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A 7=8-Approximation Algorithm for MAX 3SAT?

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A 7=8-Approximation Algorithm for MAX 3SAT? =8 A 7 -Approximation Algorithm for MAX 3SAT? y Howard Karloff Uri Zwick Abstract that the performance ratio of his algorithm is actually 2/3. Many years passed before Yannakakis [37] obtained a 3/4- We describe a randomized approximation algorithm which approximation algorithm for the problem. Goemans and takes an instance of MAX 3SAT as input. If the instanceÐa Williamson [17] then obtained a different and somewhat collection of clauses each of length at most threeÐis satis®- simpler 3/4-approximation algorithm. Their algorithm is able, then the expected weight of the assignment found is at based on a linear programming relaxation of the problem. least of optimal. We provide strong evidence (but not a In a major breakthrough, Goemans and Williamson [18] proof) that the algorithm performs equally well on arbitrary obtained a 0.878-approximation algorithm for MAX CUT MAX 3SAT instances. and MAX 2SAT, the version of MAX SAT in which each Our algorithm uses semide®nite programming and may be clause is of size at most two. Goemans and Williamson seen as a sequel to the MAX CUT algorithm of Goemans and used semide®nite relaxations of these problems. Feige and Williamson and the MAX 2SAT algorithm of Feige and Goe- Goemans [15] then obtained a 0.931-approximation algo- mans. Though the algorithm itself is fairly simple, its anal- rithm for MAX 2SAT. Using the MAX 2SAT algorithms of ysis is quite complicated as it involves the computation of Goemans and Williamson or of Feige and Goemans, slight volumes of spherical tetrahedra. improvements in the performance ratio for general MAX SAT can be made. Goemans and Williamson [18] obtained NP HÊastad has recently shown that, assuming P , no a 0.758 bound for MAX SAT. Asano [4] (following [5]) polynomial-time algorithm for MAX 3SAT can achieve a slightly improved this bound to 0.770. performance ratio exceeding , even when restricted to satis®able instances of the problem. Our algorithm is there- While semide®nite relaxations yield a huge improvement fore optimal in this sense. for MAX 2SAT (from 0.75 to 0.931), they give, so far, only a minor improvement for MAX SAT (from 0.75 to 0.770). We also describe a method of obtaining direct semide®nite The reason for this seems to be that the semide®nite relax- relaxations of any constraint satisfaction problem of the ations used till now do not directly handle clauses of length F F form MAX CSP , where is a ®nite family of Boolean three or more. functions. Our relaxations are the strongest possible within a natural class of semide®nite relaxations. An attempt to squeeze more from the MAX 2SAT algo- rithm of Feige and Goemans [15] was made by Trevisan, Sorkin, Sudan and Williamson [34]. They used an op- timal gadget, a concept formalized by Bellare, Goldre- 1 Introduction ich and Sudan [7], to reduce MAX 3SAT (the problem in which each clause has length at most three) to MAX MAX SAT is a central problem in theoretical computer sci- 2SAT, thereby obtaining a 0.801-approximation algorithm ence. As it is NP-hard and, in fact, MAX-SNP complete (Pa- for MAX 3SAT. Trevisan [33] recently obtained a 0.826- padimitriou and Yannakakis [30]), much attention has been approximation algorithm for satis®able instances of MAX devoted to approximating it. The ®rst approximation algo- 3SAT, and a 0.8-approximation algorithm for satis®able in- rithm for MAX SAT was proposed by Johnson [23]. John- stances of MAX SAT. son showed that the performance ratio of his algorithm is at In another major breakthrough, following a long line of re- least . Chen, Friesen and Zheng [12] recently showed search by many authors [16, 3, 2, 8, 7], HÊastad [20] recently College of Computing, Georgia Institute of Technology, Atlanta, Geor- showed that MAX E3SAT, the version of the MAX SAT gia 30332-0280, U.S.A. Research supported in part by NSF grant CCR- problem in which each clause is of length exactly three, can- 9319106. E-mail address: [email protected]. y not be approximated in polynomial time to within a ratio Department of Computer Science, Tel-Aviv University, Tel-Aviv NP 69978, Israel. This work was done while this author was visiting ICSI and greater than 7/8, unless P . HÊastad shows, in fact, UC Berkeley. E-mail address: [email protected]. that no polynomial-time algorithm can have a performance 1 guarantee of more than 7/8 even when restricted to just sat- is®able instances of the problem. HÊastad's result is easily X Maximize w z ij k seen to be tight: a random assignment satis®es, on average, ij k ijk to 7/8 of the total weight of a MAX E3SAT instance. sub ject v v v v In this paper we present a new approximation algorithm for 0 i j k z i j k n ij k MAX 3SAT. The algorithm takes as input an instance of v v v v 0 j i k z i j k n ij k MAX 3SAT and runs in time bounded by a polynomial in v v v v 0 i j the length of this instance. If the instance is satis®able, the k i j k n z ij k expected weight of the assignment the algorithm returns is at z i j k n ij k least of the weight of any assignment to the instance. We n v S i n conjecture (but have only strong evidence, not a proof) that i v v i n i i the same performance is achieved on arbitrary MAX 3SAT n instances. Our algorithm can possibly be derandomized us- ing the techniques of Mahajan and Ramesh [28]. The novelty of our algorithm is that it uses a direct semidef- Figure 1. A direct semide®nite relaxation of a inite relaxation of MAX 3SAT. The algorithm itself is quite MAX 3SAT instance. simple. The analysis relies, however, on two inequali- two or one (or zero). ties involving the volume function of spherical tetrahedra (Lemma 4.4 and Conjecture 4.5 below). Proving these two inequalities seems, at least for now, extremely complicated. 3 The New Approximation Algorithm for We provide a computer-assisted proof of the ®rst inequality. MAX 3SAT For the second inequality we are able to present only strong numerical evidence. We are currently working on a simpler A direct semide®nite relaxation of a generic MAX 3SAT in- proof of the ®rst inequality and a complete, and hopefully stance is presented in Figure 1. In this relaxation, we attach simple, proof of the second inequality. The ®rst inequality v i n a unit vector i to each Boolean variable, , and a alone implies that the performance ratio of the algorithm for z v scalar ij k to each clause. We also have a special vector satis®able instances of MAX 3SAT is at least 7/8. v v v n that corresponds to FALSE. The vectors are n n R We also describe a method of obtaining direct semide®- vectors on the Euclidean unit sphere S in . nite relaxations of any constraint satisfaction problem of the z The scalar ij k is meant to get the value 1 if the clause is F F F form MAX CSP or MIN CSP , where is a ®nite satis®ed and 0 if it is not satis®ed. De®ne family of Boolean functions. Such problems were studied relax v v v v i j k by Khanna, Sudan and Williamson [27] and by Khanna, Su- dan and Trevisan [26]. Our relaxations are the strongest pos- v v v v v v v v 0 i j 0 j i k k sible within a natural class of semide®nite relaxations. This min v v v v 0 i j class includes almost all the semide®nite relaxations pro- k posed to date for these problems. z The constraints on ij k in the relaxation are equivalent to the We hope that the results presented here pave the way for sim- z relax v v v v i j k constraint ij k . ilar improvements for MAX SAT. It is fairly easy to see that program presented in Figure 1 is equivalent to a semide®nite program. All we have to do is v x 2 MAX 3SAT v j ij replace each inner product i by a scalar , add the x x ij constraints ii , and require that the matrix be pos- v v ni i w n itive semide®nite. The constraint is equivalent An instance of MAX 3SAT in variables is an array ij k v v z i ni ij k i j k n of nonnegative weights, where . A valid to . (The 's can be assumed nonnega- x x x x tive but need not satisfy any semide®niteness constraints.) n assignment is a 0-1 vector such that n x f g x x n hi j k i x Why is it a relaxation of MAX 3SAT? Let ni i and , for . A clause is v x i i x x x x x x be a valid assignment. Let if j k i j satis®ed by iff i , i.e., if at least one of , v x v i i x and if . Let . or k is assigned the value 1. The weight of a valid assign- relax v v v v i j k ment is the sum of the weights of all the satis®ed clauses, It is easy to check (see Table 1) that P x x x i j k w x x x eight x w The program described in Figure 1 is therefore i j k i.e., ij k The op- ijk timal solution to the MAX 3SAT instance is an assignment a relaxation of MAX 3SAT.
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