Abstract
Given an n-vertex graph with nonnegative edge weights and a positive integer k≤n, our goal is to find a k-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(n−1/3)≤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<n/3. For k=n/2, for example, these bounds are 9/4±O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.
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