TY - JOUR

T1 - Approximating minimum subset feedback sets in undirected graphs with applications

AU - Even, Guy

AU - Naor, Joseph

AU - Schieber, Baruch

AU - Zosin, Leonid

PY - 2000/4

Y1 - 2000/4

N2 - Let G = (V, E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ⊂ V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a Δ-approximation algorithm for the subset feedback vertex set problem, where Δ is the maximum degree in G. We also consider the multicut problem and show how to achieve an O(log τ*) approximation factor for this problem, where τ* is the value of the optimal fractional solution. To achieve the O(log τ*) factor we employ a bootstrapping technique.

AB - Let G = (V, E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ⊂ V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a Δ-approximation algorithm for the subset feedback vertex set problem, where Δ is the maximum degree in G. We also consider the multicut problem and show how to achieve an O(log τ*) approximation factor for this problem, where τ* is the value of the optimal fractional solution. To achieve the O(log τ*) factor we employ a bootstrapping technique.

KW - Approximation algorithms

KW - Combinatorial optimization

KW - Feedback edge set

KW - Feedback vertex set

KW - Multicut

KW - Subset feedback set

UR - http://www.scopus.com/inward/record.url?scp=0003822874&partnerID=8YFLogxK

U2 - 10.1137/S0895480195291874

DO - 10.1137/S0895480195291874

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AN - SCOPUS:0003822874

SN - 0895-4801

VL - 13

SP - 255

EP - 267

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -