Greedily finding a dense subgraph

Abstract

Given an n-vertex graph with non-negative edge weights and a positive integer k ≤ n, we are to find a fc-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2+n/(2k))2-O(1/n) ≤ R ≤ (1/2+n/(2k))2+O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k - 1) - O(1/k) ≤ R ≤ 2(n/k - 1) + O(n/k2) for k