Graph Bisection Modeled as Binary Quadratic Task Allocation and Solved via Tabu Search

Balanced graph bisection is an NP-complete problem which partitions a set of nodes in the graph G = (N,E) into two sets with equal cardinality such that a minimal sum of edge weights exists between the nodes in the two separate sets. In this paper we transform graph bisection to capacitated task allocation using variable substitutions to remove most capacity and assignment constraints. The resulting problem is in the form of a generic xQx unconstrained quadratic binary problem, except for a single cardinality constraint. Problems are solved using tabu search employing strategic oscillation around critical events created when the single constraint is satisfied. On a set of benchmark graphs, we improve the best known solution for several problems. Comparison results with the fast, freely available multilevel balanced graph partitioning program METIS are presented on a set of random graphs. Our approach works well when balanced graph bisection is augmented to the 2-processor task allocation problem which adds node preferences for a set, as well as edge weights, to the objective function. For these problems, our approach compares favorably to Cplex and Gurobi, providing better solutions in a much shorter time.