U. Feige and L. Lovasz. Two-prover one-round proof systems: their power and their problems. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pages 733--744, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....total weight is at least 0.87856 times the optimal value. In the paper [4] Feige and Goemans proposed an approximation algorithm for MAX 2SAT which achieves 0.93109 of approximation ratio. Their algorithm based on two ideas. First, they added some constraints introduced by Feige and Lovasz in [3] to SDP relaxation problem. Next, they proposed the rotation technique which modifies the solution obtained by SDP relaxation. They calculated the approximation ratio of their algorithm numerically. Recently, Zwick refined the rotation technique and proposed an algorithm whose approximation ....

....paper [6] Goemans and Williamson relaxed the above problem by replacing each variable v i 1 with a vector on the n dimensional unit sphere S n where S n def. R v = 1 . This relaxation is proposed by Lovasz [8] originally. By introducing some valid constraints used in papers [3, 4], we obtain the following relaxation problem; P) maximize (1 4) w ij (3 v 0 v j ) subject to v 0 = 1, 0, 0) # , v i v i n = 0 (#i S n (#i . n, n 1, 2n ) #(i, j) It is well known that we can transform the above problem to a semidefinite ....

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U. Feige and L. Lovasz, "Two-prover one-round proof systems: Their power and their problems", Proc. of the 24th Annual ACM Symposium on the Theory of Computing, 1992, 733--744.

....Arora [Aro98] after failing to achieve lower bounds for traveling salesman in the plane, has developed a polynomial time approximation algorithm for this and related problems. A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov asz [CCL92, CCL90, CCL91, Fei91, LS91, FL92] have modi ed the protocol of Babai, Fortnow and Lund [BFL91] 10 to show that every language in NEXP has a two prover, one round proof systems with an exponentially small error. This problem remained so elusive because running these proof systems in parallel does not have the expected error ....

U. Feige and L. Lovasz. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th ACM Symposium on the Theory of Computing, pages 733-744. ACM, New York, 1992.

....whose weight is at least 0.79607 times the optimal value. In the paper [4] Feige and Goemans proposed an approximation algorithm for MAX DICUT which achieves 0.859387 of approximation ratio. Their algorithm based on two ideas. First, they added some constraints introduced by Feige and Lov asz in [3] to SDP relaxation problem. Next, they proposed the rotation technique which modi es the solution obtained by SDP relaxation. They calculated the approximation ratio of their algorithm numerically. Recently, Zwick re ned the rotation technique and proposed an algorithm whose approximation ....

....the paper [5] Goemans and Williamson relaxed this problem by replacing each variable v i 2 f1; 1g with a vector on the n dimensional unit sphere v i 2 S n where S n = fv 2 R j jjvjj = 1g: This relaxation is proposed by Lov asz [6] originally. And We add some valid constraints used in papers [3, 4], and obtain the following relaxation problem; DI) maximize (1=4) w ij (1 v 0 v i v 0 v j v i v j ) subject to v 0 = 1; 0; 0) v i 2 S n (8i 2 V ) v 0 v i v 0 v j v i v j 1 (8(i; j) 2 A) v 0 v i v 0 v j v i v j 1 (8(i; j) 2 A) v 0 v i v 0 ....

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U. Feige and L. Lovasz, \Two-prover one-round proof systems: Their power and their problems", Proc. of the 24th Annual ACM Symposium on the Theory of Computing, 1992, 733-744.

....proofs over a large alphabet in which number of alphabets that a verifier is allowed to probe is a parameter. This concept is an important ingredient in the recursive construction of probabilistically checkable proofs [AS92, ALMSS92, BGLR93] and is also of independent interest in complexity theory [LS91, FL92a]. The original definition of probabilistically checkable proofs is due to [AS92] based on an implicit notion in [FGLSS91] A very closely related notion that of holographic proofs appears in the work of [BFLS91] The particular choice of parameters made in the following definition is due to ....

U. Feige and L. Lovasz. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 733--744, 1992.

....time algorithms. After the work of [FGLSS] the field took off in two major directions. One was to extend the interactive proof approach to prove the non approximability of other optimization problems. Direct reductions from proofs were used to show the hardness of quadratic programming [BeRo, FeLo], Max3SAT [ALMSS] set cover [LuYa] and other problems [Be] The earlier work of Papadimitriou and Yannakakis introducing the class MaxSNP [PaYa] now came into play; by reduction from Max3SAT it implied hardness of approximation for any MaxSNP hard problem. Also, reductions from Max Clique lead ....

....proofs in PCP perspective. Constant prover proofs have been instrumental in the derivation of non approximability results in several ways. One of these is that they are a good starting point for reductions examples of such are reductions of two prover proofs to quadratic programming [BeRo, FeLo] and set cover [LuYa] However, it is a different aspect of constant prover proofs that is of more direct concern to us. This aspect is the use of constant prover proof systems as the penultimate step of the recursion, and begins with [ALMSS] It is instrumental in getting PCP systems with only a ....

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U. Feige and L. Lov' asz. Two-prover one round proof systems: Their power and their problems. Proceedings of the 24th Annual Symposium on the Theory of Computing, ACM, 1992, pp. 733-744.

....system Psi as described above, to an equation system e Psi with the desired properties. We use two techniques iteratively in the construction of e Psi. The arithmetization technique from [BFL91] summarized in lemma 13, and the curve extension technique, which is similar to [ALM 92, LS91, FL92] and is formalized in lemma 16. We also use the linearization technique, described in lemma 17. Aside from having different parameters, these techniques have a similar structure. They receive an equation system as input, and substitute each equation in it with a set of new equations, that ....

....properties of e Psi are maintained (the weight of the system relative to any assignment may change by at most a factor by this transition, as shown in the proof of proposition 28) 4. 3 The Curve Extension Lemma This lemma uses a technique called curve extension, similar to [ALM 92, LS91, FL92] to reduce the principal degree and dimension of the input system. Lemma 16 (Curve Extension) The curve extension lemma is obtained from the lemma template by setting its C The parameters of the input system should satisfy r yes ; r no ; d; D jFj such that d Delta r yes D r no . ....

U. Feige and L. Lovasz. Two-prover one-round proof systems: Their power and their problems. In Proc. 24th ACM Symp. on Theory of Computing, pages 733--741, 1992.

....most . Parameters. The parameters of interest are p, r 1 and l = jjCjj l 0 . Our transformation applies to any canonical proof system. The above theorems are obtained by plugging in speci c canonical proof systems as we now describe. The only property missing to make the proof system of [19, 14] canonical is that the second prover s answers are not expressible as a projection of those of the rst. This is simple to x. To see how, recall that the rst prover sends a sequence of polynomials A 1 ; A d : F F where d = O(jF j log 2 n) and F is the underlying eld. These ....

....the analysis a better analysis of the linearity tests, due to [10] which, although known before [7] seems to have been forgotten by them; we even reduce to 13, from the 15 in [7] the number of 3SAT clauses needed to write the quadratic test. A very skimpy description follows. We rst use the [19, 14] proof system to get a canonical two prover proof system as discussed in Section 3.1. We then convert this into a Max 3SAT by expressing the task of the extended veri er by a 3cnf formula: A linearity test requires 4 clauses, a quadraticity test 13, an output test 2 and an input test requires 4 ....

U. Feige and L. Lov asz. Two-prover one round proof systems: Their power and their problems. STOC 92.

....are omitted from this abstract, but they are obvious from reading our proofs (specifically, by noting their algorithmic nature) Past work on constant prover proof systems. The first construction of a nontrivial constant prover 1 round proof system for NP appeared in [23] others appeared in [16, 10, 34, 14, 27]. These systems could not reduce the error probability to below a constant while using O(logn) random bits (the best construction needs O(k log n) random bits to make the error probability 2 Gammak ; see [27] It was also known [15] that some obvious ideas (such as recycling randomness ) ....

U. FEIGE AND L. L OVASZ. Two-prover one-round proof systems: Their power and their problems. ACM STOC 1992.

....where the veri er uses r = O(log n j log j) random bits and the answer of each prover has length a. The values of the parameters p and a are critical for the applications. Lapidot, Shamir [LS] obtained this result with p = 4 provers and a = poly(log n; log ) answer size; Feige, Lov asz [FL] reduced the number of provers The author was visiting the Automation and Computation Institute of the Hungarian Academy of Sciences, DIMACS Center and the University of Toronto while part of this research was done. The author is partially supported by NSF grants CCR 92 00788, CCR 95 03254 and ....

....the random string R (this is similar to the de nition of a quasi oracle in [LS] But the decoding does not depend on the rst p 1 provers. Below all MQS have p = 2 provers unless otherwise stated. 4 Existence of the encoding schemes The main result of this section is a scaled down version of [FL] to achieve a MQS for any polynomial function family F . We start with a transparent proof like version of MES (Lemma 1) Take any polynomial function family F . Let the parameters of F be y, t, and z. Take a good encoding function E. We de ne a transparent encoding of the functions in F x . For ....

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U. Feige and L. Lovasz. Two-prover one-round proof systems: their power and their problems. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, 1992, 733-744.

....time can be simulated deterministically (see [IW97] Sudan et al. STV99] also give a simpler analysis with improved parameters for the self correction problem. Past work. The rst construction of a nontrivial constant prover 1 round proof system for NP appeared in [LS91] others appeared in [FL92, BGLR93, T94, FK94, R95] These systems could not reduce the error probability to below a constant while using O(log n) random bits (the best construction needs O(k log n) random bits to make the error probability 2 k ; see [R95] It was also known [FK95] that some obvious ideas (such as ....

U. Feige and L. L ovasz. Two-prover one-round proof systems: Their power and their problems. Proceedings of the 24th Annual Symposium on Theory of Computing, ACM, 1992.

....proofs over a large alphabet in which number of alphabets that a veri er is allowed to probe is a parameter. This concept is an important ingredient in the recursive construction of probabilistically checkable proofs [AS92, ALMSS92, BGLR93] and is also of independent interest in complexity theory [LS91, FL92a]. The original de nition of probabilistically checkable proofs is due to [AS92] based on an implicit notion in [FGLSS91] A very closely related notion that of holographic proofs appears in the work of [BFLS91] The particular choice of parameters made in the following de nition is due to ....

U. Feige and L. Lovasz. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 733-744, 1992.

....and Safra [6] it follows that approximating the clique number within a factor 2 ( p log N) is NP hard. The discovery of Theorem 2 inspired the search for other connections between probabilistic proof checking and non approximability (Bellare [18] Bellare and Rogaway [22] Feige and Lov asz[45], and Zuckerman [100] Another such connection is reported by Arora, Motwani, Safra, Sudan and Szegedy [5] which shows the connection between PCP s and the hardness of approximating MAX 3SAT. The following theorem summarizes this result; for a proof see Section 3. Theorem 3 ( 5] If NP ....

....Sudan [90] The in uence of the former will be apparent in Section 5, while the latter work (together with a lemma of Arora and Safra [6] is used in our analysis of a Low Degree Test described in Section 7.2. Finally, we were in uenced by work on constant prover 1 round interactive proof systems [74, 45]. In fact, our de nition of an outer veri er (De nition 8) may be viewed as a generalization of the de nition of such proof systems, and our Theorem 4 provides the rst known construction of a 2 prover 1 round proof system that uses logarithmic random bits and constant number of communication ....

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U. Feige and L. Lov asz. Two-prover one-round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992. 42

....to select a single representative from each set and maximize the number of super edges covered. Let us now recall the satisfiability (SAT ) problem. A CNF boolean formula I is given, and the question is whether there is an assignment satisfying all the clauses. The following result follows from [FL 92] It can also be deduced from [R 95] Theorem 5.2 Let I be an instance of SAT . For any 0 ffl 1, there exists a reduction of each instance of the satisfiability problem, to an instance G of Max rep of size n, such that if I is satisfiable, there is a set of unique representatives which cover ....

U. Feige L. Lova'sz, Two-prover one-round proof systems: Their power and their problems, Proc. 24th ACM Symp. on Theory of Computing, 733-741, 1992 23

....system, the proof consists of p arrays of letters from #. The verifier is only allowed to query 1 letter from each array. Since each letter of # is represented by #log # # bits, the number of bits queried may be viewed as p #log # #. Constructions of such proof systems for NP appeared in [30, 96, 52, 27, 50, 113]. Lund and Yannakakis [108] used these proof systems to prove inapproximability results for SETCOVER and many subgraph maximization problems. The best construction of such proof systems is due to Raz and Safra [114] They show that for each k # # log n, every NP language has a verifier that uses ....

U. Feige and L. Lovasz. Two-prover one-round proof systems: Their power and their problems. In Proc. 24th ACM Symp. on Theory of Computing, pages 733--741, 1992.

....and Safra [6] it follows that approximating the clique number within a factor 2 #( # log N) is NP hard. The discovery of Theorem 2 inspired the search for other connections between probabilistic proof checking and non approximability (Bellare [18] Bellare and Rogaway [22] Feige and Lovsz[45], and Zuckerman [100] Another such connection is reported by Arora, Motwani, Safra, Sudan and Szegedy [5] which shows the connection between PCP s and the hardness of approximating MAX 3SAT. The following theorem summarizes this result; for a proof see Section 3. Theorem 3 ( 5] If NP # # ....

....[90] The influence of the former will be apparent in Section 5, while the latter work (together with a lemma of Arora and Safra [6] is used in our analysis of a Low Degree Test described in Section 7.2. Finally, we were influenced by work on constant prover 1 round interactive proof systems [74, 45]. In fact, our definition of an outer verifier (Definition 8) may be viewed as a generalization of the definition of such proof systems, and our Theorem 4 provides the first known construction of a 2 prover 1 round proof system that uses logarithmic random bits and constant number of communication ....

[Article contains additional citation context not shown here]

U. Feige and L. Lovsz. Two-prover one-round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992. 43

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U. Feige and L. Lov'asz. Two-prover one-round proof systems: Their power and their problems (extended abstract). In Proceedings of the 24rd Annual ACM Symposium on Theory of Computing, Victoria, Canada, pages 733--744, 1992.

....k prover proof system. It is fortunate that we can invoke a recent theorem of Raz [17] regarding reduction of error by parallel repetition, and hence make this description largely self contained. In contrast, Lund and Yannakakis used the more complicated 2 prover proof system of Feige and Lovasz [8] (the result of [17] was not available at the time) and needed to quote its special properties without proof. In Section 3 we describe the reduction from our k prover proof system to set cover. In Section 4 we explain how to construct the partition systems mentioned above. In Section 5 we ....

U.Feige, L. Lovasz. "Two prover one round proof systems: their power and their problems ". Proc. 24th STOC, 1992, 733--744.

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U. Feige and L. Lovasz. Two-prover one-round proof systems: their power and their problems. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pages 733--744, 1992.

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U. Feige and L. Lov'asz. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, pages 733-- 744, 1992.

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U. Feige and L. Lov' asz. Two-prover one round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992.

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U. Feige and L. Lov' asz. Two-prover one round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992.

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U. Feige and L. Lov'asz. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, pages 733--744, 1992.

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U. Feige, L. Lovasz. "Two-prover one-round proof systems: their power and their problems". Procedings of the 24th Annual ACM Symposium on the Theory of Computing, 733--744, 1992.

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U. Feige and L. Lov' asz. Two-prover one round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992.

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U. Feige and L. Lov' asz. Two-prover one round proof systems: Their power and their problems. Proceedings of the Twenty Fourth Annual Symposium on the Theory of Computing, ACM, 1992.

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