P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, T. Pitassi. "The Relative Complexity of NP Search Problems," J. Comput. Syst. Sci. 57, 1, pp. 13--19, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in polynomial time. Papadimitriou, Schafffer and Yannakakis [15] show that finding a local optimum for some problems in PLS is PSPACE complete if we insist on a specific initial solution for the local search procedure. Other investigations into the relative difficulty of NP search problems include [2, 4]. The properties of the standard algorithm on hypercubes, and its ability to converge on a local optima has been by Tovey [20] 3 A Model of Energy Landscapes We model the energy landscape for a given problem as a random directed graph G = V, E) where IVI = N, IEI = k N, together with an ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazo, and T. Pitassi. The relative complexity of NP search problems. In ACM Symposium on Theory of Computing, pages 303-314, 1995.

....ed in polynomial time. Papadimitriou, Scha fer and Yannakakis [14] show that nding a local optimum for some problems in PLS is PSPACE complete if we insist on a speci c initial solution for the local search procedure. Other investigations into the relative diculty of NP search problems include [2, 4]. 3 A Model of Energy Landscapes We model the energy landscape for a given problem as a random directed graph G = V; E) where jV j = N , jEj = k N , together with an objective function f : V that imposes a total order among the vertices, so that f(v i ) f(v j ) for 1 i j N . We ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In ACM Symposium on Theory of Computing, pages 303-314, 1995.

....n ) log 2 n From Theorems 17 and 21 we know that it lies in the interval [1=3; 1=2] 5. Is it true that for m n and pj(m n) every polynomial calculus proof of onto FPHP m n over a eld of characteristic p must have degree n (1) This is known for the weaker Nullstellensatz proof system [34]. ....

Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of NP search problems. In: Proceedings of the 27th ACM STOC. (1995) 303-314

....have an obvious time consuming solution: walk from the given vertex to the other end of its path. It seems likely that PPAD is strictly contained by PPA, although an actual separation would imply P6=NP. As weaker evidence, it is known that PPA G strictly contains PPAD G for generic oracles G [4]. Roughly stated, PPAD problems may be easier because of the following slight advantage: if you jump to a random vertex, you know which way to walk to avoid returning to the original source vertex. This advantage occurs naturally in topology: several classical existence results (Sperner, ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, T. Pitassi, The relative complexity of NP search problems, J. Comput. System Sci. 57 (1) (1998) 3--19.

.... unit ideal are well known (here d is the degree and n is the number of variables of the input polynomials) These bounds are known to be sharp due to [13] for the first bound and for the second bound due to the example of LazardMora Philippon (see [2] In the proof system theory (see e.g. 3] [7], 6] 4] 14] 10] a similar question is studied when among input polynomials f 1 ; f k 2 F [X 1 ; Xn ] necessarily the polynomials X 2 i Gamma X i , 1 i n appear (let us call such a system of input polynomials a boolean system) Then the known methods [13] 5] for ....

....degree which describes a modification of the into pigeon hole principle (an exposition of this method see also in [10] It holds over an arbitrary field. But for many other systems of polynomials the issue of lower bounds still remains open. Let us also mention that in the earlier papers [3] [7], 6] 4] the methods for obtaining somewhat weaker than linear bounds were exhibited. It seems to be an interesting general question, how to obtain lower bounds for boolean Nullstellensatz refutations. In this paper we develop an approach which allows to produce explicitly a system of ....

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, T. Pitassi. The relative complexity of NP search problems. Proc. ACM STOC, 1995, p. 303--314.

....F M the claim follows. Now note that there are at most c different submodules of P oly(M; k; t) hence some t 0 t c has the above property. 33 q.e.d. Note that the theorem yields also non constant PC lower bounds for the systems encoding the pigeonhole principle (see e.g. one of [3, 21]) A linear lower bound was proved in [21] another proof of non constant lower bound is in [18] Finally we note that the same method yields Omega Gammaelds N) lower bounds for Nullstellensatz proofs. The difference allowing to save one log is the following. The space PC t (M; F ) is a span ....

P. Beame, S. A. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems. in: "Proceedings of the 27th ACM Symposium on Theory of Computing", 303-314. ACM Pres, 1995.

....knew the corresponding special cases of Theorems 3 and 4. In any event, the next section discusses all the published work known to us. 6 SAMUEL R. BUSS 2.2. Prior constructions of designs. Designs have been used to obtain several degree lower bounds for Nullstellensatz refutations. First, [1] gave designs of degree p n for polynomials which express a version of the pigeonhole principle. This thereby established a p n lower bound on the degree of Nullstellensatz refutations of the pigeonhole principle. It is still open whether this lower bound can be improved substantially, ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, in Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 303--314. LOWER BOUNDS ON NULLSTELLENSATZ PROOFS 13

....of the degree complexity of algebraic proofs has many applications. As shown in [9] degree lower bounds for certain unsatisfiable formulas imply lower bounds for bounded depth Frege proofs. Also, degree lower bounds are related to separation results for NP search classes, as shown in [18]. We review these connections, lower bounds as well as lower bound techniques. In Section 5, we discuss a stronger version of small degree algebraic proofs, Grobner proofs, which were first studied in [16] We review their results, and discuss connections and applications to Frege lower bounds. ....

....in (2) and we are left with n 1 Gamma n = 1. The more general version of the propositional pigeonhole principle states that there is no 1 1 map from n 1 to n. For each n, the general pigeonhole principle can be expressed by equations (1) and (3) above, and is denoted by :PHP n . In [18], it is shown that for all sufficiently large n, PHP n requires degree p n algebraic refutations over any field. The best upper bound is a degree n algebraic refutation, and we conjecture that this is optimal. An open problem is to show that algebraic refutations of :PHP n over any field ....

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Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., and Pitassi, T. (1995) The relative complexity of NP search problems, in Proceedings of the 27th ACM Symposium on Theory of Computing, 303--314.

..... 2 The unique endnode principle is very closely related to the search problems in the class PPA introduced by Papadimitriou [18] In a nutshell, the class PPA contains search problems where, given one node of degree one, it is required to find another node which does not have degree two. In [4], it is shown that the unique endnode principle is equivalent to the mod 2 counting principle in the setting of constant depth Frege proofs. For 0 k n, let B k = X i jk n r i;j (15) S k = X ik j n r i;j (16) Thus B k is the number of edges r i;j for i j both of whose endpoints are ....

....the latter is equivalent (by constant depth polynomial size Frege proofs) to the pigeonhole principle. Moreover, the nonunique endnode principle of the previous section is easily seen to be equivalent (by constant depth polynomial size Frege proofs) to the undirected s t connectivity principle. In [4], the nonunique endnode principle is shown to be equivalent (by constant depth polynomial size Frege proofs) to Ajtai s parity principle. By [1, 3] the parity principle is strictly stronger than the pigeonhole principle. Thus it follows from this section that the undirected s t connectivity ....

P. Beame, S. Cook., C. Papadimitriou and T. Pitassi. The relative complexity of NP search problems. Manuscript, November 1993.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, in Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 303--314.

....in their degree. Kaltofen and Trager [16] give efficient algorithms for such things as computing greatest common divisors, when the input polynomials are accessed only through queries. Another example is NP search problems, where the input is an exponentially large search space. Beame et al. [2] give a natural type 2 description of such problems. For a third example, Cook and Urquhart [7] show how to use higher type polynomial time functions to give constructive meaning to number theory theorems proved in a certain formal system. An oracle Turing machine (OTM) is a Turing machine that ....

.... [20] see also [18] and [17] For the case of the polynomial hierarchy, where the oracle represents a function whose growth does not affect the input size, this point of view was taken by Townsend [27] see also [28] and [29] For the case of NP search classes, this was done by Beame et al. [2]. Baker, Gill, and Solovay [1] defined the polynomial time hierarchy PH relativized to an oracle A. By taking our second point of view, their work defines the type 2 polynomial time hierarchy PH, in which each member relation takes an oracle A as an argument, in addition to a string argument. ....

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Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., and Pitassi T. (1995), The relative complexity of NP search problems, in "Proceedings, 27th ACM Symposium on Theory of Computing", pp. 303--314.

.... factor not dividing p then there is no proof of Count q (R) in S 2 (R) Count p (R) 10 Remarks It is interesting to compare the degree lower bound for the Nullstellensatz refutations of onto PHP N p N with the degree lower bound for PHP N s N using the quite different construction in [5]. If we take p = 2 and N = 4 Gamma 2 ) 2, then the degree lower bound from Theorem 12 is d = 2 Gamma 1 which satisfies N = d(d 1) 2, i.e. the same degree as in [5] despite the more stringent conditions required in Theorem 12. For p 2 Theorem 12 does not give as large a degree ....

.... of onto PHP N p N with the degree lower bound for PHP N s N using the quite different construction in [5] If we take p = 2 and N = 4 Gamma 2 ) 2, then the degree lower bound from Theorem 12 is d = 2 Gamma 1 which satisfies N = d(d 1) 2, i.e. the same degree as in [5] despite the more stringent conditions required in Theorem 12. For p 2 Theorem 12 does not give as large a degree bound. It would be interesting to improve the lower bound and close the gap between p Gamma 1 and 2 Gamma 1 or to reduce the size of N required to achieve it. ....

Paul W. Beame, Stephen A. Cook, Jeff Edmonds, and Russell Impagliazzo. The relative complexity of NP search problems. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, Las Vegas, NV, May 1995. To appear.

....from the initial equations in degree d. The only other lower bound known for the Polynomial Calculus, due to Kraj icek[16] uses important ideas from Ajtai [1] linking the lower bound in question to the representation theory of the symmetric group. Formulae Reference Lower bound Notes PHP [12] O(n 1=4 ) HN) nearly optimal PHP [19] O(n 1=2 ) PC) nearly optimal ontoPHP [3] O(n) HN) nearly optimal IND [7] O(logn) HN) nearly optimal Homesitting [11, 6] O(n 1=2 ) HN) Graph [14] O(n) HN) Char(F ) 6= 2 Modp [4, 1] nonconstant (HN) Modp [8] n Omega Gamma26 (HN) Modp [16] ....

Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., and Pitassi, T. The relative complexity of NP search problems, 27th ACM STOC, pp. 303-314.

....that Horn clause resolution as well as boundedclause size resolution cannot be simulated by constant degree Nullstellensatz proofs. Other degree lower bounds for Nullstellensatz were known prior to our result, but the hard examples did not separate resolution from Nullstellensatz. For example, in [1] it was shown that the pigeonhole principle from m pigeons to n holes requires degree p n Nullstellensatz refutations, and in [5] it was shown that the mod p counting principle requires degree n Omega Gamma1 Nullstellensatz refutations over GF q (p; q distinct primes) However, both of these ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, in Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 303--314.

....that there is a prime factor q of n which is not a prime factor of m. Then the size of any constant depth F proof of Count N n from instances of the axiom schema Count M m is at least 2 N Omega Gamma1 . 4. Design based lower bounds We now introduce the combinatorial notion of designs (cf. Beame et al. 1995, Clegg et al. 1996) which can be used for proving lower bounds on the degree of Nullstellensatz proofs. We ll give two applications of designs: first, we give an alternative proof of Theorem 3.2, and second, we give a linear lower bound for the degree of Nullstellensatz refutations of the (N; ....

....are examples showing that bounded depth Frege with MOD p gates is strictly stronger than Nullstellensatz, it is not possible in general to eliminate all extension axioms without too big increase of the degree. One such example is the weak pigeonhole principle PHP 2n n Paris et al. 1988) Beame et al. 1995) , and another example, the house sitting tautologies of Clegg et al. 1996) was already mentioned in Section 5: it separates the polynomial calculus from the Nullstellensatz system. We conclude by observing that it is possible to introduce extension polynomials in an unstructured way, i.e. ....

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems. In Proceedings of the 27th ACM STOC, 1995, 303--314.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, T. Pitassi. "The Relative Complexity of NP Search Problems," J. Comput. Syst. Sci. 57, 1, pp. 13--19, 1998.

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P. Beame, S. A. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The Relative Complexity of NP Search Problems, in: Proc. 27th Annual ACM Symposium on the Theory of Computing, (1995), pp. 303-314.

No context found.

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In Proceedings of 27th ACM Symposium on Theory of Computing, pages 303-314, 1995.

No context found.

P. Beame, S. Cook., C. Papadimitriou and T. Pitassi. The relative complexity of NP search problems. Manuscript, November 1993.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, Journal of Computer and System Sciences, 57 (1998), pp. 3--19.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, in Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 303--314.

No context found.

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi, The relative complexity of NP search problems, Journal of Computer and System Sciences, 57 (1998), pp. 3--19.

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P. Beame, S. A. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. Journal of Computer and System Sciences, 57:3-19, 1998.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In Proc. 27th ACM Symp. on Theory of Computing, pages 303-- 314, 1995.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In Proceedings of 27th ACM Symposium on Theory of Computing, pages 303314, 1995.

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P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In Proceedings of 27th ACM Symposium on Theory of Computing, pages 303--314, 1995.

No context found.

P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi. The relative complexity of NP search problems. In Proceedings of 27th ACM Symposium on Theory of Computing, pages 303314, 1995.

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