An approximation notion between P and FPTAS

An app ro ximation notion b et w een P and FPT AS Sam uel Bism uth #  Departmen t of Computer Science, Ariel Univ ersit y , Ariel 40700, Israel Erel Segal-Halevi #  Departmen t of Computer Science, Ariel Universit y , Ariel 40700, Israel Abstract W e presen t an approximation notion for NP-hard optimization problems represen ted by biv ariate functions. W e prov e that (assuming P  = N P ) the new notion is strictly stronger than FPT AS, but strictly w eak er than having a p olynomial-time algorithm. 2012 ACM Subject Classification Mathematics of computing → Combinatorial algorithms Keyw o rds and phrases FPT AS , algorithm, complexity , combinatorial problems, approximation F unding Samuel Bismuth : Israel Science F oundation grant no. 712/20. Er el Se gal-Halevi : Israel Science F oundation grant no. 712/20, 1092/24. 1 Intro duction When an optimization problem is found to b e NP -hard, w e assume that it cannot be solv ed exactly by a p olynomial-time algorithm, and lo ok for p olynomial-time approximation algorithms. The most efficient kind of an approximation algorithm currently known is the FPT AS (F ully Polynomial Time Appro ximation Scheme): for an y ϵ > 0 , it finds a solution that is at least (1 − ϵ ) times the optimal solution (in case of a maximization problem), and runs in time p olynomial in the input size and 1 /ϵ . But is FPT AS really the second-b est alternativ e to a p olytime algorithm? Is there a kind of approximation algorithm for NP-hard problems, that is more efficient than an FPT AS? This note describ es a new (to the b est of our kno wledge) kind of approximation algorithm. Instead of approximating the optimal solution v alue, it approximates the p erfe ct solution v alue, that is, the optimal solution of the fractional relaxation of the original maximization problem. F or any t > 0 , it chec ks whether there exists a solution with v alue at least (1 − t ) times the perfect solution v alue, and runs in time polynomial in the input size and 1 /t . W e call suc h an algorithm FFPT AS (F ractional F ully P olynomial Time Approximation Sc heme). W e prov e that FFPT AS is a strictly better approximation than FPT AS, that is: every problem that has an FFPT AS (under certain conditions) has an FPT AS, but the opp osite is not true unless P=NP . W e complement this result by showing an FFPT AS for an NP-hard problem. T ogether, our results sho w that the notion of FFPT AS lies strictly b et w een p olynomial-time solv ability and FPT AS. 1.1 Motivation Our in terest in FFPT AS comes from problems in which a p erfect solution is required. As an example, consider the problem of dividing items of differen t v alues b etw een t wo partners. It ma y b e required by la w to give each partner exactly 1 / 2 of the total v alue. Ho wev er, suc h a p erfect partition may b e imp ossible to attain if the items cannot b e split. A p ossible solution is to hav e the partner who received the higher v alue comp ensate the other partner by monetary paymen ts; but the amoun t of monetary paymen ts a v ailable might also b e b ounded. If there exists a partition in which the smallest sum is at least (1 − t ) · P E R , where P E R is the p erfect partition v alue, then it is p ossible to balance the partition b y a paymen t of size at most t · P E R . 2 An app roximation notion betw een P and FPT AS An FFPT AS may also b e useful to find a partition in which only a b ounded num b er of items is split, as we hav e shown in another pap er [ 1 ]. In that pap er, we discov ered a strong relationship b et ween problems with splitting and approximation algorithms. One may view our results as introducing an alternative kind of approximation: instead of keeping the items discrete and allowing a b ounded deviation in the ob jective v alue, we keep the ob jective v alue optimal and allow a b ounded n um b er of items to b ecome fractional. This approximation notion may b e meaningful in other optimization problems. Our results might also be interesting from the persp ective of complexit y theory . Currently , it is known that P ⊊ FPT AS (assuming P  = N P ), but it is not kno wn whether an y complexity class lies strictly b etw een these tw o. This question remains op en; see Section 4 for a discussion of why our results do not solve it. 2 Definitions W e state all our definitions and results in terms of maximization problems, but analogous results hold for minimization problems; we omit the details. ▶ Definition 1. A maximization problem is a p air ( f , g ) , wher e f is a pr e dic ate and g is a r ational-value d function; b oth f and g ac c ept a p air of ve ctors x , y with r ational entries. W e call f the fe asibility pr e dic ate and g the obje ctive function . ▶ Definition 2. F or a maximization pr oblem ( f , g ) and for any r ational ve ctor x , we denote: O P T ( x ) := the optimal value of ( f , g ) for input x , that is, the maximum value of the obje ctive function g ( x , y ) among al l in teger ve ctors y satisfying f ( x , y ) . 1 P E R ( x ) := the maximum value of g ( x , y ) among al l rational ve ctors y satisfying f ( x , y ) . Ob viously , for any maximization problem ( f , g ) , P E R ( x ) ≥ O P T ( x ) for every v ector x . W e now define several classes of algorithms for maximization problems. W e denote by len( x ) the num b er of bits in the binary representation of the integer v ector x . ▶ Definition 3. L et P b e a maximization pr oblem, P = ( f , g ) . A p olynomial-time algorithm for P is an algorithm that ac c epts as input a ve ctor x and r eturns as output the value O P T ( x ) , and runs in time p olynomial in len( x ) . A polynomial-time relaxation algorithm for P is an algorithm that ac c epts as input a ve ctor x and r eturns as output the value P E R ( x ) , and runs in time p olynomial in len ( x ) . A n FPT AS for P , denote d FPT AS[ P ] , is an algorithm that, given input ve ctor x and r e al numb er ϵ > 0 , runs in time p olynomial in len ( x ) and 1 /ϵ , and r eturns a value v such that v ≥ (1 − ϵ ) · O P T ( x ) , and — v = g ( x , y ) for some inte ger ve ctor y that satisfies f ( x , y ) . A n FFPT AS for P , denote d FFPT AS[ P ] , is an algorithm that, given input ve ctor x and r e al numb er t > 0 , runs in time p olynomial in len( x ) and 1 /t , and r eturns one of: “Y es” if some inte ger ve ctor y satisfies f ( x , y ) and has g ( x , y ) ≥ (1 − t ) · P E R ( x ) ; “No” otherwise. ▶ Rema rk 4. The ab ov e classes of algorithms can be defined to also return an output vector y that attains the optimal / approximately-optimal v alue; it will not affect our results. 1 If no y satisfies f ( x , y ) then OP T ( x ) = −∞ ; if g ( x , y ) subject to f ( x , y ) is unbounded then OP T ( x ) = ∞ . S. Bismuth, and E. Segal-Halevi 3 W e call a problem ( f , g ) r elaxable if it has a p olynomial-time relaxation algorithm, that is, if P E R ( x ) can b e computed in time p olynomial in len( x ) . 2 3 Main theo rem ▶ Theorem 5. The fol lowing hold among the class of r elaxable maximization pr oblems: (1) If a r elaxable pr oblem has a p olynomial-time algorithm, then it also has an FFPT AS. (2) Some r elaxable pr oblem has an FFPT AS but no p olytime algorithm unless P = NP . (3) If a r elaxable pr oblem has an FFPT AS, then it also has an FPT AS. (4) Some r elaxable pr oblem has an FPT AS but has no FFPT AS unless P = NP . Theorem 5 sho ws that, at least among the class of relaxable problems, FFPT AS is a strictly stronger approximation notion than FPT AS, but still strictly weak er than P . 3.1 Pro of of part (1) ▷ Claim. If a relaxable problem has a p olytime algorithm, then it has an FFPT AS. Pro of. The claim is easily prov ed by Algorithm 1 b elow. Algo rithm 1 FFPT AS [ f , g ]( x , t ) 1: Compute O P T ( x ) and P E R ( x ) . 2: If O P T ( x ) ≥ (1 − t ) P E R ( x ) return “yes”, 3: Else , return “no” . ▷ O P T ( x ) < (1 − t ) P E R ( x ) . Since ( f , g ) is relaxable, P E R ( x ) can b e computed in p olynomial time. Since ( f , g ) is p olynomial-time solv able, O P T ( x ) can b e computed in p olynomial time to o. By definition, O P T ( x ) ≥ (1 − t ) P E R ( x ) , if and only if there is an integer v ector y that satisfy f ( x , y ) and has g ( x , y ) ≥ (1 − t ) P E R ( x ) . Therefore, Algorithm 1 answers correctly . The run time is p olynomial in the length of x and in the length of t , which is in O ( log (1 /t )) , so this is indeed an FFPT AS. ◀ ▶ Remark 6. Part (1) is true for any definition of PER that can b e computed in p olytime. 3.2 Pro of of part (2) ▷ Claim. Some relaxable problem has an FFPT AS but no p olytime algo. unless P = NP . Pro of. W e use the max-min v ariant of the P ar tition problem for n = 2 bins. The input is a set of some m items with p ositive integer sizes; the goal is to partition the list into tw o bins suc h that the sum of item sizes in the bin with the smallest sum is as large as p ossible. In our notation, it can b e presented as follows. The input is an integer vector x = ( x 1 , . . . , x m ) , where x i represen ts the size of item i . The output is a v ector y = ( y 0 , y 1 , . . . , y m ) , where y 1 , . . . , y m are binary v ariables represen ting whether item i is in bin #1, and y 0 is an in teger v ariable represen ting the smallest bin sum. 2 There are many problems for which computing OP T is NP-hard whereas computing P E R is in P. Fo r example, integer linear programming is NP-hard but rational linear programming is in P. In App endix A we show that the converse is also p ossible: there are optimization problems in which computing P E R is NP-hard whereas computing OP T is in P. 4 An app roximation notion betw een P and FPT AS The feasibility function f ( x , y ) returns T rue if the follo wing three conditions hold: y 0 ≤ m X i =1 x i y i ( y 0 is at most the sum of bin #1) y 0 ≤ m X i =1 x i (1 − y i ) ( y 0 is at most the sum of bin #2) 0 ≤ y i ≤ 1 ∀ i ∈ { 1 , . . . , m } The ob jective function g ( x , y ) equals y 0 : the ob jective is to maximize y 0 sub ject to the three constraints f ( x , y ) . In this problem, O P T ( x ) is the v alue of the optimal inte gr al allo cation, that is, the maximum of g ( x , y ) sub ject to f ( x , y ) with the additional constraint that y i ∈ Z for all i ∈ { 0 , . . . , m } . This represen ts the max-min allo cation when items cannot b e split. P E R ( x ) is the v alue of the optimal fr actional allo cation, that is, the max-min allo cation when items can b e split b et ween bins. The problem is relaxable, as P E R ( x ) = P m i =1 x i / 2 , whic h can b e computed in time p olynomial in len( x ) . W e denote the pair ( f , g ) defined ab ov e by P 2 . It is well kno wn that P 2 is NP-hard but has an FPT AS. W e now sho w that it also has an FFPT AS. W e will use an auxiliary problem — a v ariant of P 2 with a critic al c o or dinate , which w e denote b y P 2 ,cc . The input to P 2 ,cc is a vector x = ( x 1 , . . . , x m , z ) , where the x i are the item sizes and z is a rational num b er representing a low er b ound on the sum of bin #1. The output is y = ( y 1 , . . . , y m ) , where y i is a binary v ariable r epresen ting whether item i is in bin #1. The feasibilit y function f ( x , y ) is defined by the following inequalities: z ≤ m X i =1 x i y i (the sum of bin #1 is at least z ) 0 ≤ y i ≤ 1 ∀ i ∈ { 1 , . . . , m } The ob jectiv e function g ( x , y ) is the sum of bin #2: g ( x , y ) = P m i =1 x i (1 − y i ) . The general scheme of W o eginger [ 6 ] can b e used to construct an FPT AS for P 2 ,cc (in fact, P 2 ,cc is equiv alent to the Subset Sum problem: the ob jective is to choose a subset of items for bin #2 with a maximum sum, sub ject to that sum b eing at most ( P m i =1 x i ) − z ). Algorithm 2 b elow uses this FPT AS to design an FFPT AS for P 2 . Algo rithm 2 FFPT AS [ P 2 ]( x , t ) 1: Set z := (1 − t ) · P E R ( x ) . 2: R un FPT AS [ P 2 ,cc ] with input ( x 1 , . . . , x m , z ) and approximation accuracy ϵ = t/ 2 . 3: Denote the sum of bin #2 in the obtained solution by b 2 . 4: if b 2 ≥ z then return “yes” . 5: else return “no” . 6: end if The run-time of Algorithm 2 is dominated b y the run-time of FPT AS[ P 2 ,cc ], which is p olynomial in len ( x ) and 1 /ϵ = 2 /t . It remains to pro ve that Algorithm 2 is indeed an FFPT AS for P 2 . If the algorithm returns “yes”, then the partition of ( x 1 , . . . , x m ) returned by FPT AS[ P 2 ,cc ] has tw o bins with sum at least z = (1 − t ) P E R ( x ) , so the “yes” answer is correct. S. Bismuth, and E. Segal-Halevi 5 Con v ersely , supp ose there exists a partition of ( x 1 , . . . , x m ) in which the sum of b oth bins is at least z . By symmetry , we can assume that the bin #1 has the smaller sum, so the sum of bin #2 is at least half the sum of all items, that is, at least P E R ( x ) . That partition is a feasible solution to P 2 ,cc , as the sum of bin #1 is at least z . Its ob jectiv e v alue is at least P E R ( x ) . Therefore, FPT AS[ P 2 ,cc ] must output a solution with ob jectiv e v alue at least (1 − ϵ ) · P E R ( x ) > z . Therefore, in line 4 the algorithm returns “y es” correctly . Algorithm 2 sho ws that there exists an FFPT AS for P 2 . On the other hand, it is known that the max-min P ar tition problem (even with n = 2 bins) is NP -hard, so, there is no p olynomial time algorithm for P 2 unless P = NP . This concludes the pro of of P art (2). ◀ ▶ Rema rk 7. P art (2) is true for any definition of PER such that, for the P ar tition problem, P E R ( x ) ≤ P m i =1 x i / 2 . 3.3 Pro of of part (3) ▷ Claim. If a relaxable problem has an FFPT AS, then it also has an FPT AS. Pro of. Consider Algorithm 3 b elo w. Algo rithm 3 FPT AS [ f , g ]( x , ϵ ) 1: Compute P E R ( x ) . 2: L ← − 0 . 3: U ← − 1 . 4: while U − L ≥ ϵ do 5: t ← − ( L + U ) / 2 . 6: If FFPT AS [ f , g ]( x , t ) returns “y es”, then U ← − t . 7: Else , L ← − t . ▷ FFPT AS [ f , g ]( x , t ) return “no” . 8: end while 9: Return (1 − U ) P E R ( x ) . W e claim that the following tw o inequalities hold at each step of the algorithm: (1 − L ) P E R ( x ) ≥ O P T ( x ) ≥ (1 − U ) P E R ( x ) . Initially L = 0 and U = 1 , so (1 − L ) P E R ( x ) = P E R ( x ) and (1 − U ) P E R ( x ) = 0 . The inequalities hold since, by definition, P E R ( x ) ≥ O P T ( x ) ≥ 0 . W e now verify that the inequalities con tin ue to hold after each up date to U or L : In step 6 the FFPT AS returns “y es”, so there is a solution with v alue at least (1 − t ) P E R ( x ) , so O P T ( x ) ≥ (1 − t ) P E R ( x ) , and the inequalit y O P T ( x ) ≥ (1 − U ) P E R ( x ) still holds when U is set to t . In step 7, the FFPT AS returns “no”, so there is no solution with v alue at least (1 − t ) P E R ( x ) , so O P T ( x ) < (1 − t ) P E R ( x ) , and the inequality O P T ( x ) ≤ (1 − L ) P E R ( x ) still holds when L is set to t . 6 An app roximation notion betw een P and FPT AS The algorithm stops when U − L < ϵ and returns (1 − U ) P E R ( x ) , which satisfies (1 − U ) P E R ( x ) > (1 − ( L + ϵ )) P E R ( x ) since U < L + ϵ = (1 − L ) P E R ( x ) − ϵP E R ( x ) ≥ O P T ( x ) − ϵOP T ( x ) since (1 − L ) P E R ( x ) ≥ O P T ( x ) and P E R ( x ) ≥ O P T ( x ) = (1 − ϵ ) OP T ( x ) . Hence, Algorithm 3 returns a v alue larger than (1 − ϵ ) OP T ( x ) , as required. W e fo cus now on the running time. In step 1, since the maximization problem ( f , g ) is relaxable, P E R ( x ) can b e computed in p olynomial time in len ( x ) . Also, the algorithm stops at some iteration k when U − L < ϵ , so after k iterations, the interv al length b ecomes ϵ : 1 2 k = ϵ ⇐ ⇒ 2 k = 1 /ϵ ⇐ ⇒ k = log 2 (1 /ϵ ) . So, the n umber of iterations is log (1 /ϵ ) , and at eac h iteration, the algorithm runs FFPT AS [ f , g ]( x , t ) whic h runs in p olynomial time in x , and 1 /t . The smallest v alue for t is ϵ since U ≥ ϵ + L and L ≥ 0 . T ogether, the algorithm takes O (log(1 /ϵ ) · p oly (len( x ) , 1 /ϵ )) = O ( p oly (1 /ϵ ) · poly (len ( x ) , 1 /ϵ )) time to find a solution that is larger than (1 − ϵ ) OP T ( x ) . This is done in p olynomial time in the num b er of bits of the input vector x and in 1 /ϵ , as requested by the definition of FPT AS. ◀ ▶ Rema rk 8. This claim is true for any definition of PER that can b e computed in p olytime and satisfies P E R ( x ) ≥ O P T ( x ) for all inputs x . 3.4 Pro of of part (4) ▷ Claim. Some relaxable problem has an FPT AS but has no FFPT AS unless P = NP . Pro of. W e use the max-min v arian t of 4-w ay P ar tition , which w e denote b y P 4 . P 4 is defined similarly to P 2 from P art (2), except that there are more binary v ariables, representing the inclusion of each item in each p ossible bin. As P E R ( x ) = P m i =1 x i / 4 , the problem is relaxable. W oeginger [ 6 ] ga ve an FPT AS for P 4 . 3 W e now prov e that 4-wa y partition has no FFPT AS unless P = NP . The pro of is by reduction from the Equal-Cardinality P ar tition problem: given a list with 2 m in tegers and sum 2 S , decide if they can b e partitioned into tw o subsets with cardinalit y m and sum S . It is prov ed to b e NP -hard in [3]. Giv en an instance X 1 of Equal-Cardinality P ar tition , we can assume w.l.o.g. that all integers in X 1 are at most S , since otherwise the answer is necessarily “no” . W e construct an instance X 2 of the Equal-Cardinality P ar tition problem by replacing eac h integer x in X 1 b y 2 x + 4 S . So X 2 con tains 2 m in tegers b etw een 4 S and 6 S . Their sum is 2 · (2 S ) + 2 m · (4 S ) = 2(4 m + 2) S . Clearly , X 1 has an equal-sum equal-cardinality 3 More generally , he gav e an FPT AS for k -wa y partition for any fixed k . S. Bismuth, and E. Segal-Halevi 7 partition (with bin sums S ) if and only if X 2 has an equal-sum equal-cardinality partition (with bin sums (4 m + 2) S ). W e construct an instance X 4 for P 4 that con tains 2(2 m + 1) items: The 2 m items in X 2 , and 2( m + 1) additional smal l items with size 4 S . So the sum of all item sizes in X 4 is: 2(4 m + 2) · S + 2( m + 1) · 4 S = 4(4 m + 3) · S. Hence, P E R ( X 4 ) = (4 m + 3) S (the sum of each bin in an optimal fractional 4-wa y partition). ▷ Claim 9. X 4 can b e partitioned into four bins with sum at least (4 m + 2) S , if-and-only-if X 2 can b e partitioned into tw o bins with cardinality m and sum exactly (4 m + 2) S . Pro of. ⇐ = : Supp ose X 2 can b e partitioned into tw o bins of sum (4 m + 2) S . The 2( m + 1) small items can b e divided into tw o additional bins of m + 1 items each, with sum ( m + 1) · 4 S = (4 m + 4) S > (4 m + 2) S, (1) so in the resulting partition of X 4 , the sum of each of the four bins is at least (4 m + 2) S . = ⇒ : Supp ose X 4 can b e partitioned into four bins with sum at least (4 m + 2) S . Let us analyze the structure of this partition. – Since the sum of eac h bin is at least (4 m + 2) S , and the av erage sum of a bin is (4 m + 3) S , the sum of every tw o bins is most 4(4 m + 3) S − 2(4 m + 2) S = 2(4 m + 4) S . – The sum of the smallest 2( m + 1) items in X 4 is exactly 2(4 m + 4) S . Hence, ev ery tw o bins must contain together at most 2( m + 1) items. – Since the total num b er of items in X 4 is 2 m + 2( m + 1) , there must b e some tw o bins that con tain together exactly 2( m + 1) items. W.l.o.g. assume that these are bins #1 and #2. – The sum of bins #1 and #2 is at most 2(4 m + 4) S , they contain exactly 2( m + 1) items, and eac h item size is at least 4 S . Therefore, bins #1 and #2 m ust con tain only items of size exactly 4 S , and we can assume w.l.o.g. that these are the 2( m + 1) smal l items. – The other t wo bins, bin #3 and bin #4, con tain together the 2 m items of X 2 , their sum is 2(4 m + 2) , and the sum of each bin is at least (4 m + 2) S , so the sum of eac h bin m ust in fact b e exactly (4 m + 2) S . But the sum of ev ery m + 1 items is at least ( m + 1)4 S > (4 m + 2) S . Hence, bins #3 and #4 must contain exactly m items each. This concludes the pro of of the auxiliary claim. ◀ W e now return to the pro of of the theorem. W e apply F F P T AS [ P 4 ] on X 4 with parameter t := 1 / (4 m + 3) . Note that (1 − t ) · P E R ( X 4 ) = (1 − t ) · (4 m + 3) S = (4 m + 2) S. Hence, F F P T AS [ P 4 ] returns “y es” if and only if there exists a partition of X 4 in to four bins with sum at least (4 m + 2) S . By the auxiliary claim, this holds if and only if the Equal-Cardinality P ar tition instance ( X 2 ) is a “yes” instance. Hence, F F P T AS [ P 4 ] exists if and only if Equal-Cardinality P ar tition can be solved in p olynomial time, whic h is imp ossible unless P = NP . ◀ ▶ Rema rk 10. The auxiliary Claim 9 do es not dep end on PER. In principle, the theorem pro of can b e extended to other definitions of PER. Let S 2 := (4 m + 2) S 1 and P E R 3 := P E R ( X 3 ) . Note that O P T ( X 3 ) ≥ (4 m + 2) S 1 , so P E R 3 ≥ S 2 . W e need t := 1 − S 2 P E R 3 1 /t = P E R 3 P E R 3 − S 2 . If P E R 3 > S 2 and the difference is sufficiently large, 1 /t is p olynomial in m , so the same pro of holds. 8 An app roximation notion betw een P and FPT AS 4 Is FFPT AS a new complexit y class? Denote by P / FFPT AS / FPT AS the class of problems that hav e a p olynomial-time algorithm / an FFPT AS / an FPT AS resp ectiv ely . Theorem 5 shows that, among the relaxable problems, P ⊊ FFPT AS ⊊ FPT AS . But, w e cannot conclude that FFPT AS is a new complexity class. The reason is that our results dep end on the representation of an optimization problem using the pair of bivariate functions f and g , rather than the univariate function O P T . F or every combinatorial optimization problem O P T , there can b e many differen t representations as a pair ( f , g ) , and for each representation, the relaxation function P E R is p oten tially different. 4 It is p ossible that one representation has an FFPT AS, and another representation of the same problem (the same O P T ) do es not hav e an FFPT AS. W e hav e tried several w a ys to define the relaxation function P E R directly based on O P T , but so far we could not pro ve that all four parts of Theorem 5 hold for any suc h definition. Hence, the following problem remains op en. ▶ Op en Question 1. Is ther e a class C of optimization pr oblems that satisfies P ⊊ C ⊊ FPT AS (assuming P  = NP )? A cknowledgments W e are grateful to Neal Y oung, Andras F arago, Edward A. Hirsch and Y ossi Azar for insigh tful commen ts and discussions. References 1 Sam uel Bismuth, Vladisla v Makarov, Erel Segal-Halevi, and Dana Shapira. Partitioning problems with splittings and interv al targets. In 35th International Symp osium on Algorithms and Computation (ISAA C 2024) , pages 12–1. Schloss Dagstuhl–Leibniz-Zen trum für Informatik, 2024. 2 W F ernandez de La V ega and George S. Lueker. Bin packing can b e solved within 1+ ε in linear time. Combinatorica , 1(4):349–355, 1981. 3 M. R. Garey and David S. Johnson. Computers and Intr actability: A Guide to the The ory of NP-Completeness . W. H. F reeman, 1979. 4 Narendra Karmarkar and Richard M Karp. An efficient approximation scheme for the one- dimensional bin-packing problem. In 23r d A nnual Symp osium on F oundations of Computer Scienc e (sfcs 1982) , pages 312–320. IEEE, 1982. 5 Theo dore S Motzkin and Ernst G Straus. Maxima for graphs and a new pro of of a theorem of T urán. Canadian Journal of Mathematics , 17:533–540, 1965. 6 Gerhard J. W oeginger. When does a dynamic programming formulation guaran tee the existence of a fully p olynomial time approximation scheme (FPT AS)? INF ORMS J. Comput. , 12(1):57–74, 2000. 4 A famous example is bin-packing: its naive representation as a linear program has a v ariable for each (item,bin) pair, but it has a more sophisticated representation known as the configur ation pr o gram . The fractional relaxation of the configuration program is substantially different (and muc h more useful for approximation algorithms ) than that of the naive representation, as shown by [ 2 ] and [ 4 ] and the many follow-up works on that problem. F or more examples, see comments by Neal Y oung here: https://cstheory.stackexchange.com/q/52830 . S. Bismuth, and E. Segal-Halevi 9 A Not Every Polynomial-time Solvable Problem is Relaxable F or many combinatorial optimization problems, computing the integral solution O P T ( x ) is NP-hard whereas computing the fractional relaxation P E R ( x ) can b e done in p olytime. As a general example, integer linear programming is NP-hard whereas its fractional relaxation is linear programming, whic h is p olytime. This might give the impression that P E R is alw ays “at least as easy” as O P T . But this impression is false. ▶ Prop osition 11. Ther e is an optimization pr oblem ( f , g ) for which O P T c an b e c ompute d in p olynomial time wher e as P E R is NP-har d. T o prov e the prop osition we use the Motzkin-Straus theorem [5]. ▶ Theo rem 12 (Motzkin and Straus, 1965) . L et ∆ n b e the standar d n -vertex simplex, ∆ n := { y ∈ R n : ∀ i : y i ≥ 0 , P i y i = 1 } . Given an undir e cte d gr aph G on vertex set { 1 , . . . , n } , define — g ( G, y ) := X ( i,j ) ∈ G y i · y j for al l y ∈ ∆ n . L et ω ( G ) denote the lar gest numb er of vertic es in a clique in G (the clique num b er of G ). Then max y ∈ ∆ n g ( G, y ) = 1 2  1 − 1 ω ( G )  . Pro of of Prop osition 11. 5 W e define an optimization problem in which The input x represen ts an undirected graph G on n v ertices (e.g. it can be a binary v ector of size n 2 , representing the adjacency matrix of G ). The output y is a vector of length n . The feasibility function f ( x , y ) = f ( G, y ) ignores x and requires just that y is in ∆ n . The ob jectiv e function g ( x , y ) = g ( G, y ) is as defined in Theorem 12 ab ov e. In this optimization problem, O P T ( G ) is the maximum v alue of g ( G, y ) o v er the vertices of the simplex ∆ n . As there are only n such vertices, O P T ( G ) can b e computed in p olynomial time. P E R ( G ) is, by the Motzkin-Straus theorem, equal to 1 2  1 − 1 ω ( G )  . Equiv alently , ω ( G ) = 1 / (1 − 2 · P E R ( G )) . Computing ω ( G ) is a w ell-known NP-hard problem — the CLIQUE problem. Hence, computing P E R ( G ) is NP-hard to o. ◀ 5 The pro of was developed in collab oration with claude.ai. The complete conv ersation is av ailable here: https://claude.ai/share/5064c5af- 74d1- 4213- b72a- 1e41fa58b74e