 
Summary: Approximations for minimum and minmax vehicle routing
problems
Esther M. Arkin # Refael Hassin + Asaf Levin #
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 # the longest distance traveled by a vehicle. We consider two types of problems: (1) Given
a bound # 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 #. (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 #, the maximum travel
distance. For all these problems we provide constant ratio approximation algorithms and prove
their NPhardness.
Keywords: Approximation algorithms, Edgerouting, Vehicle routing problem, Minmax
problems.
# Department of Applied Mathematics and Statistics, SUNY Stony Brook, Stony Brook, NY 117943600,
estie@ams.sunysb.edu. Partially supported by NSF (CCR0098172).
+ Department of Statistics and Operations Research, TelAviv University, TelAviv 69978, Israel,
hassin@post.tau.ac.il
