arc of w eight w i ¯ g i for all i . So h = ( h 0 , h 1 , . . . , h n ) is a dipath f r om S 0 to S n with w eigh t w ¯ g = P n i =1 w i ¯ g i and a ssociat ed vector g ( ¯ h ) = ¯ g . Since h is a minimum w eight dipath, w g ≤ w ¯ g and so w ( x + γ g ) ≤ w ( x + γ ¯ g ) .  Remark 3.5 (optimalit y certification in linear time) As noted in Section 2 , a feasible p oin t x in in teger program (1 ) is optima l if and only if there is no G ra ve r augmen ting step g ∈ G  A ( n )  for x . Th us, with γ := 1 , Lemma 3.4 implies that the optimalit y of a feasible p o in t x in ( 1 ) can b e determined in linear time O ( n ). The next lemma sho ws that w e can quick ly find a Grav er-b est augmentation. Lemma 3.6 A fe asible step γ g for x satisfying w ( x + γ g ) ≤ w ( x + ¯ γ ¯ g ) f o r a n y fe asible s tep ¯ γ ¯ g with ¯ γ ∈ Z + and ¯ g ∈ G  A ( n )  c an b e found in quadr atic time O ( n 2 ) . Pro of. If Z ( A ) = { 0 } then G  A ( n )  = ∅ by Lemma 3.1, so γ g := 0 will do. Otherwise, construct a set Γ of O ( n ) p ositiv e in tegers in O ( n ) time as f o llo ws: for 7 ev ery i = 1 , . . . , n a nd ev ery z ∈ Z ( A ) \ { 0 } determine the largest p ositiv e in teger γ suc h that l i ≤ x i + γ z ≤ u i and include it in Γ. No w, fo r each γ ∈ Γ , construct and solv e the corresp onding dynamic program, resulting in total of O ( n 2 ) time b y Lemma 3.4. Let γ g b e that feasible step for x whic h att ains minim um v alue w γ g among the b est steps obtained from all these dynamic pro grams, and let ¯ γ ¯ g b e that feasible step f or x whic h at t a ins minimum v alue w ¯ γ ¯ g a mong ¯ γ ¯ g with ¯ g ∈ G  A ( n )  if any . Assume that w ¯ γ ¯ g < 0 as otherwise w e are done since w γ g ≤ w γ 0 = 0. Then ¯ γ is the largest p ositiv e in teger suc h t ha t l ≤ x + ¯ γ ¯ g ≤ u since otherwise the step ( ¯ γ + 1) ¯ g will b e feasible and b etter. So for some i = 1 , . . . , n , it m ust b e that ¯ γ is the largest p ositiv e integer suc h that l i ≤ x i + ¯ γ ¯ g i ≤ u i . Since ¯ g ∈ G  A ( n )  , it follo ws from Lemma 3 .1 that ¯ g i ∈ Z ( A ). Therefore ¯ γ ∈ Γ. Now let ¯ γ ˆ g b e the b est step attained from the dynamic program o f ¯ γ . Then w γ g ≤ w ¯ γ ˆ g b y c hoice of γ g and w ¯ γ ˆ g ≤ w ¯ γ ¯ g by Lemma 3.4. Therefore w ( x + γ g ) ≤ w ( x + ¯ γ ¯ g ) as claimed.  W e next show, followin g [10 ], that rep eatedly applying Gr a v er-b est augmen ting steps, w e can augmen t an init ia l feasible p oin t for ( 1 ) to an optimal one efficien tly . Lemma 3.7 F or any fixe d bima trix A ther e is an algorithm that, given n , w , b , l , u , and fe asible p oint x fo r (1), finds an optimal solution x ∗ for (1) in time O ( n 3 L ) . Pro of. Iterate the following: find b y the a lgorithm of Lemma 3.6 a Grav er- b est step γ g for x ; if it is augmen ting then set x := x + γ g a nd rep eat, else x ∗ := x is optimal. T o b ound the n umber of it erations, follow ing [10], note that while x is no t o p- timal, and x ∗ is some optimal solution, w e hav e that x ∗ − x = P k i =1 γ i g i is a nonnegativ e inte g er com bination of Gra ver basis elemen ts g i ∈ G  A ( n )  all lying in the same o rthan t, and hence eac h x + γ i g i is feasible in (1). Moreov er, b y the inte ger Carath ´ eo dory theorem of [3, 19], w e can a ssume that k ≤ 2( nt − 1). Letting γ i g i b e a summand attaining minim um w γ i g i , and letting γ g b e a Grav er-b est augmenting step for x obtained from the algorithm of Lemma 3.6, w e find that w ( x + γ g ) − w x ≤ w ( x + γ i g i ) − w x ≤ 1 2( nt − 1) ( wx ∗ − wx ) . So the G ra ver-best step provides an improv emen t whic h is a constant fraction of the b est p ossible impro ve men t, and this can b e sho wn t o lead to a b ound of O ( nL ) on the n um b er of it erat ions to optimalit y , see [10] for mo r e details. Since each iteratio n tak es O ( n 2 ) time by Lemma 3.6, the ov erall running time is O ( n 3 L ) as claimed.  W e next sho w how to find an initial feasible p oin t for (1) with the same com- plexit y . W e fo llo w the approach of [5] using a suitable auxiliary n -fold pro g ram. Lemma 3.8 F or any fixe d bima trix A ther e is an algorithm that, given n , b , l , u , either finds a fe asible p oint x for (1) or asserts that n one exists, in time O ( n 3 L ) . 8 Pro of. Construct an auxiliary n -fo ld in teger program min  ¯ wz : ¯ A ( n ) z = b , ¯ l ≤ z ≤ ¯ u , z ∈ Z n ( t +2 r +2 s )  (5) as follow s. Fir st, construct a new fixed ( r, s ) × ( t + 2 r + 2 s ) bimatrix ¯ A with ¯ A 1 :=  A 1 I r − I r 0 r × s 0 r × s  , ¯ A 2 :=  A 2 0 s × r 0 s × r I s − I s  . No w, the n ( t + 2 r + 2 s ) v ariables z hav e a natural partition into nt original v ariables x and n (2 r + 2 s ) new auxiliary v ariables y . Keep the origina l low er and upp er b ounds on t he original v ariables and introduce low er b ound 0 and upp er b ound k b k ∞ on eac h auxiliary v ariable. Let the new ob jectiv e ¯ w z b e the sum of auxiliary v ariables. Note that the binary length of the auxiliary prog ram satisfies ¯ L = O ( L ) a nd an initial feasible p oint ¯ z with ¯ x = 0 for (5) with the original b is easy to construct. No w apply the algor it hm of Lemma 3.7 a nd find in time O ( n 3 ¯ L ) = O ( n 3 L ) a n optimal solution z f or (5). If the optimal ob jective v a lue is 0 then y = 0 and x is feasible in the original program (1) whereas if it is p ositiv e then (1) is infeasible.  W e can now o btain the main result of this article. Theorem 3.9 F or every fixe d inte ger ( r, s ) × t bimatrix A , ther e is an algorithm that, give n n , ve ctors w , l , u ∈ Z nt and b ∈ Z r + ns having bi n ary enc o ding length L := h w , b , l , u , i , solv e s in time O ( n 3 L ) the n -fold inte ger pr o gr amming pr oblem min  wx : A ( n ) x = b , l ≤ x ≤ u , x ∈ Z nt  . Pro of. Use the algor it hm of Lemma 3.8 to either detect inf easibility or obtain a feasible p oin t and aug men t it by the algorithm of Lemma 3.7 to o ptimalit y .  4 Extensio n s to nonlinear ob jec t iv es Here w e extend some of our results to progra ms with nonlinear ob j ective functions, min  f ( x ) : A ( n ) x = b , l ≤ x ≤ u , x ∈ Z nt  . (6) A function f : R nt → R is sep ar able c onvex if f ( x ) = P n i =1 f i ( x i ) = P n i =1 P t j = 1 f i j ( x i j ) with eac h f i j univ ariat e con v ex. In [16 ] it was shown that Grav er bases provide o pti- malit y certificates for problem (6) with separable conv ex functions as w ell: a feasible p oin t x is o pt ima l if and only if there is no feasible Grav er step g for x whic h satisfies f ( x + g ) < f ( x ). This w as used in [10] to provide p olynomial time pro cedures for optimalit y certification and solution of problem (6 ) with separable conv ex functions f . How ev er, this inv olv ed ag a in c hec king eac h of the O ( n g ( A ) ) elemen ts of G  A ( n )  . Our results f r o m Section 3 can b e extended to provide linear time optimality certification f or separable con vex functions and a cubic time solution of ( 6) for separable con vex piecewise affine functions. W e discuss these resp ectiv ely next. 9 Optimalit y certification for separable c on v ex ob jectiv es Here w e assume t ha t the ob jectiv e function f is presen ted b y a c omp arison or acle that, queried on t wo vec t o rs x , y , asserts whether o r no t f ( x ) ≤ f ( y ). The time complexit y no w measures the nu m b er of arithmetic o p erations and oracle queries. Theorem 4.1 F or any fixe d bimatrix A , ther e is an algorithm that, g iven n , b , l , u , sep ar a b l e c onvex f pr esente d by c omp arison or acle, and fe asible p oint x i n p r o gr am (6), either asserts that x is optimal or finds an augme nting step g for x which satisfies f ( x + g ) ≤ f ( x + ¯ g ) for any fe asib l e step ¯ g ∈ G  A ( n )  , in line ar time O ( n ) . Pro of. G iv en t he feasible p oint x , set a dynamic prog ram similar to that in Defini- tion 3.3, with γ := 1, with the o nly mo dification that the we igh t of arc ( h i − 1 , h i ) fro m h i − 1 ∈ S i − 1 to h i ∈ S i is now defined to b e f i ( x i + g i ) − f i ( x i ) with g i := h i − h i − 1 . Then, for ev ery dipath h and its asso ciated v ector g := g ( h ), w e now ha ve f ( x + g ) − f ( x ) = n X i =1  f i ( x i + g i ) − f i ( x i )  = w eight of dipat h h . W e no w claim that the desired step is the vec tor g := g ( h ) asso ciated with a minim um we ig h t dipath h in this dynamic program. Indeed, an arg umen t similar to that in the pro of Lemma 3.4 no w implies that for any feasible step ¯ g ∈ G  A ( n )  w e ha ve f ( x + g ) − f ( x ) ≤ f ( x + ¯ g ) − f ( x ) and therefore f ( x + g ) ≤ f ( x + ¯ g ) .  Optimization of separable conv ex piecewise affine ob ject iv es In [10] it w as shown that pro blem (6) can b e solv ed for a n y separable conv ex function in p olynomial t ime, but with v ery large dep endency of O ( n g ( A ) ) of the running time on n , with the exp onen t g ( A ) dep ending on the bimatrix A . Here we restrict atten tion to separable con v ex ob jectiv e functions whic h are piecewise affine, for whic h w e are able to reduce the time dep endency o n n to O ( n 3 ) indep enden t o f A . So w e assume a g ain that f ( x ) = P n i =1 f i ( x i ) = P n i =1 P t j = 1 f i j ( x i j ) with eac h f i j : R → R univ ariate con v ex. Moreo ve r, w e now also assume that f or some fixed p , eac h f i j is p - piecewis e affine, that is, the interv al b et we en the lo w er b ound l i j and upp er b ound u i j is partitioned into at most p in terv als with inte ger end-p o ints, and the restriction of f i j to eac h in terv al k is an affine function w i j,k x i j + a i j,k with all w i j,k , a i j,k in tegers. W e denote b y h f i the binary length of f whic h is the sum of binary lengths of all interv al end-p oints and w i j,k , a i j,k needed to describe it. The binary length of the input for the nonlinear problem (6) is no w L := h f , b , l , u i . Theorem 4.2 F or any fixe d p and bimatrix A , ther e is an algorithm that, given n , b , l , u , a n d sep ar able c onvex p -pie c ewise affine f , solve s i n time O ( n 3 L ) the pr o gr am min  f ( x ) : A ( n ) x = b , l ≤ x ≤ u , x ∈ Z nt  . 10 Pro of. W e need to establish analogs of some of the lemmas o f Section 3 for suc h ob jectiv e functions. First, for t he analo g of Lemma 3.4, pr o ceed a s in the pro o f of Theorem 4.1 ab ov e: g iven a feasible p oin t x and p ositive integer γ , set again a dynamic program similar to that in Definition 3.3, with the we ig h t of arc ( h i − 1 , h i ) from h i − 1 ∈ S i − 1 to h i ∈ S i defined to b e f i ( x i + γ g i ) − f i ( x i ) with g i := h i − h i − 1 . A similar argumen t to that in the pro of of Lemma 3 .4 no w show s that γ g with g := g ( h ) the v ector asso ciated with the minim um w eight dipath h is a feasible step for x which satisfies f ( x + γ g ) ≤ f ( x + γ ¯ g ) for an y feasible step γ ¯ g with ¯ g ∈ G  A ( n )  . F o r the analog of Lemma 3.6, w e construct again a set Γ of critical step sizes γ as follows. First, as in the pro of of Lemma 3.6, w e collect the critical step sizes due to the lo wer and upp er b ound constrain ts by finding, for ev ery i = 1 , . . . , n and ev ery z ∈ Z ( A ) \ { 0 } , the largest p ositiv e in teger γ suc h that l i ≤ x i + γ z ≤ u i , and include it in Γ. How eve r, in con trast to Lemma 3.6 , we are now dealing with the mo r e general class of piecewise affine ob jectiv e functions f i j . So w e mus t add also the follow ing v alues γ to Γ: for eve r y i = 1 , . . . , n , ev ery z ∈ Z ( A ) \ { 0 } , and ev ery j = 1 , . . . , t , if x i j + γ z j and x i j + ( γ + 1) z j b elong to differen t affine pieces of f i j , then γ is included in Γ. Since the n um b er p of affine pieces is constan t, the n umber of suc h v alues for eac h i , z and j is also constan t. So the total n um b er of elemen ts of Γ remains linear and it can b e constructed in linear time O ( n ) again. No w w e con tinue as in the pro of of Lemma 3.6 : for eac h γ in Γ, using the analog of Lemma 3.4 established in t he first paragraph ab o v e, w e solv e the corresp onding dynamic pro gram in O ( n ) time, resulting in tot a l of O ( n 2 ) time a g ain. Let γ g b e that feasible step for x whic h attains minim um v alue f ( x + γ g ) among the b est steps obtained from all these dynamic programs, a nd let ¯ γ ¯ g b e that feasible step for x whic h attains minimum v alue f ( x + ¯ γ ¯ g ) among ¯ γ ¯ g with ¯ g ∈ G  A ( n )  if an y . It no w f ollo ws from t he construction o f Γ that if ¯ γ ¯ g is augmenting, namely , if f ( x + ¯ γ ¯ g ) − f ( x ) < 0, then ¯ γ ∈ Γ, as otherwise the step ( ¯ γ + 1) ¯ g will b e feasible and b etter. Let ¯ γ ˆ g b e t he b est step a ttained from the dynamic program of ¯ γ . Then f ( x + γ g ) ≤ f ( x + ¯ γ ˆ g ) ≤ f ( x + ¯ γ ¯ g ) where the first inequalit y follow s from the c hoice of γ g and the second inequality follo ws from the analog of Lemma 3.4. Therefore f ( x + γ g ) ≤ f ( x + ¯ γ ¯ g ). F o r the ana log of Lemma 3.7, we use the results of [1 0 ] incorp orating the optimal- it y criterion o f [16], whic h a ssure that the n umber of iterat io ns needed when using a Grav er-b est augmenting step at eac h iteration, is b o unded by O ( n h f i ) = O ( nL ), resulting aga in in ov erall time complexit y O ( n 3 L ) fo r augmen ting an initial f easible p oin t to an optimal solution of (6). Since an initial feasible p oin t if an y can b e fo und b y Lemma 3.8 as b efore in the same complexit y , the theorem now fo llo ws.  11 5 Some consequenc es Here w e briefly discuss t wo of the ma ny consequences of n -fold in teger programming whic h, no w with our new alg orithm, can b e solv ed drastically faster than b efore. Nonlinear mu lt icommo dit y transp ortation The m ulticommo dit y tra nspo rtation problem seeks minimu m cost ro uting of l com- mo dities fr om m suppliers t o n consumers sub ject to supply , consumption a nd ca- pacit y constrain ts. F or l = 1 this is t he classical transp ortation problem whic h is efficien tly solv able b y linear programming. But already for l = 2 it is NP-hard. Here w e consider the problem with fixed (but a rbitrary) n umber l of commo dities, fixed (but arbitrary) n um b er m of suppliers, and v ariable n um b er n of consumers. This is natural in ty pical applications where few facilities serv e man y customers. The data is as follo ws. Eac h supplier i has a supply v ector s i ∈ Z l + with s i k its supply in commo dit y k . Eac h consumer j has a consumption v ector c j ∈ Z l + with c j k its consumption in commo dit y k . The amoun t o f commo dity k to b e ro uted from supplier i to consumer j is an integer decision v ariable x j i,k . The total amoun t P l k =1 x j i,k of commo dities ro uted on the c ha nnel from i to j should not exceed the c hannel capacit y u i,j , and has cost f i,j  P l k =1 x j i,k  for suitable univ ariate functions f i,j . W e can handle standa r d linear costs as w ell as more realistic, con vex piecewise affine cost functions f i,j , whic h a ccoun t for c ha nnel congestion under hea vy routing. As a corollary of Theorem 4.2, for any fixed num b ers l of commo dities and m of suppliers, the problem can b e solv ed in time cubic in the num b er n of consumers. Corollary 5.1 F or every fixe d l c ommo dities, m suppliers, and p , ther e exists an algorithm that, g i v en n c ons ume rs , supplies and dema nds s i , c j ∈ Z l + , c ap acities u i,j ∈ Z + , and c on vex p -pie c ewise affine c osts f i,j : Z → Z , solv es in time O ( n 3 L ) , with L := h s i , c j , u i,j , f i,j i , the inte ger multic o mmo dity tr a n sp o rtation p r oble m min ( m X i =1 n X j = 1 f i,j l X k =1 x j i,k ! : x j i,k ∈ Z + , X j x j i,k = s i k , X i x j i,k = c j k , l X k =1 x j i,k ≤ u i,j ) . Pro of. In tro duce new v ariables y j i and equations y j i = P l k =1 x j i,k for all i, j . Then the ob jectiv e function b ecomes P i,j f i,j ( y j i ) whic h is separable conv ex p -piecewise affine in the new v ariables, and the capa city constraints b ecome y j i ≤ u i,j whic h a r e upp er b ounds o n the new v ariables. Use u i,j as a n upp er b ound on x j i,k and 0 as a trivial low er b ound on y j i for all i, j, k . As sho wn in [1 1], arranging the original and new v ariables in a tuple z = ( z 1 , . . . , z n ) of n brick s z j ∈ Z m ( l +1) defined by z j : = ( x j 1 , 1 , . . . , x j 1 ,l , y j 1 , x j 2 , 1 , . . . , x j 2 ,l , y j 2 , . . . . . . , x j m, 1 , . . . , x j m,l , y j m ) , 12 this problem can b e mo deled as an n -fold program, resulting in solution in la rge time O  n g ( A ) L  , with exp onent dep ending on l, m . Theorem 4 .2 no w enables solution in cubic time indep enden t of the n umbers l of commo dities and m of suppliers.  Priv acy in statistical databases A common practice in the disclosure of sensitiv e data con t ained in a m ultiw ay table is t o release some of the table margins rather than the en tries of the table. Once the margins are released, the securit y of a n y sp ecific en try of the table is related to the set of p ossible v alues that can o ccur in that en try in all ta bles having the same margins as those of the source table in the database, see [7, 20] and the references therein. In particular, if this set is small or consists of a unique v alue, tha t of the source table, then this entry can b e exp osed. Thus , it is desirable to compute the minim um and maxim um inte g er v alues that can o ccur in an en try , which in particular are equal if and only if the en try v alue is unique, b efore ma r gin disclosure is enabled. Consider ( d + 1)-wa y tables of format m 0 × · · · × m d , that is, array s v = ( v i 0 ,...,i d ) indexed by 1 ≤ i j ≤ m j for all j , with all en tries v i 0 ,...,i d nonnegativ e in t egers. Our results hold for arbitra ry hierarc hical marg ins, but for simplicit y w e restrict atten tion to disclosure of d -marg ins, that is, the d + 1 man y d -w ay tables ( v i 0 ,...,i j − 1 , ∗ ,i j +1 ,...,i d ) obtained fro m v by collapsing one factor 0 ≤ j ≤ d at a time, with en tr ies give n b y v i 0 ,...,i j − 1 , ∗ ,i j +1 ,...,i d := m j X i j =1 v i 0 ,...,i j − 1 ,i j ,i j +1 ,...,i d , 1 ≤ i k ≤ m k , 0 ≤ k ≤ d , k 6 = j . The problem is then to compute the minim um and maximum inte ger v alues tha t can o ccur in an en try sub ject to the margins of the source ta ble in t he database. This problem is NP-hard already for 3 - w ay tables o f format n × m × 3, see [6]. Ho we ver, as a corollary of Theorem 3 .9, if only one side n of the table is v aria ble, the problem can b e solv ed in cubic time regardless of the other sides m i as follows. Corollary 5.2 F or every fixe d d, m 1 , . . . , m d , ther e is an alg o ri thm that, give n n , inte g er d -mar gins ( v ∗ ,i 1 ,...,i d ) , . . . , ( v i 0 ,...,i d − 1 , ∗ ) , and index ( k 0 , . . . , k d ) , determines, in time O ( n 3 L ) , with L the binary l e n gth of the given mar gins, the minimum and maximum values of entry x k 0 ,...,k d among al l tables with these mar gins, that is, solves min / max  x k 0 ,...,k d : x ∈ Z n × m 1 ×···× m d + , ( x i 0 ,...,i j − 1 , ∗ ,i j +1 ,...,i d ) = ( v i 0 ,...,i j − 1 , ∗ ,i j +1 ,...,i d ) ∀ j  . Pro of. Let u b e the maxim um v alue of any en try in the given marg ins and use it as an upp er b ound o n ev ery v ariable. As sho wn in [5], this problem can b e mo deled as an n -f old in t eger programming problem, resulting in solution in large running time O  n g ( A ) L  , with exp onen t whic h depends on m 1 , . . . , m d . Theorem 3.9 no w 13 enables to solv e it in cubic time indep enden t of the table dimensions m 1 , . . . , m d .  W e note t ha t long tables, with one side m uc h larger tha n the others, often arise in practical applications. F or instance, in health statistical tables, the long factor ma y b e the age o f an individual, whereas other factors may b e binary ( y es-no) or ternary (subnormal, normal, and supnormal). Moreo ver, it is alw ays p ossible to merge catego r ies o f f a ctors, with t he resulting coarser t ables appro ximating the original ones, making the algorithm of Corollary 5.2 applicable. W e also note that, b y rep eatedly incremen ting a low er b ound and decremen ting an upp er b ound on the en try x k 0 ,...,k d , and computing its new minim um and maxi- m um v alues sub ject to these b ounds, w e can pro duce the entire set of v alues that can o ccur in that entry in time prop ortio nal to the n umber of such v a lues. 6 Solv abili ty o v er b i matrices with v ariabl e entries The drop o f the Grav er complexit y from the exp onen t of n to the constan t multiple also leads t o the first p olynomial time solution of n -fold in t eger programming with v aria ble bimatrices. Of course, by the univ ersality of n -fold in t eger prog ramming, the v aria bilit y of the bimatrices m ust b e limited. In what follows , w e fix the dimensions r , s, t of the input bimatrix A , and let the en tries v ary . W e sho w tha t, giv en as par t of the input an upp er b o und a on the absolute v alue of ev ery en try of A , w e can solv e the problem in time p olynomial in a , that is, p olynomial in the unary length of a . This holds for linear as w ell as separable con v ex piecewise affine o b jectiv es. W e ha ve the following theorem, with L := h f , a, b , l , u i t he length o f the input. Theorem 6.1 F or any fixe d r, s, t and p , ther e is an algorithm that, giv e n n , a , ( r , s ) × t bimatrix A with al l en tries b ounde d by a in absolute v alue, b , l , u , and sep ar a b l e c onvex p -p i e c ewise affine f , i n p olynomial time O ( a 3 t ( rs + st + r + s ) n 3 L ) , solve s min  f ( x ) : A ( n ) x = b , l ≤ x ≤ u , x ∈ Z nt  . Pro of. Let G ( A 2 ) b e the Grav er basis of the s × t second blo ck A 2 of A , let p := |G ( A 2 ) | b e its cardinalit y , and arrange its elemen ts as the columns o f a t × p matrix G 2 . Since r , s, t are fixed, it follows from b ounds on Grav er bases (see e.g. [17, Section 3.4]) that ev ery g ∈ G ( A 2 ) satisfies k g k ∞ = O ( a s ) and hence p = O ( a st ). No w, it is kno wn (see [13 , 18] or [17, Section 4.1]) that the Grav er complexit y g ( A ) of A is equal to the maximum v alue k v k 1 of any elemen t v in t he Grav er basis G ( A 1 G 2 ) of the r × p matrix A 1 G 2 . Since the en tries of A 1 G 2 are b ounded in absolute v alue b y O ( a s +1 ), the b ounds on Grav er bases (see again [1 7 , Section 3.4]) imply that k v k 1 = O ( p · ( a s +1 ) r ) for ev ery v ∈ G ( A 1 G 2 ) and hence g ( A ) = O ( a r s + st + r ). 14 No w, consider ag ain the fo llowing set defined in (4 ) in Lemma 3.1, Z ( A ) :=  z ∈ Z t : z is the sum of at most g ( A ) elemen ts of G ( A 2 )  . F o r eac h z ∈ Z ( A ) we hav e that k z k ∞ is b ounded by g ( A ) times t he maxim um v alue of k g k ∞ o ve r all g ∈ G ( A 2 ), and therefore k z k ∞ = O ( g ( A ) · a s ) = O ( a r s + st + r + s ). So the cardinality o f Z ( A ) satisfies | Z ( A ) | = O (( a r s + st + r + s ) t ) = O ( a t ( rs + s t + r + s ) ). No w, suitable analogs of some o f the lemmas of Section 3 go through, except tha t the complexities now dep end on the v ariable size of the set Z ( A ), as f o llo ws. The time complexit y o f t he algorithm of Lemma 3.4 b ecomes O ( | Z ( A ) | 2 n ). The size of the set Γ o f critical v alues is no w O ( | Z ( A ) | n ) and therefore the t ime complexit y of the algorithm of Lemma 3.6 no w b ecomes O ( | Z ( A ) | 3 n 2 ). The n umber of iterations needed to augmen t a n initial feasible p oin t to an optimal solution remains O ( nL ) as b efo re and therefore the t ime complexit y of the algo rithm of Lemma 3.7 now b ecome O ( | Z ( A ) | 3 n 3 L ). T o find an initial f easible p oint, one can use the algorithm of Lemma 3.8, but then t would ha ve to b e replaced b y t + 2 r + 2 s for the auxiliary bimatrix ¯ A , r esulting in a somewhat larger expo nent fo r a in the running time. Ho we ver, it is p ossible t o find an initial feasible p o in t in an alt ernativ e, somewhat more inv olved w ay , k eeping the origina l system with the bimatrix A , as follo ws. First find a n inte ger solution to the system of equations o nly (without t he lo wer and upp er b ounds) using the Hermite no rmal fo rms of the blo c ks A 1 and A 2 . Second, relax the b o unds so as to make that p oint feasible. Third, minimize the follow ing auxiliary ob jective function whic h is separable conv ex 3-piecewise affine, with f i j ( x i j ) :=    l i j − x i j , if x i j ≤ l i j , 0 , if l i j ≤ x i j ≤ u i j , x i j − u i j , if x i j ≥ u i j . If the optimal v alue is zero then the optimal auxiliary solution is feasible in the original problem, whereas if it is p ositiv e then the o riginal problem is infeasible. Since this minimization can b e done using the separable conv ex piecew ise affine analog o f Lemma 3 .7 describ ed in the pro of of Theorem 4.2 in the same complexit y O ( | Z ( A ) | 3 n 3 L ), the o verall running time is O ( a 3 t ( rs + st + r + s ) n 3 L ) as claimed.  7 P arametrization and appro ximation hierarc h y W e conclude with a short discussion o f the univ ersalit y of n -fold integer programming and the resulting parametrization and simple approximation hierarc hy for all of in teger prog ramming. As men tioned in the introduction, ev ery in teger progra m is an n -fold pro g ram f or some m o v er the bimatr ix A ( m ) having first blo c k t he iden tity matrix I 3 m and second blo c k t he (3 + m ) × 3 m incidence matrix of K 3 ,m . It is con v enien t and illuminating to in tro duce also the f o llo wing description. 15 Consider the follow ing sp ecial form of the n -fo ld pro duct o p erator. F or an s × t matrix D , let D [ n ] := A ( n ) where A is the ( t, s ) × t bimatrix A with first blo c k A 1 := I t the t × t iden tit y matrix and second blo c k A 2 := D . W e consider suc h m -fold pro ducts of the 1 × 3 matrix (1 1 1). Note that (1 1 1) [ m ] is precisely the (3 + m ) × 3 m incidence matrix of the complete bipartite g raph K 3 ,m . F or instance, (1 1 1) [2] =       1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1       . The followin g theorem was established in [6]. The Univ ersality Theorem [6] Eve ry (b ounde d) inte ger pr o gr amming pr oblem min { cy : y ∈ Z k + , V y = v } is p olynomial time e quivalent to some in te ger pr o gr am min  wx : x ∈ Z 3 mn + , (1 1 1 ) [ m ][ n ] x = b  ∼ = min  wx : x ∈ Z 3 mn + , A ( m ) ( n ) x = b  . This theorem pro vides a new, v ariable dimension, parametrization of in teger programming: for eac h fixed v alue of the parameter m , the r esulting pro grams ab ov e with n v ariable liv e in v ariable dimension 3 mn and include natural mo dels suc h as those describ ed in Section 5, and can b e solved in cubic time O ( n 3 L ) by Theorem 3.9; and when the pa r a meter m v aries, eve ry in t eger program app ears for some m . Our new algorithm suggests a natural simple a pproximation hierarc h y for integer programming, parameterized by degree d , a s f ollo ws. Fix an y d . Then g iv en any m , let A := A ( m ), so t = 3 m , A 1 = I 3 m , and A 2 is the incidence ma t rix of K 3 ,m . Define the approximation at degree d of the set Z ( A ) in equation (4) in Lemma 3.1 b y Z d ( m ) :=  z ∈ Z 3 m : z is the sum of at most d elemen ts of G ( A 2 )  . (7) Since A 2 is totally unimo dular, G ( A 2 ) consists of the O ( m 3 ) ve cto r s in { 0 , ± 1 } 3 m supp orted on circuits of K 3 ,m with alternating ± 1 and hence | Z d ( m ) | = O ( m 3 d ). No w, g iv en a feasible p oin t x in the univ ersal program ab o ve, and p o sitive in teger γ , set a dynamic prog ram similar t o that in D efinition 3.3, with the only mo dification that the sets S i are defined using the approximation Z d ( m ) of Z ( A ). W eak er forms of Lemmas 3.4 a nd 3.6 now assert that in time O ( | Z d ( m ) | 3 n 2 ) = O ( m 9 d n 2 ), whic h is p olynomial in b oth m and n , we can find a go o d feasible step γ g . W e use this iterativ ely to a ug men t an initial feasible p oint to one whic h is a s go o d as p ossible and output it. How ev er, Lemma 3.1 no longer holds and not all bric k sums of elemen ts of the G ra ve r basis G ( A ( n ) ) lie in Z d ( m ). So the b ounds on the num b er of iterations and total running time are no longer v alid and the output p oint ma y b e non optimal. 16 By increasing the degree d w e can get b etter approx imat io ns at increasing run- ning times, and when d = g ( A ) w e get the t rue optimal solutio n. These appro xima- tions are curren tly under study , implemen tation and testing. They sho w promising b eha vior already a t degree d = 3 and will b e discuss ed in mor e detail elsewhe re. F o r m = 3, discussed in Example 3.2, for whic h the univ ersal problem is equiv a- len t to opt imizatio n ov er 3-w ay n × 3 × 3 ta bles, the approxim ation Z 3 (3) at degree d = 3 con tains only 811 ve cto r s out of t he 42931 ve ctors in the true Z ( A ), suc h as  − 3 2 1 2 − 3 1 1 1 − 2  . References [1] Aoki, S., T akem ura, A.: Minimal basis for connected Marko v c hain ov er 3 × 3 × K con tingency tables with fixed tw o-dimensional ma r ginals. Austr. New Zeal. J. 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