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28
HardCore Distributions for Somewhat Hard Problems
 In 36th Annual Symposium on Foundations of Computer Science
, 1995
"... Consider a decision problem that cannot be 1 \Gamma ffi approximated by circuits of a given size in the sense that any such circuit fails to give the correct answer on at least a ffi fraction of instances. We show that for any such problem there is a specific "hardcore" set of inputs whic ..."
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Cited by 118 (13 self)
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Consider a decision problem that cannot be 1 \Gamma ffi approximated by circuits of a given size in the sense that any such circuit fails to give the correct answer on at least a ffi fraction of instances. We show that for any such problem there is a specific "hardcore" set of inputs which is at least a ffi fraction of all inputs and on which no circuit of a slightly smaller size can get even a small advantage over a random guess. More generally, our argument holds for any nonuniform model of computation closed under majorities. We apply this result to get a new proof of the Yao XOR lemma [Y], and to get a related XOR lemma for inputs that are only kwise independent. 1 Introduction If you have a difficult computational problem, is it always the case that several independent instances of the problem are proportionately harder than a single instance? In particular, if any algorithm taking less than R resources has failure probability at least ffi for a particular problem on a certai...
Randomized Rounding without Solving the Linear Program
 In Proceedings of the Sixth Annual ACMSIAM Symposium on Discrete Algorithms
, 1995
"... We introduce a new technique called oblivious rounding  a variant of randomized rounding that avoids the bottleneck of first solving the linear program. Avoiding this bottleneck yields more efficient algorithms and brings probabilistic methods to bear on a new class of problems. We give oblivious ..."
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Cited by 89 (6 self)
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We introduce a new technique called oblivious rounding  a variant of randomized rounding that avoids the bottleneck of first solving the linear program. Avoiding this bottleneck yields more efficient algorithms and brings probabilistic methods to bear on a new class of problems. We give oblivious rounding algorithms that approximately solve general packing and covering problems, including a parallel algorithm to find sparse strategies for matrix games.
Gradientbased algorithms for finding nash equilibria in extensive form games
 In Proceedings of the Eighteenth International Conference on Game Theory
, 2007
"... We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires ..."
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Cited by 42 (16 self)
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We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires O(1/ɛ) iterations to compute an ɛequilibrium, and the work per iteration is extremely low. These features enable us to find approximate Nash equilibria for sequential games with a tree representation of about 10 10 nodes. This is three orders of magnitude larger than what previous algorithms can handle. We present two heuristic improvements to the basic algorithm and demonstrate their efficacy on a range of realworld games. Furthermore, we demonstrate how the algorithm can be customized to a specific class of problems with enormous memory savings. 1
On the number of iterations for dantzigwolfe optimization and packingcovering approximation algorithms
 In Proceedings of the 7th International IPCO Conference
, 1999
"... We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed d ..."
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Cited by 22 (2 self)
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We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed decision
A GameTheoretic Classification of Interactive Complexity Classes (Extended Abstract)
 IN PROCEEDINGS OF THE TENTH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 1995
"... Gametheoretic characterizations of complexity classes have often proved useful in understanding the power and limitations of these classes. One wellknown example tells us that PSPACE can be characterized by twoperson, perfectinformation games in which the length of a played game is polynomial i ..."
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Cited by 21 (1 self)
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Gametheoretic characterizations of complexity classes have often proved useful in understanding the power and limitations of these classes. One wellknown example tells us that PSPACE can be characterized by twoperson, perfectinformation games in which the length of a played game is polynomial in the length of the description of the initial position [Chandra et al., Journal of the ACM, 28 (1981), pp. 114133]. In this paper, we investigate the connection between game theory and interactive computation. We formalize the notion of a polynomially definable game system for the language L, which, informally, consists of two arbitrarily powerful players P 1 and P 2 and a ...
Lagrangian Relaxation Based Algorithms for Convex Programming Problems
, 2004
"... This thesis deals with a class of Lagrangian relaxation based algorithms developed in the computer science community in last couple of decades. We present a unified framework for designing such algorithms for a large family of convex programming problems. Our algorithms are based on exponential pote ..."
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Cited by 21 (2 self)
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This thesis deals with a class of Lagrangian relaxation based algorithms developed in the computer science community in last couple of decades. We present a unified framework for designing such algorithms for a large family of convex programming problems. Our algorithms are based on exponential potential functions and given any 2 (0; 1), compute (1 + )approximate solutions in number of iterations proportional to .
On the complexity of succinct zerosum games
 IEEE Conference on Computational Complexity
, 2005
"... We study the complexity of solving succinct zerosum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXPhardness of computing the exact value of a succinct zerosum game by several results on approximating ..."
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Cited by 15 (0 self)
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We study the complexity of solving succinct zerosum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXPhardness of computing the exact value of a succinct zerosum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zerosum game to within an additive factor is complete for the class promiseS p 2, the. To the best of our knowledge, it is “promise ” version of S p 2 the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zerosum game. As a corollary, we obtain, in a uniform fashion, several complexitytheoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT [7] and a recent result by Cai [9] that S p 2 ⊆ ZPPNP. (3) We observe that approximating the value of a succinct zerosum game to within a multiplicative factor is in PSPACE, and that it cannot be in promiseS p 2 unless the polynomialtime hierarchy collapses. Thus, under a reasonable complexitytheoretic assumption, multiplicativefactor approximation of succinct zerosum games is strictly harder
A Competitive Approach to Game Learning
 In Proceedings of the Ninth Annual Conference on Computational Learning Theory
, 1996
"... Machine learning of game strategies has often depended on competitive methods that continually develop new strategies capable of defeating previous ones. We use a very inclusive definition of game and consider a framework within which a competitive algorithm makes repeated use of a strategy learning ..."
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Cited by 15 (4 self)
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Machine learning of game strategies has often depended on competitive methods that continually develop new strategies capable of defeating previous ones. We use a very inclusive definition of game and consider a framework within which a competitive algorithm makes repeated use of a strategy learning component that can learn strategies which defeat a given set of opponents. We describe game learning in terms of sets H and X of first and second player strategies, and connect the model with more familiar models of concept learning. We show the importance of the ideas of teaching set [9] and specification number [2] k in this new context. The performance of several competitive algorithms is investigated, using both worstcase and randomized strategy learning algorithms. Our central result (Theorem 4) is a competitive algorithm that solves games in a total number of strategies polynomial in lg(jHj), lg(jX j), and k. Its use is demonstrated, including an application in concept learning with...
New Limits to Classical and Quantum Instance Compression
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 112
, 2012
"... Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression s ..."
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Cited by 13 (0 self)
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Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression scheme for SAT is a polynomialtime reduction R that maps (ψ 1,...,ψ t) to a string z, such that z lies in some “target ” language L ′ if and only if ∨ j [ψj ∈ SAT] holds. (Here, L ′ can be arbitrarily complex.) ANDcompression schemes are defined similarly. A compression scheme is strong if z  is polynomially bounded in n = maxj ψ j , independent of t. Strong compression for SAT seems unlikely. Work of Harnik and Naor (FOCS ’06/SICOMP ’10) and Bodlaender, Downey, Fellows, and Hermelin (ICALP ’08/JCSS ’09) showed that the infeasibility of strong ORcompression for SAT would show limits to instance compression for a large number of natural problems. Bodlaender et al. also showed that the infeasibility of strong ANDcompression for SAT would have consequences for a different list of problems. Motivated by this, Fortnow and Santhanam (STOC ’08/JCSS ’11) showed that if SAT is strongly ORcompressible,