Annealing-Assisted Column Generation for Inequality-Constrained Combinatorial Optimization Problems

TL;DR

This paper introduces a hybrid annealing-assisted column generation method leveraging Ising machines to efficiently solve inequality-constrained combinatorial optimization problems like CVRP, extending Ising machine applications.

Contribution

It presents a novel hybrid algorithm combining column generation with annealing-assisted Ising machines to handle inequality constraints in complex optimization problems.

Findings

Converges to better lower bounds on the problem.

Demonstrates reduced computation time with Ising machines.

Extends Ising machine applicability to inequality-constrained problems.

Abstract

Ising machines are expected to solve combinatorial optimization problems faster than the existing integer programming solvers. These problems, particularly those encountered in practical situations, typically involve inequality constraints. However, owing to the hardware limitations of the current Ising machines, solving combinatorial optimization problems with inequality constraints remains challenging. The Capacitated Vehicle Routing Problem (CVRP) is a typical example of a problem with inequality constraints. The objective function of the CVRP is to minimize the total distance traveled by each vehicle while limiting the total demand of customers served by a single vehicle to the vehicle's capacity. The CVRP is classified as NP-hard and, thus, is commonly solved using heuristic algorithms, such as column generation. Column generation attempts to iteratively generate only the promising routes, as the number of feasible routes increases exponentially. Within this framework, the CVRP is formulated as a set cover problem. The corresponding dual solutions are used to define the pricing subproblem, which is intended to create a new route. By applying Ising machines to this pricing subproblem, the overall computation time can be reduced. This study aims to solve combinatorial optimization problems with inequality constraints using a hybrid algorithm that combines column generation and Ising machines, thereby extending the applications of the latter. We parameterize the difficulty of the inequality constraints and demonstrate that our annealing-assisted column generation can converge to a better lower bound.