通过Lovász扩展探索子图密度-大小权衡
Exploring the Subgraph Density-Size Trade-off via the Lovász Extension
摘要：
给定一个无向图，稠密k子图问题（DkS）寻求找到k个顶点的子集，使得相应子图中的边权重之和最大化。这个问题是NP难的，在最坏的情况下也很难近似。在本文中，我们提出了一个新的凸松弛的问题。我们的主要思想是将DkS重新表述为最小化一个基数约束下的子模函数。利用子模函数具有凸的、连续的扩张（称为Lovász扩张）这一事实，我们提出在基数约束的凸包上最小化Lovász扩张.虽然Lovász扩展的次模函数不承认一般的分析形式，DkS，我们表明，它。我们利用这一结果开发一个高度可扩展的算法的基础上交替方向乘法（ADMM）解决放松的问题。再加上一对偶然简单的舍入方案，我们证明了我们的方法优于现实世界的图上现有的基线，并可以产生高质量的次优解决方案，通常是一个后验不差于65-80\%的最佳密度。
英文摘要：
Given an undirected graph, the Densest-k-Subgraph problem (DkS) seeks to find a subset of k vertices such that the sum of the edge weights in the corresponding subgraph is maximized. The problem is known to be NP-hard, and is also very difficult to approximate, in the worst-case. In this paper, we present a new convex relaxation for the problem. Our key idea is to reformulate DkS as minimizing a submodular function subject to a cardinality constraint. Exploiting the fact that submodular functions possess a convex, continuous extension (known as the Lovász extension), we propose to minimize the Lovász extension over the convex hull of the cardinality constraints. Although the Lovász extension of a submodular function does not admit an analytical form in general, for DkS we show that it does. We leverage this result to develop a highly scalable algorithm based on the Alternating Direction Method of Multipliers (ADMM) for solving the relaxed problem. Coupled with a pair of fortuitously simple rounding schemes, we demonstrate that our approach outperforms existing baselines on real-world graphs and can yield high quality sub-optimal solutions which typically are a posteriori no worse than 65-80\% of the optimal density.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
