An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs

摘要

Given a positive integer p and a complete graph with non-negative edge weights that satisfy the triangle inequality, the remote-clique problem is to find a subset of p vertices having a maximum-weight induced subgraph. A greedy algorithm for the problem has been shown to have an approximation ratio of 4, but this analysis was not shown to be tight. In this paper, we present an algorithm called d-GREEDY AUGMENT that generalizes this greedy algorithm (they are equivalent when d = 1). We use the technique of factor-revealing linear programs to prove that d-GREEDY AUGMENT, which has a running time of O(pdn~d), achieves an approximation ratio of (2p — 2)/(p + d — 2). Thus, when d = 1, d-GREEDY AUGMENT achieves an approximation ratio of 2 and runs in time O(pn), making it the fastest known 2-approximation for the remote-clique problem. The usefulness of factor-revealing LPs in the analysis of d-GREEDY AUGMENT suggests possible applicability of this technique to the study of other approximation algorithms.