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

TL;DR

This paper develops a constrained shortest-path reformulation framework that leverages decision diagrams to encode discrete two-stage optimization problems. By embedding side constraints either as linear inequalities in the DD polyhedron or as state variables in a dynamic-programming-based shortest-path, the authors address interdiction-type and robust optimization classes where traditional duality approaches falter. They provide two solution pathways: a polyhedral MILP reformulation and a DP-based constrained-path model, applying them to competitive project selection and robust TSP with time windows, respectively. Empirical results demonstrate substantial computational gains over state-of-the-art bilevel and MILP methods, highlighting the practical impact for large-scale, structured discrete optimization. The methods generalize to broader two-stage and robust settings, offering a versatile toolkit for tightening formulations and accelerating solution times in complex decision 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.

Figures (6)

• Figure 1: (a) A decision diagram for the knapsack instance in Example \ref{['ex:dd']} and (b) the expanded network for Example \ref{['ex:dp']}. Green-shaded arcs represent the longest-path and optimal solution (in colour.)
• Figure 2: A decision diagram corresponding to a follower's problem with 3 projects.
• Figure 3: (Coloured) Scatter plots comparing runtimes between B&C and DDR based on the constraint tightness and the number of variables. Dashed lines (in blue) represent the time-limit mark at 3,600 seconds. Both horizontal and vertical coordinates are in logarithmic scale.
• Figure 4: (Coloured) Performance profile plot for varying uniform discrete distribution of the knapsack constraint coefficients $a^{F}$ and $a^{L}$.
• Figure 5: A decision diagram for a TSP problem with $n=4$ cities for Example \ref{['ex:rptswp']}.

Theorems & Definitions (24)

• Example 1
• Proposition 1
• Proposition 2
• Example 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Example 3