An ODE approach to multiple choice polynomial programming

TL;DR

This paper introduces an ODE-based method for solving multiple choice polynomial programming problems, leveraging thermal equilibrium concepts to efficiently approximate solutions with competitive quality.

Contribution

It presents a novel ODE approach that transforms MCPP into a differential system, enabling efficient approximation of solutions for complex combinatorial problems.

Findings

Validates the ODE approach on MAX-$k$-CUT and star discrepancy problems.

Achieves solution quality comparable to state-of-the-art heuristics.

Demonstrates the first continuous algorithm for star discrepancy approximation.

Abstract

We propose an ODE approach to solving multiple choice polynomial programming (MCPP) after assuming that the optimum point can be approximated by the expected value of so-called thermal equilibrium as usually did in simulated annealing. The explicit form of the feasible region and the affine property of the objective function are both fully exploited in transforming the MCPP problem into the ODE system. We also show theoretically that a local optimum of the former can be obtained from an equilibrium point of the latter. Numerical experiments on two typical combinatorial problems, MAX-$k$-CUT and the calculation of star discrepancy, demonstrate the validity of our ODE approach, and the resulting approximate solutions are of comparable quality to those obtained by the state-of-the-art heuristic algorithms but with much less cost. This paper also serves as the first attempt to use a continuous algorithm for approximating the star discrepancy.