Approximation Algorithms for the Maximum-Weight Cycle/Path Packing Problems

Abstract

Given an undirected complete graph [Formula: see text] on [Formula: see text] vertices with a non-negative weight function on [Formula: see text], the maximum-weight [Formula: see text]-cycle ([Formula: see text]-path) packing problem aims to compute a set of [Formula: see text] vertex-disjoint cycles (paths) in [Formula: see text] containing [Formula: see text] vertices so that the total weight of the edges in these [Formula: see text] cycles (paths) is maximized. For the maximum-weight [Formula: see text]-cycle packing problem, we develop an algorithm achieving an approximation ratio of [Formula: see text], where [Formula: see text] is the approximation ratio for the maximum traveling salesman problem. For the case [Formula: see text], we design a better [Formula: see text]-approximation algorithm. When the weights of edges obey the triangle inequality, we propose a [Formula: see text]-approximation algorithm and a [Formula: see text]-approximation algorithm for the maximum-weight [Formula: see text]-cycle packing problem with [Formula: see text] and [Formula: see text], respectively. For the maximum-weight [Formula: see text]-path packing problem with [Formula: see text] (or [Formula: see text]) with the triangle inequality, we devise an algorithm with approximation ratio [Formula: see text] and give a tight example.