Approximation algorithms for group prize-collecting and location-routing problems

Abstract

In this paper we develop approximation algorithms for generalizations of the following three known combinatorial optimization problems, the Prize-Collecting Steiner Tree problem, the Prize-Collecting Travelling Salesman Problem and a Location-Routing problem. Given a graph G = ( V , E ) on n vertices and a length function on its edges, in the grouped versions of the above mentioned problems we assume that V is partitioned into k + 1 groups, { V 0 , V 1 , … , V k } , with a penalty function on the groups. In the Group Prize-Collecting Steiner Tree problem the aim is to find S , a collection of groups of V and a tree spanning the rest of the groups not in S , so as to minimize the sum of the costs of the edges in the tree and the costs of the groups in S . The Group Prize-Collecting Travelling Salesman Problem, is defined analogously. In the Group Location-Routing problem the customer vertices are partitioned into groups and one has to select simultaneously a subset of depots to be opened and a collection of tours that covers the customer groups. The goal is to minimize the costs of the tours plus the fixed costs of the opened depots. We give a ( 2 − 1 n − 1 ) I -approximation algorithm for each of the three problems, where I is the cardinality of the largest group.