We give a formal specification for a strategic network routing problem known as the convoy movement problem (CMP) and establish that the corresponding feasibility problem is NP-complete. We then introduce an integer programming (IP) model based on the concept of a time-space network and apply a Lagrangian relaxation to this model. We discuss how the dual function may be evaluated using a modified version of Dijkstra’s algorithm suitable to very large, implicitly defined graphs and show how heuristic solutions to the primal problem may be obtained. We present results for a number of instances of the CMP, most of which are based on real-world problems. The number of convoys in these instances varies between 15–25, and their movement time requires up to several thousand time units in networks ranging in size from a few dozen to several thousand vertices and edges. The most difficult instance tested involves 17 long convoys each taking four times the average link travel time to pass through a point in the network. This instance is solved within 3.3% of optimality in less than 3.5 hours of computing time on a Dell Precision 420 dual processor computer. Every other test instance is solved within 2% of the optimal value in less than 20 minutes of computing time

A new heuristic algorithm, PROBE_BA, which is based on the recently introduced metaheuristic paradigm population- reinforced optimization-based exploration (PROBE), is proposed for solving the Graph Partitioning Problem. The "exploration" part of PROBE_BA is implemented by using the differential-greedy algorithm of Battiti and Bertossi and a modification of the Kernighan-Lin algorithm at the heart of Bui and Moon's genetic algorithm BFS _GBA. Experiments are used to investigate properties of PROBE and show that PROBE_BA compares favorably with other solution methods based on genetic algorithms, randomized reactive tabu search, or more specialized multilevel partitioning techniques. In addition, PROBE_BA finds new best cut values for 10 of the 34 instances in Walshaw's graph partitioning archive

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An implementation using systolic array logic of Aitken's method of iterated interpolation is described. The proposed design has a simple, linear topology, requires no clock, and makes only modest demands on the host computer. By overlapping the computation of successive function values, a processing element utilization of approximately 1/2 is achieved. The paper illustrates how “mathematical hardware” packages, as well as software library routines, may be part of the mathematical problem solver's tool kit in the future.