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 "hard-core" 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 non-uniform 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 k-wise 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...

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.

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We present a computational approach to the saddle-point formulation for the Nash equilibria of two-person, zerosum sequential games of imperfect information. The algorithm is a first-order gradient method based on modern smoothing techniques for non-smooth convex optimization. The algorithm requires O(1/ɛ) iterations to compute an ɛ-equilibrium, 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 real-world games. Furthermore, we demonstrate how the algorithm can be customized to a specific class of problems with enormous memory savings. 1

Game-theoretic characterizations of complexity classes have often proved useful in understanding the power and limitations of these classes. One well-known example tells us that PSPACE can be characterized by two-person, perfect-information 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. 114--133]. 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 ...

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 .

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 worst-case 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...

We study the complexity of solving succinct zero-sum 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 EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approxi-mating the value of a succinct zero-sum game to within an additive factor is complete for the class promise-S 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 zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic 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 suc-cinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-S p 2 unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative-factor approximation of succinct zero-sum games is strictly harder

We study the complexity of refereed games, in which two computationally unlimited players play against each other, and a polynomial time referee monitors the game and announces the winner. The players may exchange messages with the referee in private, resulting in a game of perfect recall but incomplete information. We show that any EXPTIME statement can be efficiently transformed into a refereed game in which if the statement is true, the first player wins with overwhelming probability, and if the statement is false, the second player wins with overwhelming probability. We also prove matching PSPACE upper and lower bounds on the complexity of statements that have refereed games that take one round of communication.

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