H. Karloff and U. Zwick. A 7=8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pages 406--415, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....solves one of Lutz and Mayordomo s Twelve Problems on Resource Bounded Measure (1999) 1 Introduction MAX3SAT is a well studied optimization problem. Tight bounds on its polynomial time approximability are known: 1. There exists a polynomial time approximation algorithm (Karlo# and Zwick [5, 3]) 2. If P NP, then for all # 0, there does not exist a polynomial time ( #) approximation algorithm (Hastad [4] Recently there has been some investigation of approximating MAX3SAT in exponential time. For example, for any # (0, Dantsin, Gavrilovich, Hirsch, and Konev [2] give a ....

....measure provides strong, reasonable hypotheses which seem to have more explanatory power than weaker, traditional complexity theoretic # This research was supported in part by National Science Foundation Grant 9988483. An algorithm with conjectured performance ratio 8 was given in [5], and this conjecture has since been proved according to [3] hypotheses. The hypothesis that NP does not have p measure 0, p (NP) 0, implies P NP and is known to have many plausible consequences that are not known to follow from P NP. Resource bounded dimension was recently introduced by ....

H. Karlo# and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In 38th Annual Symposium on Foundations of Computer Science, pages 406--415, 1997.

....evi Throughout this paper we deal only with maximization problems and by an factor approximation we mean a solution whose value is at least times that of an optimum solution. Consequently, all factors of approximation we discuss will be less than 1. dence, the best proven bound is 0:87868 [11, 17], which is slightly better than the Goemans and Williamson approximation guarantee for Max Cut [5] For satis able instances of Max NAE E3 Sat, an approximation ratio of 0:91226 can be achieved in polynomial time [17] Turning again to set splitting problems, H astad [9] has shown the tight ....

....considered in the literature. Known results on approximating Max k NM Sat: Clearly, any algorithm that approximates Max k Sat within factor k also approximates Max k NM Sat within the same factor; in particular approximation factors of 2 = 0:931 and 3 = 7=8 can be achieved in this way [4, 11]. For Max Ek NM Sat, an approximation factor of 1 2 can be achieved trivially for all k 3, by simply picking a random truth assignment. There are no algorithms known which perform any better on non mixed clauses than on general satis ability instances. For k 4, a recent result of H astad ....

H. Karlo and U. Zwick. A (7=8 ")-approximation algorithm for MAX 3SAT? In Proceedings of the 38th FOCS, 1997.

....result for Max 3 Sat and an easy 2 gadget from Max 3 Sat to Max NAE E3 Sat [11, 15] On the algorithmic side, the best known approximation algorithm, due to Zwick [16] achieves a ratio of 0:908. This bound is as yet only based on numerical evidence, the best proven bound is 0:87868 [8, 15], which is slightly better than the Goemans and Williamson approximation guarantee for Max Cut [3] For satis able instances of Max NAEE3 Sat, an approximation ratio of 0:91226 can be achieved in polynomial time [15] Turning again to set splitting problems, H astad [5] has shown the tight ....

....considered in the literature. Known results on approximating Max k NM Sat: Clearly, any algorithm that approximates Max k Sat within factor k also approximates Max k NM Sat within the same factor; in particular approximation factors of 2 = 0:931 and 3 = 7=8 can be achieved in this way [2, 8]. For Max Ek NM Sat, an approximation factor of 1 2 can be achieved trivially for all k 3, by simply picking a random truth assignment. There are no algorithms known which perform any better on non mixed clauses than on general satis ability instances. For k 4, a recent result of H astad ....

H. Karlo and U. Zwick. A (7=8 ")-approximation algorithm for MAX 3SAT? In Proceedings of the 38th FOCS, 1997, pp. 406-415.

....of solving a semidefinite relaxation and then rounding the resulted O(n 2 ) dimensional fractional solution to a binary n vector has been applied in the last few years to a large variety of combinatorial optimization problems. Among these we mention the 7 8 approximation for MAX 3 SAT by [113], a 1 2 approximation for MAX CSP(AND 3 ) and a 2 3 approximation for MAX CSP(MAJ 3 ) by [168] Let us add that for the latter problem a robust but weaker 40 67 approximation can be obtained using pseudo Boolean techniques without solving a large semidefinite relaxation [26] On the ....

Karloff, H., and U. Zwick. A 7=8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, 1997, pp. 406-415.

....NAE SAT in which all literals appear unnegated. A small modi cation of Algorithm MSS described in Section 2 will also give a 0:7499 approximation algorithm for Max NAE SAT. This can be seen as follows. If a variable x i occurs negated in a clause, we de ne a new variable xm i which equals to x i ([11]) This means that we will have 2m variables and then the corresponding SDP relaxation has an unknown variable which is a 2m 2m matrix. The SDP relaxation for Max NAE SAT is: w SDP : Maximize n X j=1 w j z j (MNS SDP) subject to 1 jS j j 1 X i;k2S j ji k 1 X ik 2 z j for any S j ....

H. Karlo and U. Zwick, \A 7/8-approximation algorithm for MAX 3SAT?" 38th FOCS, (1997), 406-415.

....s 1=2. These beliefs were bolstered by the strong algorithmic techniques, based on semidefinite programming , introduced in the work of Goemans and Williamson [17] Hastad s results thus brought about (yet another) unexpected settlement of these conjectures. Subsequently, Karloff and Zwick [22] used semidefinite programming methods to show the optimality of Hastad s results by showing that PCP 1;1=2 [O(log n) 3] P. Our lectures will unfortunately not be able to go into this phase of developments in the constructions of PCPs; however, we will attempt to provide pointers to this in the ....

Howard Karloff and Uri Zwick. A 7/8-approximation algorithm for MAX 3SAT? In 38th Annual Symposium on Foundations of Computer Science, pages 406-415, Miami Beach, Florida, 20-22 October 1997.

....the reciprocal of the measure used in [2] 4 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON bound from the LP s dual. Subsequent work. Subsequent to the original presentation of this work [17] the approximability results presented in this paper have been superseded. Karlo and Zwick [10] present a 7 8 approximation algorithm for MAX 3SAT. This result is tight unless NP=P [9] The containment result PCP c;s [log; 3] P has also been improved by Zwick [19] and shown to hold for any c=s 2. This result is also tight, again by [9] Finally, the gadget construction methods of this ....

H. Karlo and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In Proc. of the 38th IEEE Symposium on Foundations of Computer Science, pages 406-415, 1997.

....rounding technique, GW95] obtain a 0.87856 approximation ratio for the Max Cut problem. A number of other approximation algorithms for various problems, including Max Cut, have been designed using semidefinite programming and variations of the random hyperplane rounding technique (for example [FG95, KMS98, FJ97, KZ97, Zwi99, FKL00]) In some of these algorithms, the vectors v 1 : v n are rearranged and only then rounded using a random hyperplane. In others, the vectors v 1 : v n are rounded using the random hyperplane technique and then the combinatorial solution obtained is changed in order to improve the value of ....

H. Karloff and U. Zwick. A 7=8-approximation algorithm for Max-3-Sat? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pages 406--415, 1997.

....addition of such triangle constraints to our semidefinite relaxation is crucial to the success of our algorithm. As in the algorithm of [GW95] many other algorithms based on semidefinite programming use the random hyperplane rounding technique, and are analyzed in a local manner (for instance [FG95, KZ97, Zwi99]) i.e. for each edge a local expected approximation ratio is computed, this ratio then holds as an expected approximation ratio for the algorithm as a whole due to linearity of expectation. In our algorithm for the Max Cut problem on graphs with maximal degree three, we present an analysis in ....

B. Karloff and U. Zwick. A 7=8-approximation algorithm for Max-3-Sat? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pages 406--415, 1997.

....one, an example to this phenomenon is the well known Knapsack problem. On the other hand, unless (P=NP) there is no r approximation algorithm for the Max Clique problem for any constant r 0. There are also many problems with less extreme results such as the Max 3SAT problem. Karloff and Zwick [KZ97] have shown the existence of a polynomial approximation algorithm based 2 on semidefinite programming which obtains an approximation ratio of 7 8 on the Max 3SAT problem. This result is of exceptional interest due to the result stated in [Has97] that it is NP Hard to approximate Max 3SAT beyond ....

B. Karloff, U. Zwick. A (7=8 \Gamma ")-approximation algorithm for Max-3-SAT? Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, p. 406-415, 1997.

....solves one of Lutz and Mayordomo s Twelve Problems on Resource Bounded Measure (1999) 1 Introduction MAX3SAT is a well studied optimization problem. Tight bounds on its polynomial time approximability are known: 1. There exists a polynomial time 7 8 approximation algorithm (Karlo and Zwick [5, 3]) 1 2. If P 6= NP, then for all 0, there does not exist a polynomial time ( 7 8 ) approximation algorithm (H astad [4] Recently there has been some investigation of approximating MAX3SAT in exponential time. For example, for any 2 (0; 1 8 ] Dantsin, Gavrilovich, Hirsch, and ....

....measure provides strong, reasonable hypotheses which seem to have more explanatory power than weaker, traditional complexity theoretic This research was supported in part by National Science Foundation Grant 9988483. 1 An algorithm with conjectured performance ratio 7 8 was given in [5], and this conjecture has since been proved according to [3] 1 hypotheses. The hypothesis that NP does not have p measure 0, p (NP) 6= 0, implies P 6= NP and is known to have many plausible consequences that are not known to follow from P 6= NP. Resource bounded dimension was recently ....

H. Karlo and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In 38th Annual Symposium on Foundations of Computer Science, pages 406-415, 1997.

....N=1:44 , where jF j is the length of representation of the input. 2 NP complete 6 and MAX SNP complete, even if each clause contains at most two literals (MAX 2 SAT ; see, e.g. 31, Theorem 13.11] MAX SAT and MAX 2 SAT are well studied in the context of approximation algorithms (see, e.g. [2,11,17,20,25,38]) Recently, numerous results appeared in the domain of worst case time bounds for the exact solution of MAX SAT and MAX 2 SAT [4,11,19,21,22,28,30] The currently best bounds for MAX SAT are 2 K=2:36 and 2 L=6:89 [4] For MAX 2 SAT, the considerably better bounds 2 K=2:88 [30] and 2 ....

H. Karlo and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, FOCS'97, pages 406-415, 1997.

.... This research was supported in part by National Science Foundation Grant 9988483. 1 1 Introduction MAX3SAT is a well studied optimization problem. Tight bounds on its polynomial time approximability are known: 1. There exists a polynomial time 7 8 approximation algorithm (Karlo and Zwick [5, 3]) 1 2. If P 6= NP, then for all 0, there does not exist a polynomial time ( 7 8 ) approximation algorithm (H astad [4] Recently there has been some investigation of approximating MAX3SAT in exponential time. For example, for any 2 (0; 1 8 ] Dantsin, Gavrilovich, Hirsch, and Konev ....

....p (NP) 0, we give an exponential time lower bound for approximating MAX3SAT beyond the known polynomial time achievable ratio of 7 8 on all but a subexponentially dense set of satis able instances. Put another way, we prove: 1 An algorithm with conjectured performance ratio 7 8 was given in [5], and this conjecture has since been proved according to [3] 2 If dim p (NP) 0, then any approximation algorithm A for MAX3SAT must satisfy at least one of the following: 1. For some 0, A uses at least 2 n time. 2. For all 0, A has performance ratio less than 7 8 on an ....

H. Karlo and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In 38th Annual Symposium on Foundations of Computer Science, pages 406-415, 1997. 10

....7 8 th of all clauses. Can this optimal approximation ratio be achieved in the more general case of MAX 3SAT (when the clauses may contain 1, 2 or 3 literals) Of course, Hastad s negative result remains valid. Using semidefinite optimization, this was answered in the a#rmative by Karlo# and Zwick [46]: Theorem 6.2 There is a polynomial time approximation algorithm for MAX 3 SAT with an approximation ratio of 7 8. Let us sketch this algorithm. First, we give a quadratic programming formulation. Let x 1 , x n be the original variables, where we consider TRUE=1 and FALSE=0. Let x n i = ....

H. Karlo# and U. Zwick: A 7/8-approximation algorithm for MAX 3SAT? in: Proc. of the 38th Ann. IEEE Symp. in Found. of Comp. Sci. (1997), 406--415.

....that uses f free bits, and whose soundness is less than 2 f 2 2 f 1 can only capture P. Source of our improvement. We adapt the previously described reductions and algorithms. 4 Independent and Subsequent Results. In an independent and simultaneous research, Karlo# and Zwick [KZ97] found a new semidefinite relaxation of the Max 3SAT problem, and a new way of analysing the randomized rounding of solutions of the relaxation. As a consequence of their new technique, they were able to present a (7 8 #) approximate algorithm for Max 3SAT, for any # 0. Such an algorithm is ....

....a consequence of their new technique, they were able to present a (7 8 #) approximate algorithm for Max 3SAT, for any # 0. Such an algorithm is the best possible, since we recall that (7 8 #) approximating Max 3SAT is NP hard [Has97] More recently, Zwick [Zwi98] applied the techniques of [KZ97] to the study of Max 3CSP, and came up with a 1 2 approximate algorihtm, which is again the best possible. Using ideas from the present paper, Zwick [Zwi98] also improved our approximation of satisfiable instances of Max 3CSP, developing a 5 8 approximate algorithm for this restricted problem. ....

B. Karlo# and U. Zwick. A (7/8 - #)-approximation algorithm for MAX 3SAT? In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, 1997.

....achieves approximation ratio 1.13 for MAX CUT and MAX 2SAT. The best algorithms before that point could only achieve approximation ratios 2 and 1.33 respectively. Their chief tool, semidefinite programming [70] has since been applied to other constraint satisfaction problems. Karlo# and Zwick [83] have used it to achieve an approximation ratio 8 7, for MAX3SAT. This is an example of a tight result, since Hastad had already shown that achieving an approximation ratio 8 7 # is hard. Thus Karlo# and Zwick knew the correct approximation ratio to shoot for, which must have helped. ....

H. Karlo# and U. Zwick. A 7/8-Approximation Algorithm for MAX 3SAT? Proceedings of the Thirty Eigth Annual Symposium on the Foundations of Computer Science, IEEE, 1997 .

No context found.

H. Karloff and U. Zwick. A 7=8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pages 406--415, 1997.

No context found.

H. Karloff and U. Zwick. A 7/8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pages 406--415, 1997.

No context found.

H. Karloff and U. Zwick. A 7=8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pages 406--415, 1997.

No context found.

H. Karloff and U. Zwick. A (7=8 \Gamma ffl)-approximation algorithm for MAX 3SAT? In Proceedings of the 38rd Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, 1997. To appear.

....algorithm for satisfiable instances of MAX f3g SAT can have a performance guarantee of more than 7=8. MAX f3g SAT is the subproblem of MAX SAT in which each clause is of size exactly three. An instance is satisfiable if there is an assignment that satisfies all its clauses. Karloff and Zwick [KZ97] obtained recently an optimal 7 8 approximation algorithm for MAX 3 SAT, the version of MAX SAT in which each clause is of size at most three. This claim appears in [KZ97] as a conjecture. It has since been proved. Their algorithm uses semidefinite programming. A much simpler approximation ....

....of size exactly three. An instance is satisfiable if there is an assignment that satisfies all its clauses. Karloff and Zwick [KZ97] obtained recently an optimal 7 8 approximation algorithm for MAX 3 SAT, the version of MAX SAT in which each clause is of size at most three. This claim appears in [KZ97] as a conjecture. It has since been proved. Their algorithm uses semidefinite programming. A much simpler approximation algorithm has a performance guarantee of 7=8 if all clauses are of size at least three. If all clauses are of size at least three then a random assignment satisfies, on the ....

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H. Karloff and U. Zwick. A 7=8-approximation algorithm for MAX 3SAT? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pages 406--415, 1997.

No context found.

H. Karloff and U. Zwick, A 7/8-approximation algorithm for MAX 3SAT?, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Miami Beach, FL, IEEE, 1997, pp. 406--415.

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H. KARLOFF and U. ZWICK. A 7/8-approximation algorithm for MAX 3SAT? In Proc. 38th Symp. on Found. of Comp. Sci., pages 406--415, 1997.

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H. Karloff and U. Zwick. A 7=8-approximation algorithm for Max-3-Sat? In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pages 406-- 415, 1997.

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B. Karloff, U. Zwick. A (7=8 \Gamma ")-approximation algorithm for Max-3-SAT? Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, p. 406-415, 1997.

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