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 -