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Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem

Siddharth Barman

2015
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Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
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We present algorithmic applications of an approximate version of Carathéodory's theorem. The theorem states that given a set of vectors X in R d , for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references
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... which can be found in the work of Pisier (1981) and Carl (1985) . However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n × n payoff matrices A, B, if the number of non-zero entries in any column of A + B is at most s then an ε-Nash equilibrium of the game can be computed in time n O( log s ε 2 ) . This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games-since s can be at most n-the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003) . The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the normalized densest k-subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a subgraph with exactly k vertices with normalized density within ε (in the additive sense) of the optimal in time n O( log d ε 2 ) . Additionally, we show that a similar approximation result can be achieved for the problem of finding a k × k-bipartite subgraph of maximum normalized density. * California Institute of Technology. barman@caltech.edu 1 This bound of d + 1 is tight. 1 Specifically, this approximate version establishes that given a set of vectors X in the p-unit ball 2 with norm p ∈ [2, ∞), for every vector µ in the convex hull of X there exists an ε-close-under the p-norm distance-vector µ ′ that can be expressed as a convex combination of 4p ε 2 vectors of X. A notable aspect of this result is that the number of vectors of X that are required to express µ ′ , i.e., 4p ε 2 , is independent of the underlying dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier [30] and Carl [10] . However, in this paper we present a self-contained proof of this result, which we proceed to outline below. The author was made aware of the connection with Maurey's lemma after a preliminary version of this work had appeared. To establish the approximate version of Carathéodory's theorem we use the probabilistic method. Given a vector µ in the convex hull of a set X ⊂ R d , consider a convex combination of vectors of X that generates µ. The coefficients in this convex combination induce a probability distribution over X and the mean of this distribution is µ. The approach is to draw b independent and identically distributed (i.i.d.) samples from this distribution and show that with positive probability the sample mean, with an appropriate number of samples, is close to µ under the p-norm distance, for p ∈ [2, ∞). Therefore, the probabilistic method implies that these exists a vector close to µ that can be expressed as a convex combination of at most b vectors, where b is the number of samples we drew. Note that in this context applying the probabilistic method is a natural idea, but a direct application of this method will not work. Specifically, a dimension-free result is unlikely if we first try to prove that the ith component of the sample mean vector is close to the ith component of µ, for every i ∈ [d]; since this would entail a union bound over the number of components d. Bypassing such a component-wise analysis requires the use of atypical ideas. We are able to accomplish this task and, in particular, bound (in expectation) the p-norm distance between µ and the sample mean vector via an interesting application of Khintchine inequality (see Theorem 1). Given the significance of Carathéodory's theorem, this approximate version is interesting in its own right. The key contribution of the paper is to substantiate the algorithmic relevance of this approximate version by developing new algorithmic applications. Our applications include additive approximation algorithms for (i) Nash equilibria in two-player games, and (ii) the densest subgraph problem. These algorithmic results are outlined below. Algorithmic Applications Approximate Nash Equilibria. Nash equilibria are central constructs in game theory that are used to model likely outcomes of strategic interactions between self-interested entities, like human players. They denote distributions over actions of players under which no player can benefit, in expectation, by unilateral deviation. These solution concepts are arguably the most well-studied 2 That is, X is contained in the set {v ∈ R d | v p ≤ 1}.

doi:10.1145/2746539.2746566
dblp:conf/stoc/Barman15
fatcat:x2lt5v63fve73k33mva47x7ytu