Constrained Shortest-Path Reformulations via Decision Diagrams for Structured Two-stage Optimization Problems

TL;DR

This paper introduces a decision diagram-based framework for constrained shortest-path reformulations in two-stage discrete optimization problems, enabling more efficient solutions for interdiction and robust optimization scenarios.

Contribution

It develops a novel decision diagram approach that incorporates side constraints directly, applicable to problems where traditional methods are ineffective.

Findings

Significant computational improvements over existing methods.

Effective reformulation of interdiction constraints as network flows.

Successful application to project selection and robust TSP problems.

Abstract

Many discrete optimization problems are amenable to constrained shortest-path reformulations in an extended network space, a technique that has been key in convexification, bound strengthening, and search. In this paper, we propose a constrained variant of these models for two challenging classes of discrete two-stage optimization problems, where traditional methods (e.g., dualize-and-combine) are not applicable compared to their continuous counterparts. Specifically, we propose a framework that models problems as decision diagrams and introduces side constraints either as linear inequalities in the underlying polyhedral representation, or as state variables in shortest-path dynamic programming models. For our first structured class, we investigate two-stage problems with interdiction constraints. We show that such constraints can be formulated as indicator functions in the arcs of the diagram, providing an alternative single-level reformulation of the problem via a network-flow representation. Our second structured class is classical robust optimization, where we leverage the decision diagram network to iteratively identify label variables, akin to an L-shaped method. We evaluate these strategies on a competitive project selection problem and the robust traveling salesperson with time windows, observing considerable improvements in computational efficiency as compared to general methods in the respective areas.