Parallel algorithms for vehicle routing problems

In a complete directed weighted graph there are jobs located at nodes of the graph. Job i has an associated processing time or handling time h_i, and the job must start within a prespecified time window [r_i, d_i]. A vehicle can move on the arcs of the graph, at unit speed and that has to execute the jobs within their respective time windows. We consider three different problems on the CREW PRAM. (1) Find the minimum cost routes between all pairs of nodes in a network. We give an O(log³ n) time algorithm with n⁴/log² n processors. (2) Services all locations in minimum time. The general problem is NP-complete but O(n²) time algorithms are known for a special case; for this case we obtain an O(log³ n) time parallel algorithm using n⁴/log² n processors and a linear time optimal parallel algorithm. (3) Minimize the sum of waiting times at all locations. The general problem is NP-complete but O(n²) time algorithm are known for a special case; for this case, we obtain an O(log² n) time algorithm with n³/log n processors and also a linear time optimal parallel algorithm