A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs

TL;DR

This paper introduces a linear programming heuristic framework for solving complex min-max regret combinatorial problems with interval costs, improving solution quality over existing exact and approximation methods.

Contribution

It proposes a novel heuristic approach that leverages dual information from relaxed models to efficiently find near-optimal solutions for interval robust-hard problems.

Findings

The heuristic finds optimal or near-optimal solutions.

It outperforms state-of-the-art exact algorithms.

It improves primal bounds for tested problems.

Abstract

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.