TY - JOUR

T1 - Partial multicuts in trees

AU - Levin, Asaf

AU - Segev, Danny

PY - 2006/12/15

Y1 - 2006/12/15

N2 - Let T = (V, E) be an undirected tree, in which each edge is associated with a non-negative cost, and let { s1, t1 }, ..., { sk, tk } be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the well-known multicut on a tree problem, in which we are required to disconnect all given pairs. The main contribution of this paper is an (frac(8, 3) + ε{lunate})-approximation algorithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed ε{lunate} > 0. This result is achieved by introducing problem-specific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prize-collecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2-approximation algorithm for multicut on a tree, based on LP-rounding.

AB - Let T = (V, E) be an undirected tree, in which each edge is associated with a non-negative cost, and let { s1, t1 }, ..., { sk, tk } be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the well-known multicut on a tree problem, in which we are required to disconnect all given pairs. The main contribution of this paper is an (frac(8, 3) + ε{lunate})-approximation algorithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed ε{lunate} > 0. This result is achieved by introducing problem-specific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prize-collecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2-approximation algorithm for multicut on a tree, based on LP-rounding.

KW - Approximation algorithms

KW - Lagrangian relaxation

KW - Multicut

UR - http://www.scopus.com/inward/record.url?scp=33750999950&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2006.09.018

DO - 10.1016/j.tcs.2006.09.018

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AN - SCOPUS:33750999950

SN - 0304-3975

VL - 369

SP - 384

EP - 395

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1-3

ER -