An approximation algorithm for the k-prize-collecting multicut on a tree problem

Abstract

In this paper, we consider the k-prize-collecting multicut on a tree (k-PCM(T)) problem. In this problem, we are given an undirected tree $T=(V,E)$, a set of m distinct pairs of vertices $P=\{(s_1,t_1),\dots ,(s_m,t_m)\}$ and a parameter k with $k\le m$. Every edge in E has a nonnegative cost $c_e$. Every pair $(s_i,t_i)$ in P has a nonnegative penalty cost ${\pi }_i$. Our goal is to find a multicut M that separates at least k pairs in P such that the total cost, including the edge cost of the multicut M and the penalty cost of the pairs not separated by M, is minimized. This problem generalizes the well-known multicut on a tree problem. Our main work is to present a $(4+\epsilon )$-approximation algorithm for the k-PCM(T) problem via the methods of primal-dual and Lagrangean relaxation, where ε is any fixed positive number.

Introduction

The main focus of this paper is the k-prize-collecting multicut on a tree (k-PCM(T)) problem. In the k-PCM(T) problem, we are given an undirected tree $T=(V,E)$ with non-negative costs $c_e$ for the edges $e\in E$, a set of m distinct pairs of vertices $P=\{(s_1,t_1),\dots ,(s_m$, $t_m)\}$ with nonnegative penalty costs ${\pi }_i$ for the pairs $(s_i,t_i)$, and a parameter k at most m. For $1\le i\le m$, the pair $(s_i,t_i)$ is said to be separated by the edge set $M\subseteq E$ if it is not contained in a single connected component of $T\setminus M$, in other words, the removal of M disconnects $s_i$ and $t_i$. Our goal is to find a multicut M that separates at least k pairs of vertices such that the total cost, including the edge cost of the multicut M and the penalty cost of the pairs not separated by M, is minimized.

The k-PCM(T) problem contains as special cases the multicut on a tree, the prize-collecting multicut on a tree and the partial multicut on a tree problems. For the multicut on a tree problem, Garg et al. [3] presented a prime-dual algorithm that constructs a feasible solution whose cost is at most twice the optimum. For the prize-collecting multicut on a tree problem, Levin and Segev [7] proved that it is equivalent to the multicut on a tree problem by modifying the original tree and set of pairs, and gave a 2-approximation algorithm. For the partial multicut on a tree problem, Levin and Segev [7] presented an $(\frac{8}{3}+\epsilon )$-approximation algorithm by using the Lagrangian relaxation technique, where ε is any fixed positive number.

An algorithm for a prize-collecting problem is called Lagrangian-multiplier preserving (LMP) [9], if it satisfies the so-called Lagrangian multiplier preserving property, that is it satisfies the initial objective cost plus r times the penalty cost is at most r times the dual objective where r is some desired approximation ratio. LMP algorithms have been designed for the k-median problem [5], k-MST problem [2], k-Steiner tree [1], k-prize-collecting Steiner tree [4] and partial covering [6] problems.

Liu et al. [8] introduced the multicut on a tree problem with submodule penalties, which is a generalization of the prize-collecting multicut on a tree problem, and presented a 3-approximation algorithm based on the primal-dual method. Motivated by the above work, in this paper, we first restate the algorithm as an algorithm for the prize-collecting multicut on a tree problem and show that it is LMP, and then present an approximation algorithm with an approximate ratio of $4+\epsilon$ for the k-PCM(T) problem via the methods of primal-dual and Lagrangean relaxation, where ε is any fixed positive number.

This paper is organized as follows: In Section 2, we recall the prize-collecting multicut on a tree problem. In Section 3, we present the LP relaxation and dual program of the k-PCM(T) problem. In Section 4, we describe our algorithm for the k-PCM(T) problem. In Section 5, we analyze the approximate ratio of our algorithm.