用Carathéodory定理的近似形式逼近Nash平衡和稠密子图
Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carathéodory's Theorem
摘要：
我们提出的算法应用的近似版本的Carathéodory定理。该定理指出，给定$\mathbb{R}^d$中的一组向量$X$，对于$X$的凸包中的每个向量，存在一个$\varepsilon$ -闭（对于$2\leq p < \infty$，在$p$ -范数距离下）向量，该向量可以表示为$X$的至多$b$向量的凸组合，其中，约束$b$取决于$\varepsilon$和范数$p$，并且与维度$d$无关。这个定理可以通过实例化Maurey引理来推导，早期的参考文献可以在Pisier（1981）和Carl（1985）的工作中找到。然而，在本文中，我们提出了一个独立的证明这一结果。利用这个定理，我们建立了在一个具有$ n \times n$支付矩阵$A, B$的双矩阵博弈中，如果$A+B$的任何一列中的非零元素的个数至多为$s$，那么这个博弈的$\varepsilon$ -纳什均衡可以在时间上计算出$n^{O\left(\frac{\log s }{\varepsilon^2}\right)}$ .这，特别是，给我们一个多项式时间近似计划纳什均衡在游戏中固定列稀疏$s$。此外，对于任意双矩阵游戏-因为$s$最多可以是$n$ -我们算法的运行时间与Lipton，Markakis和Mehta（2003）获得的最著名的上限相匹配。近似Carathéodory定理也导致了一个加性近似算法的规范化denominationPDMATHABMZ-子图问题。给定一个图的$n$顶点和最大程度的$d$，该算法确定一个子图的$k$顶点与规范化的密度内$\varepsilon$（在添加剂的意义上）的最佳时间$n^{O\left( \frac{\log d}{\varepsilon^2}\right)}$。此外，我们发现，一个类似的近似结果可以找到一个最大归一化密度的$k \times k$ -二部子图的问题。
英文摘要：
We present algorithmic applications of an approximate version of Carathéodory's theorem. The theorem states that given a set of vectors $X$ in $\mathbb{R}^d$, for every vector in the convex hull of $X$ there exists an $\varepsilon$-close (under the $p$-norm distance, for $2\leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ depends on $\varepsilon$ and the norm $p$ and is independent of the dimension $d$. This theorem can be derived by instantiating Maurey's lemma, early references to 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 \times n$ payoff matrices $A, B$, if the number of non-zero entries in any column of $A+B$ is at most $s$ then an $\varepsilon$-Nash equilibrium of the game can be computed in time $n^{O\left(\frac{\log s }{\varepsilon^2}\right)}$. 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 $\varepsilon$ (in the additive sense) of the optimal in time $n^{O\left( \frac{\log d}{\varepsilon^2}\right)}$. Additionally, we show that a similar approximation result can be achieved for the problem of finding a $k \times k$-bipartite subgraph of maximum normalized density.
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