An Interior Point Heuristic for the Hamiltonian Cycle Problem
via Markov Decision Processes

Technical Report, School of Mathematics, The University of South Australia

V. Ejov, J. Filar, J. Gondzio

We consider the Hamiltonian cycle problem embedded in a singularly perturbed Markov decision process (MDP). More specifically, we consider the HCP as an optimization problem over the space of long-run state-action frequencies induced by the MDP's stationary policies. We show that Hamiltonian cycles (if any) correspond to the global minima of a suitably constructed indefinite quadratic programming problem over the frequency space. We show that the above indefinite quadratic can be approximated by quadratic functions that are ``nearly convex" and as such suitable for the application of logarithmic barrier methods. We develop an interior-point type algorithm that involves an arc elimination heuristic that appears to perform rather well in moderate size graphs. The approach has the potential for further improvements.

Key words: Hamiltonian cycles, Markov Decision Processes, Interior Point Methods, Non-convex Optimization.

PDF TR03-xxx.pdf.

Written: April 3, 2003.
Published: Journal of Global Optimization 29 (2004) No 3, pp. 315-334.
Related Software:
HOPDM Higher Order Primal Dual Method.