TY - GEN

T1 - Greedily finding a dense subgraph

AU - Asahiro, Yuichi

AU - Iwama, Kazuo

AU - Tamaki, Hisao

AU - Tokuyama, Takeshi

N1 - Funding Information:
1Research supported in part by Science Research Grant 07458061, Ministry of Education, Japan. 2 Main part of the work done while author was at IBM Tokyo Research Laboratory.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1996.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1996

Y1 - 1996

N2 - 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 < 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 shows that this simple algorithm is better than the best previously known algorithm at least by a factor of 2 when k ≥ n11/18.

AB - 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 < 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 shows that this simple algorithm is better than the best previously known algorithm at least by a factor of 2 when k ≥ n11/18.

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U2 - 10.1007/3-540-61422-2_127

DO - 10.1007/3-540-61422-2_127

M3 - Conference contribution

AN - SCOPUS:84867999934

SN - 3540614222

SN - 9783540614227

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 136

EP - 148

BT - Algorithm Theory - SWAT 1996 - 5th Scandinavian Workshop on Algorithm Theory, Proceedings

A2 - Karlsson, Rolf

A2 - Lingas, Andrzej

PB - Springer Verlag

T2 - 5th Scandinavian Workshop on Algorithm Theory, SWAT 1996

Y2 - 3 July 1996 through 5 July 1996

ER -