The Delivery Man Problem and Cumulative Matroids

Given a completed directed graph G=(V,A), the Delivery Man Problem (DMP) consists of determining a Hamiltonian circuit minimizing the sum of distances from a given vertes v1 to every vertex of V, including v1 itself. There exists a number of applications of the DMP in the fields of distribution and of machine scheduling. The DMP is NP-hard. The objective of this paper is to develop new theoretical results and an exact algorithm for the problem. New integer linear programminig formulations are provided, and results on the matroidal structure of a class of combinatorial problems are developed. These are used to derive lower bounds for the DMP. These bounds are embedded into an enumerative algorithm. Computational results indicate that the proposed algorithm can solve to optimality problems involving up to 60 vertices. This compares favourably with previously published methods. (A)

[1]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[2]  Shen Lin.  Computer solutions of the traveling salesman problem , 1965 .

[3]  Abilio Lucena,et al.  Time-dependent traveling salesman problem-the deliveryman case , 1990, Networks.

[4]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..

[5]  T. B. Tate,et al.  Machine Scheduling Problems , 1978 .

[6]  Lucio Bianco,et al.  The traveling salesman problem with cumulative costs , 1993, Networks.

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  John N. Tsitsiklis,et al.  Special cases of traveling salesman and repairman problems with time windows , 1992, Networks.

[9]  David Simchi-Levi,et al.  Minimizing the Total Flow Time of n Jobs on a Network , 1991 .

[10]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[11]  Matteo Fischetti,et al.  A new dominance procedure for combinatorial optimization problems , 1988 .

[12]  Thomas L. Magnanti,et al.  A Dual-Ascent Procedure for Large-Scale Uncapacitated Network Design , 1989, Oper. Res..

[13]  Matteo Fischetti,et al.  An Efficient Algorithm for the Min-Sum Arborescence Problem on Complete Digraphs , 1993, INFORMS J. Comput..

[14]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[15]  E. Minieka.  The delivery man problem on a tree network , 1990 .

[16]  Maurice Queyranne,et al.  The Time-Dependent Traveling Salesman Problem and Its Application to the Tardiness Problem in One-Machine Scheduling , 1978, Oper. Res..

[17]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[18]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .