Approximations for minimum and min-max vehicle routing problems

  • Authors:

  • Esther M. Arkin;Refael Hassin;Asaf Levin

  • Affiliations:

  • Department of Applied Mathematics and Statistics, SUNY Stony Brook, Stony Brook, NY 11794-3600, USA;Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv 69978, Israel;Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv 69978, Israel

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2006

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Abstract

We consider a variety of vehicle routing problems. The input to a problem consists of a graph G=(N,E) and edge lengths l(e), e@?E. Customers located at the vertices have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and @l the longest distance traveled by a vehicle. We consider two types of problems. (1) Given a bound @l on the length of each path, find a minimum sized collection of paths that cover all the vertices of the graph, or all the edges from a given subset of edges of the input graph. We also consider a variation where it is desired to cover N by a minimum number of stars of length bounded by @l. (2) Given a number k find a collection of k paths that cover either the vertex set of the graph or a given subset of edges. The goal here is to minimize @l, the maximum travel distance. For all these problems we provide constant ratio approximation algorithms and prove their NP-hardness.