Continuous-Time Systems for Solving 0-1 Integer Linear Programming Feasibility Problems

TL;DR

This paper introduces a continuous-time dynamical system with an impulse algorithm to efficiently solve 0-1 integer linear programming feasibility problems, outperforming traditional methods in escaping local minima and reducing solution time.

Contribution

It presents a novel continuous-time approach with an impulse algorithm that improves solution efficiency and escape from local traps in 0-1 ILP feasibility problems.

Findings

Impulse algorithm outperforms randomization in escaping traps.

Time-to-solution distribution is favorable compared to exhaustive search.

Basins of attraction for the global minimum are significantly larger than random chance.

Abstract

The 0-1 integer linear programming feasibility problem is an important NP-complete problem. This paper proposes a continuous-time dynamical system for solving that problem without getting trapped in non-solution local minima. First, the problem is transformed to an easier form in linear time. Then, we propose an "impulse algorithm" to escape from local traps and show its performance is better than randomization for escaping traps. Second, we present the time-to-solution distribution of the impulse algorithm and compare it with exhaustive search to see its advantages. Third, we show that the fractional size of the basin of attraction of the global minimum is significantly larger than $2^{-N}$, the corresponding discrete probability for exhaustive search. Finally, we conduct a case study to show that the location of the basin is independent of different dimensions. These findings reveal a better way to solve the 0-1 integer linear programming feasibility problem continuously and show that its cost could be less than discrete methods in average cases.