M. Bellare and M. Sudan. Improved non-approximability results. In Proc. 26th ACM Symp. on Theory of Computing, pages 184--193, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....vertices. The set c 1 (i) is called the color class i of the coloring c and the list (S 1 , S k ) of all color classes completely determines the coloring c. It is wellknown that it is NP complete to decide whether, for a given graph G and an integer k there exists a k coloring of G [1] [2]. The least possible number k of colors for which a graph G has a k coloring is called the chromatic number of G and is denoted by #(G) The graph coloring problem is the problem of finding, for a given graph G, an optimal coloring, that is a coloring with #(G) colors. Since this problem is not ....

M. Bellare and M. Sudan, Improved non-approximability results, Proceedings 26th Ann. ACM Symposium on Theory of Computation, ACM, 1994, 184-193.

....when it comes to proving lower bounds for Max Clique, but also the gap, the quotient of the soundness and the completeness. We see above that an exponential increase in the gap does not give us anything if the free bit complexity and the number of random bits increase linearly. Bellare and Sudan [5] recognized that the interesting parameter is f log s 1 in the case of perfect completeness. This parameter was later named the amortized free bit complexity and denoted by f . Note that the above gap amplification does not change f . Two methods which do improve the lower bound in the ....

....some arbitrarily small constant # 0. 3.2 Connection Between PCPs and Min Chromatic Number Lund and Yannakakis [19] reduced Min Chromatic Number to Max Clique. This reduction, which did not preserve the ratio of approximation, was improved by Khanna, Linial, and Safra [17] and Bellare and Sudan [5]. As a final improvement, Furer [12] constructed a randomized reduction showing that if Max Clique cannot be approximated within n 1 (1 f) then Min Chromatic Number cannot be approximated within n min 1 2,1 (1 2f) o(1) In the reduction used above to prove strong lower bounds on the ....

Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193, Montreal, Quebec, Canada, 23--25 May 1994.

....algorithm, for 1, if it returns an independent set of size at least fl= where fl is the size of a maximum independent set of R. Although it is known that no polynomial time Omega Gamma n 1=4 ) approximation algorithm exists for maximum independent sets in arbitrary graphs [1], no such lower bound is known for intersection graphs of rectangles. In this paper we present an O(n log n) time (log n) approximation algorithm for rectangles. For the case that all rectangles in R have the same height, we describe an (1 1=k) approximation algorithm whose running time is ....

M. Bellare and M. Sudan. Improved non-approximability results. In Proc. 26th Annu. ACM Sympos. Theory Comput., pages 184--193, 1994.

.... and because it preserves many canonical properties of the proof system (e.g. zero knowledge) Since the appearance of a preliminary version of this paper in STOC95 [40] our results were used by [9, 32] to improve many of the hardness results for approximation of optimization problems (see also [2, 3, 24, 11, 30, 31]) In particular, 32] uses our results to prove very impressive optimal inapproximability results for some of the most basic optimization problems (e.g. for Max 3 SAT) Our result was also used by [42] to prove a new direct sum theorem for probabilistic communication complexity. The reader can ....

M. BELLARE AND M. SUDAN, Improved non-approximability results, in Proc. STOC 1994, ACM, New York, pp. 184--193.

.... is a general notion has been implicit in the context of pcp: It is understood that low degree tests as used in this context are actually codeword tests (in this case of BCH codes) and that such tests can be defined and performed also for other error correcting codes such as the Hadamard Code [3, 10, 11, 8, 9, 29, 32], and the Long Code [9, 25, 26, 32] For as much as error correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of familiar objects whose properties we may wish to investigate. That is, one can certainly encode any given object by an ....

....context of linearity testing. For a function f : f0; 1g 7 f0; 1g (as above) one may define ffl lin (f) to be its distance from the set of linear functions and ffi lin (f) to be the fraction of pairs, x; y) 2 f0; 1g Theta f0; 1g for which f(x) f(y) 6= f(x Phi y) A sequence of works [13, 10, 11, 8] has demonstrated a fairly complex behavior of the relation between ffi lin and ffl lin . The interested reader is referred to [8] Previous Bound on ffi M (f) This paper is the journal version of [21] In [21] a weaker version of Theorem 2 was proved. In particular it was shown that ffi M (f) ....

M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of STOC94, pages 184--193, 1994.

....(log n) 2 ) 12] and thus it is of the form n 1 o(1) This is not an easy result but note that an approximation factor of n is trivial since the clique cannot contain more than all n nodes and any set of a single node is a clique. On the negative side, there has been a sequence of papers [11, 17, 2, 1, 8, 18, 9, 7] giving stronger and stronger inapproximability results based on very plausible complexity theoretic assumptions. The strongest lower bound is by Bellare, Goldreich and Sudan [7] who prove (under the assumption that NP#=ZPP that, for any # 0, MC cannot be e#ciently approximated within n ....

....V by V as discussed above, i.e. running V k times with independent random choices. In this case f is replaced by kf while s is replaced by s . Thus the number of amortized free bits is preserved. The connection between inapproximability and amortized free bits is now the following [17, 18, 9, 34]; suppose any NP statement admits a PCP which has a polynomial time verifier which uses logarithmic randomness and uses k amortized free bits. Then for any # 0, unless NP=ZPP, MC cannot be approximated with n # in polynomial time. Bellare, Goldreich and Sudan [7] proved that in fact we ....

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the 26th Annual ACM Symposium on Theory of Computation, Montreal, 1994, pp 184-193.

....number of random coins. This result implies that there is some constant C 1 such that Max 3 Sat could not be approximated within C unless NP=P. The first explicit constant was given by Bellare et al. [7] and based on slightly stronger hypothesis they achieved the constant 94 93. Bellare and Sudan [8] improved this to 66 65 and the strongest result prior to our results here are by Bellare, Goldreich and Sudan [6] obtaining the bound 27 26. This paper [6] also studied the problem of linear equations mod 2 and obtained inapproximability constant 8 7. The improvement in the constants has many ....

....and a new inner verifier. Thus we once more display the power of proof composition introduced in [2] although in an unusually simple form. Also note that the idea of using a multi prover proof system and coding the answers is not novel to this paper and was used in more or less explicit form in [1, 7, 8, 6]. The power of this approach was increased and the proof complexity decreased by the wonderful parallel repetition theorem [22] of Raz. Our main results are obtained by designing a PCP where the verifier reads bits, each random but correlated, of the supposed long codes of the answers of the ....

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of 26th Annual ACM Symposium on Theory of Computation, Montreal, 1994, pp 184-193.

....n (log n) 2 ) 12] and thus it is of the form n 1 o(1) This is not an easy result but note that an approximation factor of n is trivial since the clique cannot contain more than all n nodes and any set of a single node is a clique. On the negative side, there has been a sequence of papers [11, 17, 2, 1, 8, 18, 9, 7] giving stronger and stronger inapproximability results based on very plausible complexity theoretic assumptions. The strongest lower bound is by Bellare, Goldreich and Sudan [7] who prove (under the assumption that NP6=ZPP 1 ) that, for any 0, MC cannot be eciently approximated within n ....

....V by V (k) as discussed above, i.e. running V k times with independent random choices. In this case f is replaced by kf while s is replaced by s k . Thus the number of amortized free bits is preserved. The connection between inapproximability and amortized free bits is now the following [17, 18, 9, 34]; suppose any NP statement admits a PCP which has a polynomial time veri er which uses logarithmic randomness and uses k amortized free bits. Then for any 0, unless NP=ZPP, MC cannot be approximated with n 1 k 1 in polynomial time. Bellare, Goldreich and Sudan [7] proved that in fact we ....

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the 26th Annual ACM Symposium on Theory of Computation, Montreal, 1994, pp 184-193.

....number of random coins. This result implies that there is some constant C 1 such that Max 3 Sat cannot be approximated within C unless NP=P. The rst explicit constant was given by Bellare et al. 10] and based on a slightly stronger hypothesis they achieved the constant 94 93. Bellare and Sudan [11] improved this to 66 65 and the strongest result prior to our results here is by Bellare, Goldreich and Sudan [9] obtaining the bound 80=77 for any 0. The last two papers, 11, 9] use a similar approach to ours and let us describe this approach. The starting point is an ecient multiprover ....

....Bellare et al. 10] and based on a slightly stronger hypothesis they achieved the constant 94 93. Bellare and Sudan [11] improved this to 66 65 and the strongest result prior to our results here is by Bellare, Goldreich and Sudan [9] obtaining the bound 80=77 for any 0. The last two papers, [11, 9], use a similar approach to ours and let us describe this approach. The starting point is an ecient multiprover protocol, which in our case and in [9] comes naturally from a combination of the basic PCP by Arora et al. mentioned above and the wonderful parallel repetition theorem of Raz [32] ....

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M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of 26th Annual ACM Symposium on Theory of Computation, Montreal, 1994, pp 184-193.

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M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the Twenty Sixth Annual Symposium on the Theory of Computing, ACM, 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the Twenty Sixth Annual Symposium on the Theory of Computing, ACM, 1994.

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Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193, Montreal, Quebec, Canada, 23-25 May 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the Twenty Sixth Annual Symposium on the Theory of Computing, ACM, 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the Twenty Sixth Annual Symposium on the Theory of Computing, ACM, 1994.

....H such that Rej G;H ( 4 ) 8 , but Rej G;H ( 3 ) 9 . The threshold of x = 4 turns out to be signi cant in this example and an important parameter that emerges in the study of linearity testing is how low Rej G;H (x) can be for x 4 . In this paper we call this parameter, identi ed in [2, 6, 7, 8], the knee of the curve. Formally: Knee G;H = minf Rej(x) x 4 g : 1.2 Error detection in Hadamard codes In this paper we look at the performance of the BLR test when the underlying groups are G = GF(2) and H = GF(2) for some positive integer n. For notational simplicity we now drop ....

....Goldwasser, Lund and Russell [6] in the context of improving the performance of PCP systems. They showed that Rej G;H (x) 3 x 6 x . It turns out that, with very little e ort, the result of [9] can be used to show that Rej G;H (x) 9 for x 3 . This claim appears in Bellare and Sudan [8], without proof. A proof is included in the appendix of this paper, for the sake of completeness. Of the three bounds above, the last two bounds supercede the rst, so that the following theorem captures the state of knowledge. Theorem 1.1 [6, 9, 10] Let G; H be arbitrary nite groups. Then: 1) ....

[Article contains additional citation context not shown here]

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the 26th Annual Symposium on Theory of Computing, ACM, 1994.

....roughly as follows. First, one exhibits an appropriate proof system for NP. Then one applies the FGLSS reduction. The factor indicated hard depends on the proof system parameters. A key element in getting better results has been the distilling of appropriate pcp parameters. The sequence of works [FGLSS, ArSa, ALMSS, BGLR, FeKi1, BeSu] lead us through a sequence of parameters: query complexity, free bit complexity and, finally, for the best known results, amortized free bit complexity. The connection in terms of amortized free bits can be stated as follows: if NP reduces to FPCP[log; f ] then NP also reduces to Gap MaxClique ....

....for s = 0:794. 4) NP PCP 1;s [log; 3] for any s 0:85. Some of these results are motivated by applications, others purely as interesting items in proof theory. The search for proof systems of low amortized free bit complexity is motivated of course by the FGLSS reduction. Bellare and Sudan [BeSu] have shown that NP FPCP[ log; 3 ffl ] for every ffl 0. The first result above improves upon this, presenting a new proof system with amortized free bit complexity 2 ffl. The question of how low one can get the (worst case and average) query complexity required to attain soundness error ....

[Article contains additional citation context not shown here]

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the 26th Annual Symposium on the Theory of Computing, ACM, 1994, pp. 184--193.

....called Max kCSP, is known to be hard to approximate within a factor 2 Gamma:4k [26] and a 2 Delta 2 Gammak approximation algorithm is also known [25] We prove that Max kCSP is NP hard to approximate within a factor of roughly 2 Gamma2k=3 . 1 Introduction PCP characterizations of NP [6, 5, 12, 4, 3, 8, 13, 9, 7, 15, 16, 26] are the best known tool to prove results about the hardness of approximation of combinatorial optimization problems. Progress in this area has been driven by the goal of characterizing NP via increasingly more efficient PCP verifiers, under various formalizations of the notion of efficiency, and ....

.... case, the methods used for recycling randomness while reducing error in general probabilistic computation [1, 17] turns out to be quite useful and are used, for instance, in [3, 28, 2] In the latter case (minimizing free bits ) also, the notion of recycling can be analyzed and the works of [9, 7, 14, 15] show that this method leads to significant benefits. Our task, however differs from the previous cases in some critical aspects. For instance, in the context of recycling randomness, a random bit is counted as a recycled bit, if it is obtained by applying some (arbitrary) function to the ....

[Article contains additional citation context not shown here]

M. Bellare and M. Sudan. Improved non-approximability results. STOC, 1994.

....Rej G;H ( 1 4 ) 3 8 , but Rej G;H ( 2 3 ) 2 9 . The threshold of x = 1 4 turns out to be signi cant in this example and an important parameter that emerges in the study of linearity testing is how low Rej G;H (x) can be for x 1 4 . In this paper we call this parameter, identi ed in [2, 6, 7, 8], the knee of the curve. Formally: Knee G;H def = minf Rej(x) x 1 4 g : 1.2 Error detection in Hadamard codes In this paper we look at the performance of the BLR test when the underlying groups are G = GF(2) n and H = GF(2) for some positive integer n. For notational simplicity we now ....

....Goldwasser, Lund and Russell [6] in the context of improving the performance of PCP systems. They showed that Rej G;H (x) 3 x 6 x 2 . It turns out that, with very little e ort, the result of [9] can be used to show that Rej G;H (x) 2 9 for x 1 3 . This claim appears in Bellare and Sudan [8], without proof. A proof is included in the appendix of this paper, for the sake of completeness. Of the three bounds above, the last two bounds supercede the rst, so that the following theorem captures the state of knowledge. Theorem 1.1 [6, 9, 10] Let G; H be arbitrary nite groups. Then: 1) ....

[Article contains additional citation context not shown here]

M. Bellare and M. Sudan. Improved non-approximability results. Proceedings of the 26th Annual Symposium on Theory of Computing, ACM, 1994.

....complexity (or equivalently the proof size) or the query complexity. Phase 3. Examination of the (non asymptotic) tightness of the parameters of the PCP theorem was initiated by Bellare, Goldwasser, Lund and Russell [10] Several intermediate results improved the constants in the parameters [15, 11, 9]. Eventually near tight results which optimize both these parameters (but not simultaneously ) were shown. Specifically: ffl Polishchuk and Spielman [23] showed that Sat 2 PCP Theta (1 ) log n; O(1) for every 0. ffl It is a folklore result that the number of queries required in the ....

....other words the PCP Theorem We would thus get: Theorem 6 ( 3, 2] There exists an 0 such that NP = PCP 1;1 Gamma [O(log n) 34] 4. Towards Optimal PCPs There are a number of respects in which one can hope to improve Theorem 6. This has been the focus of a large body of works including [15, 11, 9, 20, 21, 19, 28, 27, 26]. One specific question, for example, is: What is the minimum number of queries required to obtain a desired soundness error The quest for better (and optimal) PCP constructions has also been motivated by applications to hardness of approximations where improvements in the underlying PCPs often ....

Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of 37 38 MADHU SUDAN, PROBABILISTICALLY CHECKABLE PROOFS Computing, pages 184-193, Montreal, Quebec, Canada, 23-25 May 1994.

....function Rej G;H ( Delta) for given G; H. Much of the work that has been done provides information about various aspects of this function. THE KNEE OF THE CURVE. In particular, one parameter has emerged as an important one in connection with MaxSNP hardness results. This parameter, identified in [2, 6, 7, 8], is a single number, which we call here the knee of the curve. It is defined as the minimum rejection probability when the distance (of the function being tested from the space of linear functions) is at least 1=4: Knee G;H def = minf Rej(x) x 1=4 g : Improvements (increases) in the lower ....

.... (of the function being tested from the space of linear functions) is at least 1=4: Knee G;H def = minf Rej(x) x 1=4 g : Improvements (increases) in the lower bound that can be shown on Knee G;H translate directly into improved (increased) nonapproximability factors for MaxSNP problems via [6, 7, 8]. Exactly how or why this is the case is outside the scope of this paper, and we refer the reader to the works in question. 1.2 Previous work The first investigation of the shape of the linearity testing curve, by Blum, Luby and Rubinfeld [9] was in the general context where G; H are ....

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M. BELLARE AND M. SUDAN. Improved nonapproximability results. Proceedings of the 26th Annual Symposium on Theory of Computing, ACM, 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. In Proc. 26th ACM Symp. on Theory of Computing, pages 184--193, 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193, 1994.

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Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193. Montr eal, Qu ebec, Canada, 23--25 May 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193, 1994.

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M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 184--193, 1994.

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