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34
The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
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Cited by 227 (14 self)
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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
Lower Bounds For The Polynomial Calculus
, 1998
"... We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumpt ..."
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Cited by 49 (5 self)
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We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.
Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting
"... We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, us ..."
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Cited by 31 (9 self)
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We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constantdepth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss
Cutting Planes, Connectivity, and Threshold Logic
 ARCHIVE FOR MATHEMATICAL LOGIC
, 1996
"... Originating from work in operations research the cutting plane refutation system CP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the nonexistence of boolean solutions to associated families of linear inequalitie ..."
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Cited by 17 (3 self)
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Originating from work in operations research the cutting plane refutation system CP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the nonexistence of boolean solutions to associated families of linear inequalities. Polynomial size CP proofs are given for the undirected st connectivity principle. The subsystems CPq of CP , for q 2, are shown to be polynomially equivalent to CP , thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem for CP2proofs and thereby for arbitrary CP proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extension CPLE + , introduced in [9] and there shown to p simulate Frege systems, is proved to be polynomially equivalen...
On the Degree of Ideal Membership Proofs From Uniform Families of Polynomials Over a Finite Field
"... Let f0 ; f1 ; : : : ; fk be nvariable polynomials over a finite prime field Fp . A proof of the ideal membership f0 2 hf1 ; : : : ; fk i in polynomial calculus is a sequence of polynomials h1 ; : : : ; h t such that h t = f0 , and such that every h i is either an f j , j 1, or obtained from h1 ..."
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Cited by 17 (2 self)
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Let f0 ; f1 ; : : : ; fk be nvariable polynomials over a finite prime field Fp . A proof of the ideal membership f0 2 hf1 ; : : : ; fk i in polynomial calculus is a sequence of polynomials h1 ; : : : ; h t such that h t = f0 , and such that every h i is either an f j , j 1, or obtained from h1 ; : : : ; h i\Gamma1 by one of the two inference rules: g1 and g2 entail any Fplinear combination of g1 , g2 , and g entails g \Delta g 0 , for any polynomial g 0 . The degree of the proof is the maximum degree of h i 's. We give a condition on families ffN;0 ; : : : ; fN;k N gN!! of nN variable polynomials of bounded degree implying that the minimum degree of polynomial calculus proofs of fN;0 from fN;1 ; : : : ; fN;k N cannot be bounded by an independent constant and, in fact, is\Omega\Gamma/31 (log(N))). In particular, we obtain an\Omega\Gamma/19 (log(N))) lower bound for the degrees of proofs of 1 (so called refutations) of the (N; m)  system (defined in [4]) formalizing ...
More on the Relative Strength of Counting Principles
 In: Proceedings of the DIMACS workshop on Feasible Arithmetic and Complexity of Proofs
, 1997
"... this paper we attempt to give as complete a presentation as possible. We now outline the structure of the argument, giving references for the key techniques. We use the notion of a kevaluation due to Kraj'icek, Pudl'ak, and Woods [13], incorporating 2 the matching decision trees of Pitassi, Beame, ..."
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Cited by 17 (7 self)
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this paper we attempt to give as complete a presentation as possible. We now outline the structure of the argument, giving references for the key techniques. We use the notion of a kevaluation due to Kraj'icek, Pudl'ak, and Woods [13], incorporating 2 the matching decision trees of Pitassi, Beame, Impagliazzo [15], and built for any small Frege proof using a switching lemma proved with the methods of Beame [4]. Then, as in the argument of Riis [16] and Beame and Pitassi [8], we show that having a kevaluation implies the existence of a certain forest of matching decision trees. Following this we show, using a reduction analogous to that of Beame, Impagliazzo, Kraj'icek, Pitassi, and Pudl'ak [6], that the existence of such a forest implies a small degree Nullstellensatz refutation of an associated family of polynomials. Finally, the proof that such a small degree refutation does not exist is analogous to that of Buss, Impagliazzo, Kraj'icek, Pudl'ak, Razborov, and Sgall [10]. This last is the main new technical contribution and the reader who is familiar with the other aspects of this paper may wish to skip directly to section 8. In section 9 we combine the arguments from the previous sections to show our main results. 2 Frege Proofs and Counting Principles
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
Good Degree Bounds on Nullstellensatz Refutations of the Induction Principle
 IEEE Conference on Computational Complexity
, 1996
"... This paper gives nearly optimal, logarithmic upper and lower bounds on the minimum degree of Nullstellensatz refutations (i.e., polynomials) of the propositional induction principle. 1 Introduction A new propositional proof system based on Hilbert's Nullstellensatz was recently introduced in [2]. ..."
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Cited by 12 (2 self)
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This paper gives nearly optimal, logarithmic upper and lower bounds on the minimum degree of Nullstellensatz refutations (i.e., polynomials) of the propositional induction principle. 1 Introduction A new propositional proof system based on Hilbert's Nullstellensatz was recently introduced in [2]. (See [9] for a subsequent, more general treatment of algebraic proof systems.) In this system, one begins with an initial set of polynomial equations and the goal is to prove that they are not simultaneously solvable over a field such as GF 2 . A proof of unsolvability is simply a linear combination of the initial polynomial equations plus propositional equations x 2 \Gamma x = 0 for all variables x, such that the linear combination is the unsatisfiable equation 1=0. The coefficients of the linear combination may be arbitrary polynomials; the inclusion of the propositional equalities x 2 \Gamma x = 0 restricts the variables x to take on only propositional values 0 and 1. If such a linear...
UNIFORM FAMILIES OF POLYNOMIAL EQUATIONS OVER A FINITE FIELD AND STRUCTURES ADMITTING AN EULER CHARACTERISTIC OF DEFINABLE SETS
, 2000
"... ..."
Proof complexity of pigeonhole principles
 IN PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON DEVELOPMENTS IN LANGUAGE THEORY
, 2002
"... The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m> n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studie ..."
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Cited by 10 (1 self)
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The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m> n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studied combinatorial principle. In this survey we try to summarize what is known about its proof complexity, and what we would still like to prove. We also mention some applications of the pigeonhole principle to the study of efficient provability of major open problems in computational complexity, as well as some of its generalizations in the form of general matching principles.