M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3 colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3]. Another aspect in which our proof is different is that using the PCP theorem we can show that 4 coloring of 3 colorable graphs remains NP hard even on bounded degree graphs (this hardness result does not seem to follow from the earlier reduction of [19] We point out that such graphs can ....

....Not relying on PCP machinery implies that this hardness result could have been obtained almost three decades ago, long before the arrival of the PCP theorem. In contrast the hardness result (for approximating within n for example) for general graph coloring implies some form of PCP [3]; our result therefore also highlights a qualitative difference between the hardness of general graph coloring and coloring 3 colorable graphs. As in essentially all previous reductions showing hardness of graph coloring, our reduction too starts from the hardness of Independent Set (Max Clique) ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

....analyzed in [17] We apply parallel repetition to the inner veri er, and we show that we can drive the soundness down at an optimal rate within the inner veri er. Composition gives a PCP construction with the right soundness. For readers who are familiar with the constructions and terminology of [6, 17], we remark that our analysis takes advantage of the way in which the long code over a big alphabet can be folded. 4.1 Proof or Theorem 4 4.1.1 Background Let us start with an abstract view of the proof system of Raz[26] The veri er is given a 3SAT instance and has oracle access to two ....

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....Furthermore, all inapproximability results from now on are based on the assumption = NP. This will also be implicitly assumed in a number of places. In particular, Theorem 4.5 would be referred to as stating that Max E3 Sat cannot be approximated within 7 8 #. The following theorem, implicit in [21], has been a source of a large number of inapproximability results. It provides a connection between the tight inapproximability result of Theorem 4.4 and the gadget construction techniques of Trevisan et al. 81] described in the next section) 4.2. Finding optimal gadget reductions 33 ....

....techniques of Trevisan et al. 81] described in the next section) 4.2. Finding optimal gadget reductions 33 De#nition 4.6. The constraint families PC 0 and PC 1 (PC for Parity Check) are de#ned through the relations PC i (x, y, z) 1 if x y z = i, 0 otherwise. Theorem 4.7. [21] If there exists an # 0 gadget reducing PC 0 to and an # 1 gadget reducing PC 1 to , then for all # 0,MAX(F)is hard to approximate within 1 #0 #1 #. For many problems this theorem gives the best lower bounds currently known, often close to the performances of the best approximation ....

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Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCP's and non-approximability # towards tight results. SIAM Journal of Computing, 27(3):804#915, 1998.

....Max Cut, Max E3 Set Splitting, etc) The problem is that when the target problem is monotone, one cannot convert a PC 1 constraint to a PC 0 constraint by simply negating a variable, and one has to pay an explicit cost in the gadget for negating a variable. This error occurs in early versions of [2] and in [10, 1] For the case of Max Cut, the error can be (and was) xed in [2] who construct a PC 1 gadget from a PC 0 gadget by negating a variable at a unit extra cost. One can similarly x the error for Max E3 Set Splitting by incurring an extra cost of 4 for the PC 1 gadget, and this gives a ....

....is monotone, one cannot convert a PC 1 constraint to a PC 0 constraint by simply negating a variable, and one has to pay an explicit cost in the gadget for negating a variable. This error occurs in early versions of [2] and in [10, 1] For the case of Max Cut, the error can be (and was) xed in [2] who construct a PC 1 gadget from a PC 0 gadget by negating a variable at a unit extra cost. One can similarly x the error for Max E3 Set Splitting by incurring an extra cost of 4 for the PC 1 gadget, and this gives a hardness result for approximating Max E3 Set Splitting to better than a factor ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

.... in the gadget for negating a variable (e.g. to simulate x, one has to introduce an auxiliary variable, say t, that will play the role of x, and then one has to introduce suitable constraints in the gadget at additional cost to enforce that t = x) This error occurs in early versions of [1] and in [7] For the case of Max Cut, the error can be (and was) xed in [1] who construct a PC 1 gadget from a PC 0 gadget by negating a variable at a unit extra cost. One can similarly x the error for Max E3 Set Splitting by incurring an extra cost of 4 for the PC 1 gadget. These gadgets ....

.... introduce an auxiliary variable, say t, that will play the role of x, and then one has to introduce suitable constraints in the gadget at additional cost to enforce that t = x) This error occurs in early versions of [1] and in [7] For the case of Max Cut, the error can be (and was) xed in [1] who construct a PC 1 gadget from a PC 0 gadget by negating a variable at a unit extra cost. One can similarly x the error for Max E3 Set Splitting by incurring an extra cost of 4 for the PC 1 gadget. These gadgets together with H astad s optimal inapproximability result for 3 parity constraints ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....of nearly linear size induced matchings. The existence of such graphs is essential to our exponentially improved bounds on linearity testing. For completeness we sketch the construction of [17] in the appendix. For a more thorough discussion of PCPs and their properties we refer to the papers [7], 12] and for a discussion of the history of the current problem we refer to [16] 2 Preliminaries Here we recall the Fourier transform over a eld of two elements, which will be needed both for linearity testing over this eld, as well as for PCPs. All our Boolean functions map into 1 where we ....

....of the clauses are satis ed is bounded by d u c for some absolute constant d c 1. This two prover protocol is now turned into a PCP by, for each question to either P 1 or P 2 writing down the answer in coded form. As many other papers we use the marvelous long code introduced by Bellare et al. [7]. De nition 4.1 The long code of an assignment x 2 f1; 1g t is obtained by for each function f : f1; 1g t 7 f1; 1g writing down the value f(x) Thus the long code of a string of length t is a string of length 2 2 t . Note that even though a prover is supposed to write down a long code ....

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....discovery of the PCP Theorem ( 2] 1] opened up a whole new fascinating direction for proving various inapproximability results. In the last eight years, quantitative improvement in the efficiency of PCP verifiers has led to (in many cases optimal) hardness results for many optimization problems ([3], 5] 9] 10] 15] 7] For different applications, different aspects of the given PCP need to be optimized. For a detailed discussion of various parameters we refer to [3] In the current paper we are mostly concerned with making efficient use of queries, i.e. to obtain very strong proofs ....

....in the efficiency of PCP verifiers has led to (in many cases optimal) hardness results for many optimization problems ( 3] 5] 9] 10] 15] 7] For different applications, different aspects of the given PCP need to be optimized. For a detailed discussion of various parameters we refer to [3]. In the current paper we are mostly concerned with making efficient use of queries, i.e. to obtain very strong proofs where the verifier reads very few symbols in the proof. Samorodnitsky and Trevisan [15] obtained very strong results along these lines, giving a PCP where the verifier reads 2k k ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998.

....algorithms for Max SAT and its restricted versions Max 2SAT and Max 3SAT have been developed; we summarize such previous results in Table 1. There is a corresponding history of continuous improvements in the non approximability; we do not mention it here (the interested reader can find it in [BGS98] and we only recall that the best known hardness is 7 8 # due to Hastad [Has97] and it still holds when restricting to satisfiable instances with exactly three literals per clause. Our results. We present a polynomial time algorithm that, given a satisfiable Max SAT instance, satisfies a ....

....that, given a satisfiable Max SAT instance, satisfies a fraction .8 of the total weight of clauses, and an algorithm that, given a satisfiable Max 3SAT instance, satisfies a fraction .826 of the total weight of clauses. 2 Satisfiable instances Arbitrary instances Due to .125 . 125 [PY91] 299 [BGS98] 25 [Tre96] 367 .367 [TSSW96] 514 This paper Table 2: Evolution of the approximation factors for Max 3CSP with and without the satisfiability promise. Source of our improvement. In both cases, we show how to reduce the given instance to an instance without unit clauses. The reduction ....

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M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

....by a connection that was initiated by [FGL 91] and then stretched to an amazing extent. In the last seven years, research on probabilistically checkable proofs has focused on achieving quantitative strengthening of the PCP Theorem, involving increasingly e#cient verifiers [BGLR93, FK94, BS94, BGS98, Has96, Has97, Tre98, ST98] and also, in a somewhat di#erent direction, RS97, AS97, DFK 99] Such improvements of the PCP Theorem have been mostly driven by the search for improved non approximability results, and some of the e#ciency measures used for verifiers have been tailored to the ....

.... to 3SAT) on the other hand, it is possible to get verifiers having an arbitrarily small error s 0, and having query complexity O(log(1 s) Furthermore, one can show that, in a PCP characterization of NP, a verifier having error s must have query complexity at least log(1 s) unless P=NP) BGS98, Tre96] So the best we can hope for (in terms of trade o# between error probability and number of queries) is to construct verifiers having error s and query complexity q log 1 s where q 1 is some constant that we would like to be as small as possible. The parameter q (the ratio between ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

....NC through the techniques of Karger and Kholler [KK94] 4 This nice idea is due to Michel Goemans. 5 Negative Results: Hardness of Approximation We first define Probabilistically Checkable Proof Systems and Multi Prover OneRound Proof Systems. Our notation merges the notations of [BGLR93] and [BGS96]. For an integer d, we denote by [d] # the set of all strings over [d] Definition 16 (Verifier) A verifier V for a language L is a randomized polynomial time oracle Turing machine. V receives in input a string x and has oracle access to a string # that is an alleged proof that x # L. ....

Bellare, M., Goldreich, O., and Sudan, M. Free bits, PCP's and nonapproximability -- towards tight results (4th version). Technical Report TR95-24, ECCC (1996) Preliminary version in Proc. of FOCS'95.

....theorem, while the one in [18] does. This highlights a qualitative difference between the known hardness result for coloring 3 colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3]. Another aspect in which our proof is different is that using the PCP theorem we can show that 4 coloring of 3 colorable graphs remains NP hard even on bounded degree graphs (this hardness result does not seem to follow from the earlier reduction of [18] We point out that such graphs can ....

....Not relying on PCP machinery implies that this hardness result could have been obtained almost three decades ago, long before the arrival of the PCP theorem. In contrast the hardness result (for approximating within n for example) for general graph coloring implies some form of PCP [3]; our result therefore also highlights a qualitative difference between the hardness of general graph coloring and coloring 3 colorable graphs. As in essentially all previous reductions showing hardness of graph coloring, our reduction too starts from the hardness of INDEPENDENT SET (MAX ....

[Article contains additional citation context not shown here]

M. BELLARE, O. GOLDREICH, AND M. SUDAN. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-- 915, 1998. Preliminary version in Proc. of FOCS'95.

....proof systems. To a reader familiar with PCPs we rst give a preview of our constructions. Both our PCPs (of this section and the next) go through the standard path. We start with strong 2 prover 1 round proof systems of Raz [28] apply the composition paradigm [5] and then use the long code of [8] at the bottom level. One warning: in the literature it is common to use a variant of the long code called the folded long code we do not use the folded version. Readers unfamiliar with the terms above may nd elaborations in Section 3.1. As usual, the interesting aspects in the ....

....has the nice feature that Xor just becomes multiplication. For any domain D, denote by FD the space of all Boolean functions f : D f1; 1g. For any set D, jDj denotes its cardinality. We now describe a very redundant error correcting code, called the long code. The long code was rst used by [8], and has been very useful in most PCP constructions since. The long code of an element x in a domain D, denoted LONG(x) is simply the evaluations of all the 2 jDj Boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998.

....question. A PCP verifier uses q amortized query bits if, for some t, it makes qt queries and has error probability at most 2 Gammat . A PCP characterization of NP using 2:5 amortized query bits is known [26] and, unless P=NP, no such characterization is possible using 1 amortized query bits [7]. We present a PCP characterization of NP that uses roughly 1.5 amortized query bits. Our result has two main implications. Separating PCP from 2 Provers 1 Round: In the 2 Provers 1 Round (2P1R) model the verifier has access to two oracles (or provers) and can make one query to each oracle. Each ....

....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 ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998.

....Thus a little bit of randomness does not increase the power of the PCP verifiers in terms of the languages for which they can verify membership. However it does allow them to be significantly more efficient. A collection of these and other such folklore results about PCPs may be found in [9]. Phase 1. The first non trivial result on PCPs did not talk about the class NP but rather about the class NEXP. This result, due to Babai, Fortnow, and Lund [6] showed that NEXP = PCP[poly(n) poly(n) Note that the traditional verifier of NEXP languages looks at a proof in exponentially many ....

....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 ....

[Article contains additional citation context not shown here]

Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proceedings of FOCS'95.

....Else if F is 1 valid then MIN ONES (F) is poly APX complete (6) Else finding any feasible solution to MIN ONES (F) is NP hard. Techniques As in the work of Khanna et al. 15] two simple ideas play an important role in this paper. 1) The notion of implementations from [15] also known as gadgets [3, 26]) which shows how to use the constraints of a family F to enforce constraints of a different family F 0 , thereby laying the groundwork of a reduction from MIN CSP(F 0 ) to MIN CSP(F) 2) The idea of working with weighted versions of minimization problems. Even though our theorems only make ....

....0 valid and a not 1 valid function. Then (1) If F contains a function that is not C closed, then F perfectly implements the unary constraints x and ( x) 2) Otherwise, F perfectly implements the binary constraints (x y = 1) and (x = y) One relevant consequence (that also uses an idea from [3]) is the following. Lemma 23 Let F be a family that contains a not 0 valid and a not 1 valid function. Then MIN WEIGHT CSP(F [ fx; x)g) is A reducible to MIN WEIGHT CSP(F) Proof: If F contains a function that is not C closed, then x and ( x) can be perfectly implemented using constraints from ....

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results (3rd version). Technical Report TR95-24, Electronic Colloquium on Computational Complexity, 1995. Preliminary version in Proc. of FOCS'95.

....if c coloring 2 colorable r uniform hypergraphs is NP hard for every constant c, then NP cPCP 1; 1 k [O(log n) r] for every constant k 1. 2.3 Preliminaries on Long Code We now describe a very redundant error correcting code, called the long code. The long code was first used by [7], and has been very useful in all PCP constructions since. The long code of an element x in a domain D, denoted LONG(x) is simply the evaluations of all the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , ....

....the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , so that A(f) f(a) Folding of Long Codes: A Discussion. A function A : FD f1; Gamma1g is said to be folded if A(f) GammaA( Gammaf ) for all f 2 FD [7]. A codeword of the long code is clearly folded (since A(f) f(a) Gamma( Gammaf (a) GammaA( Gammaf ) One can many times assume that the proofs which are purportedly long codes are folded since for any A : FD f1; Gamma1g, one can define a new function A 0 by: A 0 (f) A(f) if f(ff ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....if c coloring 2 colorable r uniform hypergraphs is NP hard for every constant c, then NP cPCP 1; 1 k Theta O(log n) r for every constant k 1. 2.2 Preliminaries on Long Code We now describe a very redundant error correcting code, called the long code. The long code was first used by [7], and has been very useful in all PCP constructions since. We first develop some notation. We represent boolean values by the set f1; Gamma1g with 1 standing for FALSE and Gamma1 for TRUE. This representation has the nice feature that XOR just becomes multiplication. For any domain D, denote by ....

....the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , so that A(f) f(a) Folding of Long Codes: A Discussion. A function A : FD f1; Gamma1g is said to be folded if A(f) GammaA( Gammaf ) for all f 2 FD [7]. A codeword of the long code is clearly folded (since A(f) f(a) Gamma( Gammaf (a) GammaA( Gammaf ) One can assume that the proofs which are purportedly long codes are folded since for any A : FD f1; Gamma1g, one can define a new function A 0 by: A 0 (f) A(f) if f(ff 0 ) 1 ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....if c coloring 2 colorable r uniform hypergraphs is NP hard for every constant c, then NP cPCP 1; 1 k O(log n) r for every constant k 1. 2.2 Preliminaries on Long Code We now describe a very redundant error correcting code, called the long code. The long code was first used by [7], and has been very useful in all PCP constructions since. We first develop some notation. We represent boolean values by the set f1; 1g with 1 standing for FALSE and 1 for TRUE. This representation has the nice feature that XOR just becomes multiplication. For any domain D, denote by FD stands ....

....of all the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , so that A(f) f(a) Folding of Long Codes: A Discussion. A function A : FD f1; 1g is said to be folded if A(f) A( f) for all f 2 FD [7]. A codeword of the long code is clearly folded (since A(f) f(a) f(a) A( f) One can assume that the proofs which are purportedly long codes are folded since for any A : FD f1; 1g, one can define a new function A 0 by: A 0 (f) A(f) if f( 0 ) 1 and A 0 (f) A( f) if f( 0 ) ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

....proof systems. To a reader familiar with PCPs we rst give a preview of our constructions. Both our PCPs (of this section and the next) go through the standard path. We start with strong 2 prover 1 round proof systems of Raz [28] apply the composition paradigm [5] and then use the long code of [8] at the bottom level. One warning: in the literature it is common to use a variant of the long code called the folded long code we do not use the folded version. Readers unfamiliar with the terms above may nd elaborations in Section 3.1. 8 As usual, the interesting aspects in the ....

....has the nice feature that Xor just becomes multiplication. For any domain D, denote by FD the space of all Boolean functions f : D f1; 1g. For any set D, jDj denotes its cardinality. We now describe a very redundant error correcting code, called the long code. The long code was rst used by [8], and has been very useful in most PCP constructions since. The long code of an element x in a domain D, denoted LONG(x) is simply the evaluations of all the 2 jDj Boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998.

....if c coloring 2 colorable r uniform hypergraphs is NP hard for every constant c, then NP cPCP 1; 1 k [O(log n) r] for every constant k 1. 2.3 Preliminaries on Long Code We now describe a very redundant error correcting code, called the long code. The long code was first used by [7], and has been very useful in all PCP constructions since. The long code of an element x in a domain D, denoted LONG(x) is simply the evaluations of all the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , ....

....of all the 2 jDj boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , so that A(f) f(a) Folding of Long Codes: A Discussion. A function A : FD f1; 1g is said to be folded if A(f) A( f) for all f 2 FD [7]. A codeword of the long code is clearly folded (since A(f) f(a) f(a) A( f) One can many times assume that the proofs which are purportedly long codes are folded since for any A : FD f1; 1g, one can define a new function A 0 by: A 0 (f) A(f) if f( 0 ) 1 and A 0 (f) A( ....

[Article contains additional citation context not shown here]

M. Bellare, O. Goldreich and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95. 3 This is only place in the analysis where we use the fact that A is folded. 19

No context found.

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998.

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M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and nonapproximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

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M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

No context found.

M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability -- towards tight results. SIAM Journal on Computing, 27(3):804--915, 1998. Preliminary version in Proc. of FOCS'95.

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M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability { towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. Preliminary version in Proc. of FOCS'95.

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