An approximation algorithm for the Generalized k-Multicut problem
Discrete Applied Mathematics
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Given a tree $T=(V, E)$ with costs defined on edges, a positive integer $k$, and $l$ terminal sets $\{S_1, S_2, \dots, S_l\}$ with every $S_i \subseteq V$, the generalized $k$-Multicut in trees problem ($k$-GMC(T)) asks to find an edge subset in $E$ at the minimum cost such that its removal cuts at least $k$ terminal sets. The $k$-GMC(T) problem is a natural generalization of the classical Multicut in trees problem and the Multiway Cut in trees problem. This problem is hard to be approximated within $O(n^{\frac16 - \epsilon})$ for some small constant $\epsilon 0$ (Zhang, CiE'07). Based on a greedy approach and a rounding technique in linear programming, we give a bicriteria approximation algorithm for $k$-GMC(T). Our algorithm outputs in polynomial time a solution which cuts at least $(1-\epsilon)k$ terminal sets and whose cost is within $\sqrt{2 / \epsilon \cdot l}$times of the optimum for any small constant $\epsilon 0$, and hence gives sublinear approximation ratio for $k$-GMC(T).