L.G. Valiant and V.V. Vazirani, "NP is as easy as detecting unique solutions", Theoret. Comput. Sci., Vol. 47, pp. 85--93, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... has no satisfying assignment then 62 B. Now suppose has exactly one satisfying assignment. Then i 2 1SAT for all i, and A ( 1 ; n ) will be the satisfying assignment of , and hence 2 B. This shows that B 2 P is a solution of (1SAT; SAT) By Valiant and Vazirani s result [60] we get SAT 2 RP. Proof of Theorem 5. Let GI 2 P mc(c log n) for some c via f . We show GI 2 RP. Let (G 0 ; G 1 ) be a pair of graphs given as input. We run Algorithm 1. We claim that the algorithm will never accept non isomorphic graphs and will accept isomorphic graphs with probability at least ....

L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Comput. Sci., 47:85-93, 1986.

....counting property of circuits is Few hard, thus raising the lower bound. We first prove that for any nontrivial property A, there exists a predicate Q such that at least one of Counting(A) and Counting(A) is m hard for USATQ , where, for any boolean predicate Q, USATQ is defined (see [VV86] as follows. #USATQ (x) # SAT (x) if #(x) 1 , Q(x) otherwise. 5 The flavor of the following lemma, which we prove here for completeness, is implicit in the comments at the end of Section 5.1 of [BS00] Lemma 3.1 Let A N. m Counting(A) m Counting(A) Proof ....

.... 0, let f(e(x) k, for some k such that 1 m. Then for all j such that k j m, # c (q j ) a, and thus, for k j m, q j Counting(A) Also, # c (q k ) a 1, and hence q k Counting(A) and by the definition of n, n = k, and so we have n = f(e(x) q Valiant and Vazirani [VV86] prove that for for every Q, USATQ is hard for NP, where randomized is as per Valiant Vazirani ( VV86] see also [BS00] So the following result (which is a more refined, detailed statement of the flavor of of [BS00] Theorem 5.2) follows from Lemma 3.1. Proposition 3.4 [BS00] Let A ....

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(3):85--93, 1986. 13

....These reductions are based on the Isolation Lemma of Mulmuley, Vazirani and Vazirani. Combinatorially our results can be viewed as simple (logspace) transformations of existential quantifiers into counting quantifiers in graphs and shallow circuits. 1 Introduction Valiant and Vazirani [VV] gave a randomized reduction from NP to PhiP , using universal hash functions. PhiP is the counting version of NP , also defined by polynomial time nondeterministic Turing machnines. A language F belongs to PhiP if there is a polynomial time nondeterministic Turing machine M such that F ....

L. G. Valiant and V.V. Vazirani, "NP is as easy as detecting unique solutions," Theoretical Computer Science, 47 (1986), pp. 85-93.

....the truth assignment that satis es the maximum number of a set of clauses is unique [16] As for USAT and UOASAT, UMINSAT is easily proved co NP hard, but it is not clear how to reduce SAT to it. USAT is complete for the class D [24] under the randomized reduction vv m of Valiant Vazirani [31] and, as recently proved, USAT is not in co D , which contains NP[co NP, unless the polynomial hierarchy collapses [8, 7] We now show that UMINSAT, and thus CWA consistency checking, is at least as hard as USAT. Lemma 3.6 USAT m UMINSAT. Proof: Let E(x 1 : x n ) be a Boolean ....

L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(1):85-93, 1986.

....of W to Unique W It would be interesting to know quite a bit more than we presently do about the calculus of the operators 9 Delta, 8 Delta, BP Delta, RP Delta and Delta over the G[t] classes. For example, do the following analogs of the theorems (respectively) of Valiant and Vazirani [VV86] and Toda [Tod91] hold (1) N [t] BP Delta (2) H[t] BP Delta Analogs in parameterized complexity (if they exist) of familiar structural theorems generally present significant and novel difficulties and are in most cases not presently known. A parameterized analog of Ladner s density ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science 47 (1986), 85--93. 16

.... given a graph G, is G uniquely k colourable for example is NP hard [2] the problem given a graph G and a k colouring of G, is there another k colouring of G is NP complete [6] If there exists a polynomial time algorithm which finds a 3 colouring of a uniquely 3 colourable graph then NP=RP [24]. For complexity classes related to uniqueness problems and a survey of related results, see [15, 25] We close by stating some open questions: ffl What is the smallest f(k) such that k Colourability on graphs of maximum degree Delta f(k) is still NP complete ffl Or, more general, for given ....

Valiant, L.G., Vazirani, V., NP is as easy as detecting unique solutions, Theoret. Comput. Sci. 47, 1986, 85-93.

....[Siv91] showed that if SAT is O(log n) mc, then UniqueSAT is in P, i.e. there exists a polynomial time algorithm that distinguishes the cases of a formula having no satisfying assignments from a formula having exactly one satisfying assignment. It is known that if UniqueSAT is in P, then NP = RP [VV86]. Questions regarding reductions to pselective sets and membership comparable sets have connections with several interesting questions about function classes. See the survey by Buhrman, Fortnow, and Torenvliet [BFT97] for details. Beigel [Bei90] showed that if NP contains a P bi immune set then, ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986. 9

....other application areas such as artificial intelligence and optimization research. It is important to emphasize that the new method is completely general and can be easily adapted to watermarking any other NP complete problem. Our method is based on a combinatorial result of Valiant and Vazirani [16] that randomly reduces a given instance of SAT to an instance which has (with high probability) Sig. Len (bits) 4 8 12 16 20 24 28 32 Ave. # of Sols 4.1 2.1 1.3 0.5 0.2 0.3 0.1 0.0 Min Max Sols 2 5 1 4 0 3 0 1 0 1 0 1 0 1 0 0 Ave. Discrep. 5 .6 .72 .5 .32 .42 .18 0 Table 1: The relationship ....

.... to solve the SAT problem and many efficient public domain packages are available [1, 2, 11, 10] The economic importance of efficient SAT solving is well illustrated by a number of FPGA based application specific computers exclusively built for solving SAT problems [9] Valiant and Vazirani [16] proved that the number of solutions of a NP complete problem, which can vary from zero to exponentially many, does not impact its inherent intractability: in fact, if there is a polynomial time algorithm for finding solution to SAT instances having a unique solution, then NP=RP. The main ....

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L.G. Valiant and V.V. Vazirani. "NP is as easy as detecting unique solutions". Theoretical Computer Science, vol.47. pp. 85-93. 1986.

....set. We have shown that extra queries to B allow us to compute extra functions in polynomial time, provided that P 6= NP. However, it is not known whether extra queries to B allow us to solve extra decision problems in polynomial time. For example, Blass and Gurevich [BG82] Valiant and Vazirani [VV86], and Papadimitriou and Yannakakis [PY84] have considered the Others have defined NP hardness in terms of Turing reductions. The results to follow do not apply to that kind of NP hardness. class D = fL 1 Gamma L 2 j L 1 ; L 2 2 NPg; it is not known whether P 6= NP implies that D 6= co NP ....

L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

....and Sivakumar establishes immediately the following corollary in the light of two results discussed in the rst part of this article ( GH00] see there for a discussion of attribution of the rst of these results) namely, 1. If (9Q) USATQ 2 P] then P = Few (and thus P = UP and P = FewP) 2. VV86] If (9Q) USATQ 2 P] then R = NP. Corollary 1.2 If SAT disjunctively reduces to a sparse set, then P = Few and R = NP. Furthermore, Arvind, K obler, and Mundhenk [AKM96] prove that if SAT disjunctively reduces to a sparse set, then PH = P . However, in light of Corollary 1.2, clearly the ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.

....that P (x 1 ; x n ) f(x 1 ; x n ) is at least 1 Gamma ffl. Definition 3. A perceptron of size s and order m is a depth 2 circuit with a threshold gate at the top and s AND gates of fanin m on the bottom. We will make use of the following result due to Valiant and Vazirani [17]: Theorem 4. Let S f0; 1g be nonempty. Suppose w 0 ; w 1 ; w k are randomly chosen from . Let S 0 = S and let S i be fv 2 S : v Delta w 0 j v Delta w 1 j Delta Delta Delta j v Delta w i j 0 (mod 2)g for each i 2 f0; kg, where Delta denotes inner product. Let Pr k ....

L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Comput. Sci., 47:85--93, 1986.

....a set L in NP that is NP complete under reductions that make one query to L but not under traditional many one reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the surprising power of solving satisfiability on formulae with at most one satisfying assignment or equivalently detecting unique solutions to NP problems. On sabbatical from Yale 1996 97. Address: Dept. of Computer Science, University of Maryland at College Park, College Park, MD ....

....only if M (x) accepts. We say A is 1 tt complete if M can make only one query to A. We say two sets A and B are isomorphic if A m reduces to B via a polynomial time computable function that is one to one, onto and polynomial time invertible. 3 Detecting Unique Solutions Valiant and Vazirani [VV86] show how to randomly map satisfiable formula to those with unique satisfying assignments. Lemma 3.1 There exists a probabilistic polynomial time function f such that for all boolean formulae OE in n variables ffl If OE 62 SAT then f(OE) is never satisfiable. ffl If OE 2 SAT then with ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

....gates elsewhere that computes fn : f0; 1g f0; 1g with error probability at most 1=4. It will be clear that the proof works for any error probability bounded away from 1=2. By using a standard probabilistic simulation of AND OR by MODm gates (we can either use Valiant and Vazirani s lemma [VV86] as in [AH90] or the RazborovSmolensky simulation [Raz87, Smo87] and by the argument explained above, obtain N = O(n) deterministic ACC circuits C 1 ; CN consisting of MODm gates and fan in log n AND gates such that for every x 2 f0; 1g N ; Thus, if we let fg i ; ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoret. Comput. Sci., 47:85--93, 1986.

....other application areas such as artificial intelligence and optimization research. It is important to emphasize that the new method is completely general and can be easily adapted to watermarking any other NP complete problem. Our method is based on a combinatorial result of Valiant and Vazirani [16] that randomly reduces a given instance of SAT to an instance which has (with high probability) exactly one satisfying assignment. The construction successively conjoins constraints to the original formula to produce a series of formulas that have a monotonically decreasing number of solutions. By ....

.... to solve the SAT problem and many efficient public domain packages are available [1, 2, 11, 10] The economic importance of efficient SAT solving is well illustrated by a number of FPGA based application specific computers exclusively built for solving SAT problems [9] Valiant and Vazirani [16] proved that the number of solutions of a NP complete problem, which can vary from zero to exponentially many, does not impact its inherent intractability: in fact, if there is a polynomial time algorithm for finding solution to SAT instances having a unique solution, then NP=RP. The main ....

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L.G. Valiant and V.V. Vazirani. "NP is as easy as detecting unique solutions". Theoretical Computer Science, vol.47. pp. 85-93. 1986.

....promise that L has exactly zero or one vector of length less than r. 1 Introduction Is it easier to decide instances of NP hard problems when they are given with the additional promise that the associated search problem has exactly zero or one solution Over a decade ago, Valiant and Vazirani [VV86] proved a beautiful result that shows that this is not the case. More formally, they gave a probabilistic many one reduction from the NP complete Boolean formula satisfiability problem to the problem of deciding whether a Boolean formula is satisfiable under the promise that it has either zero or ....

.... Is there a non zero vector v 2 L such that kvk 2 r In particular, the non parsimoniousness of the reduction used in the NP completeness of the short lattice vector problem rules out the following straightforward reduction from SAT to USVP: given an instance of SAT, apply the reduction of [VV86] and produce an instance of SAT that has either zero or one satisfying assignment, then map it to an instance (L; r) of the shortest lattice vector problem by applying Ajtai s reduction. The difficulty is that (L; r) may have more than one vector of length r even though has only one ....

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986. 7

....From here on, rather than pursue quasilinear classes for their own sake, we emphasize similarities and differences in three important areas that have received much attention in the polynomial case. Section 3 shows that the randomized reduction from NP to parity given by Valiant 2 and Vazirani [VV86] and used by Toda [Tod91] which was previously proved by constructions that run in quadratic time (see [VV86, Tod91, CRS93, Gup93, Cha94] can be made to run in quasilinear time. Our construction also markedly improves both the number of random bits needed and the success probability, and uses ....

....in three important areas that have received much attention in the polynomial case. Section 3 shows that the randomized reduction from NP to parity given by Valiant 2 and Vazirani [VV86] and used by Toda [Tod91] which was previously proved by constructions that run in quadratic time (see [VV86, Tod91, CRS93, Gup93, Cha94] can be made to run in quasilinear time. Our construction also markedly improves both the number of random bits needed and the success probability, and uses errorcorrecting codes in an interesting manner. However, whether quasilinear analogues of the full Toda ....

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comp. Sci., 47:85--93, 1986.

....for the membership in SAT of the given instances, then UniqueSAT 2 P. Recall that UniqueSAT is the promise problem [ESY84] of distinguishing between unsatisfiable Boolean formulas and Boolean formulas with exactly one satisfying assignment. By the seminal result of Valiant and Vazirani [VV86] UniqueSAT is NP hard under randomized reductions. Prior to our work, three sets of authors [Ogi94, BKS94, AA94] independently showed that if SAT is c log n membership comparable for c 1, then NP = P. The question of whether their result can be extended to O(log n) membership comparability ....

....for the polynomial P a ; by checking if (the coefficients of) one of these polynomials, when interpreted as a 01 assignment, satisfies the input instance of UniqueSAT, we can discover a satisfying assignment of if one exists. By applying the randomized reduction of Valiant and Vazirani [VV86] we obtain: Corollary. If SAT is O(log n) membership comparable, then NP = RP. If SAT is polynomial time truth table reducible to a p selective set, then SAT is O(log n) membership comparable [Ogi94, BKS94, AA94] Therefore, we obtain: Corollary. Tod91, Bei88] If SAT is reducible to a ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986. 9

....of the Cook vs. weaker reductions question in this context is to ask if search reduces to decision for NP problems via non adaptive Cook reductions, that is, Cook reductions in which all questions to the decision oracle are submitted simultaneously. The seminal result of Valiant and Vazirani [VV86] demonstrates, among other things, a probabilistic non adaptive Cook reduction from the search versions for standard NP complete languages to their decision versions. We offer the notion of a partially publishable proof system that we study in this paper as a meaningful, interesting, and natural ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986. 12

....M and a string x, if M on input x has exactly one accepting path, then the path is computable in deterministic polynomial time. For M and x as above, the nondeterministic computation of M on x can be viewed as a (binary) tree T . Using the randomized hashing technique of Valiant and Vazirani [VV86] one can construct subtrees T 1 ; Tm of T , all having the same root as T , such that if T has an accepting path then, say, m=4 of T 1 ; Tm have exactly one accepting path. Then from property ( for the T k s having exactly one accepting path, one can compute this path. Thus, ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science 47:85\Gamma93, 1986.

....n k . Consider the NP language L consisting of tuples h0 n ; i; j; r 1 ; r n ki such that there exists a computation path p of M such that 1. p is an accepting computation path of M(0 n ) 2. p r = 1 for all , 1 j and 3. The ith bit of p is one. Valiant and Vazirani [VV86] show that if M(0 n ) accepts and the r s are chosen at random then with probability at least one fourth there is a j, 0 j n k such that exactly one path p ful lls conditions 1 and 2 above. Consider the polynomial time computable distribution with 0 (h0 n ; i; j; r 1 ; ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.

....correctly computes all accepting computations with non trivial probability. SEPARATION OF NP COMPLETENESS NOTIONS 72 We can prove that Turing completeness is di erent from truth table completeness in NP under the above hypothesis. The proof uses the randomized reduction of Valiant and Vazirani [84] that isolates the accepting computations. We de ne L as in the proof of Theorem 7.2. Let S = fh0 n ; k; r 1 ; r 2 ; r k ; ii j 9v such that v is an accepting computation of M , v:r 1 = v:r 2 = v:r k = 0; and the ith bit of v = 1g where v:r i denotes the inner product over ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.

....difference of two NP sets. But it is not known to be complete for D P . Blass and Gurevich [BG82] showed that USAT is complete for D P if and only if it is hard for NP. Furthermore, they constructed an oracle such that USAT is not complete for D P . Note, however, that Valiant and Vazirani [VV86] showed that USAT is NP hard under randomized many one reductions. As a lower bound, USAT is coNP hard, but it is not known to belong to coNP. In fact, USAT is not in coD P , unless the Polynomial Hierarchy collapses [CKR95] It is widely conjectured that USAT is an intermediate problem with ....

Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theoretical Computer Science 47(1) (1986) 85-93

....0, no 2 n e time bounded Turing machine correctly computes all accepting computations with non trivial probability. We can prove that Turing completeness is different from truth table completeness in NP under the above hypothesis. The proof uses the randomized reduction of Valiant and Vazirani [VV86] that isolates the accepting computations. We define L as in the proof of Theorem 2. Let S = #0 n , k,r 1 , r 2 , r k , i# #v such that v is an accepting computation of M, v.r 1 = v.r 2 = v.r k = 0, and the ith bit of v = 1 where v.r i denotes the inner product over ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

....We obtain better bounds on CD complexity using extractor graphs. These graphs are usually used for derandomization. However these improved bounds only apply to most of the strings. We also give a new simpler proof of Sipser s Lemma and show how it implies the important Valiant Vazirani lemma [VV85] that randomly isolates satisfying assignments. Surprisingly, Sipser s paper predates the result of Valiant and Vazirani. We define CND complexity, a variation of CD complexity where we allow nondeterministic computation. We prove a lower bound for CND complexity where we show that there exists ....

....x 2 A such that CND p;A (x) O(log(jxj) for some polynomial p, contradicting the fact that for all x 2 A, CND 2 p jxj ;A (x) jxj=4. 2 8. 3 Isolating Satisfying Assignments In this section we take a Kolmogorov complexity view of the statement and proof of the famous Valiant Vazirani lemma [VV85]. The Valiant Vazirani lemma gives a randomized reduction from a satisfiable formula to another formula that with a non negligible probability has exactly one satisfying assignment. We state the lemma in terms of Kolmogorov complexity. Lemma 37 There is some polynomial p such that for all OE in ....

L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. In Proceedings of the 17th ACM Symposium on the Theory of Computing, pages 458--463, 1985.

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L.G. Valiant and V.V. Vazirani, "NP is as easy as detecting unique solutions", Theoret. Comput. Sci., Vol. 47, pp. 85--93, 1986.

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L.G. Valiant and V.V. Vazirani, "NP is as easy as detecting unique solutions", Theoret. Comput. Sci., Vol. 47, pp. 85--93, 1986.

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L. Valiant and V. Vazirani, NP is as easy as detecting unique solutions, Proc. ACM Symp. on Theory of Computing, 1985.

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L.G.Valiant and V.V.Vazirani, "NP is as easy as detecting unique solutions,"Theoretical Computer Science 47 (1986), 85-93.

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L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions, Theoret. Comp. Sci. 47(3):85--93, 1986.

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Leslie Valiant and Vijay Vazirani. NP is as easy as detecting unique solutions. Theoretical

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

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Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(1):85--93, 1986.

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986. 10

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

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R.G. Valiant and Vazirani V.V. Np is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986.

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L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions, Theoret. Comp. Sci. 47(3):85--93, 1986.

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. In Proceedings of the 17th ACM Symposium on the Theory of Computing, pages 458-463, 1985.

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Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(1):85--93, 1986.

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L. Valiant and V.V. Vazirani, "NP is as Easy as Detecting Unique Solutions", Theoretical Computer Science, Vol. 47, 1986, pp. 85--93.

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Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(1):85--93, 1986.

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L.Valiant, V.Vazirani, NP is as easy as detecting unique solutions, Theoretical Computer Science, 47 (1986) 85-93

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986. 12

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85--93, 1986. 9

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Valiant L G and Vazirani V V. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(1):85-93, 1986

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(3):85--93, 1986. 21

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L. Valiant and V. Vazirani, NP is as easy as detecting unique solutions. In Theoretical Computer Science 47, (1986), 85-93.

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L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. In Proc. 17th ACM Symp. Theory of Computing, pages 458--463, 1985.

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L. Valiant and V. Vazirani, NP is as easy as detecting unique solutions,The- oret. Comput. Sci., 47 (1986), pp. 85--93.

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L.Valiant, V.Vazirani, NP is as easy as detecting unique solutions, Theoretical Computer Science, 47 (1986) 85-93

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