TY - JOUR

T1 - Approximation algorithms and hardness results for cycle packing problems

AU - Krivelevich, Michael

AU - Nutov, Zeev

AU - Salavatipour, Mohammad R.

AU - Yuster, Jacques Verstraete

AU - Yuster, Raphael

PY - 2007/11/1

Y1 - 2007/11/1

N2 - The cycle packing number v e(G) of a graph G is the maximum number of pairwise edgedisjointcycles in G. Computing v e(G) is an NP-hard problem.We present approximation algorithms for computing v e(G) in both undirected and directed graphs. In the undirected case we analyze a variant of the modified greedy algorithm suggested by Caprara et al. [2003] and showthat it has approximation ratio Θ(√log n), where n = |V(G)|. This improves upon the previous O(log n) upper bound for the approximation ratio of this algorithm. In the directed case we present √n-approximation algorithm. Finally, we give an O(n 2/3)- approximation algorithm for the problem of finding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices.We also study generalizations of these problems. Our approximation ratios are the currently best-known ones and, in addition, provide upper bounds on the integrality gap of standard LP-relaxations of these problems. In addition, we give ower bounds for the integrality gap and approximability of v e(G) in directed graphs. Specifically, we prove a lower bound of Ω( log n log log n) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate v e(G) within a factor of O(log 1-ε n) for any constant ε > 0. This improves upon the previously known APX-hardness result for this problem.

AB - The cycle packing number v e(G) of a graph G is the maximum number of pairwise edgedisjointcycles in G. Computing v e(G) is an NP-hard problem.We present approximation algorithms for computing v e(G) in both undirected and directed graphs. In the undirected case we analyze a variant of the modified greedy algorithm suggested by Caprara et al. [2003] and showthat it has approximation ratio Θ(√log n), where n = |V(G)|. This improves upon the previous O(log n) upper bound for the approximation ratio of this algorithm. In the directed case we present √n-approximation algorithm. Finally, we give an O(n 2/3)- approximation algorithm for the problem of finding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices.We also study generalizations of these problems. Our approximation ratios are the currently best-known ones and, in addition, provide upper bounds on the integrality gap of standard LP-relaxations of these problems. In addition, we give ower bounds for the integrality gap and approximability of v e(G) in directed graphs. Specifically, we prove a lower bound of Ω( log n log log n) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate v e(G) within a factor of O(log 1-ε n) for any constant ε > 0. This improves upon the previously known APX-hardness result for this problem.

KW - Approximation algorithms

KW - Cycle packing

KW - Edge-disjoint

KW - Hardness of approximation

KW - Integrality gap

UR - http://www.scopus.com/inward/record.url?scp=36549060611&partnerID=8YFLogxK

U2 - 10.1145/1290672.1290685

DO - 10.1145/1290672.1290685

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AN - SCOPUS:36549060611

SN - 1549-6325

VL - 3

SP - 48

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 4

M1 - 1290685

ER -