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MORBDD: Multiobjective Restricted Binary Decision Diagrams by Learning to Sparsify

Rahul Patel, Elias B. Khalil, David Bergman

TL;DR

This work tackles the challenge of efficiently approximating the Pareto frontier in multiobjective ILPs by leveraging Binary Decision Diagrams (BDDs). It introduces MORBDD, a learning-based sparsification pipeline that prunes nodes in an exact BDD and then stitches the remaining subgraph to preserve connectivity, yielding small restricted BDDs with strong Pareto-quality trade-offs. The approach is demonstrated on multiobjective knapsack problems with up to seven objectives, showing that MORBDD substantially speeds frontier enumeration while recovering a large fraction of Pareto-optimal solutions and achieving competitive hypervolume compared to NSGA-II and width-restricted BDDs. The results suggest a practical pathway for integrating ML with combinatorial optimization representations, enabling fast, scalable decision-support for many-objective problems.

Abstract

In multicriteria decision-making, a user seeks a set of non-dominated solutions to a (constrained) multiobjective optimization problem, the so-called Pareto frontier. In this work, we seek to bring a state-of-the-art method for exact multiobjective integer linear programming into the heuristic realm. We focus on binary decision diagrams (BDDs) which first construct a graph that represents all feasible solutions to the problem and then traverse the graph to extract the Pareto frontier. Because the Pareto frontier may be exponentially large, enumerating it over the BDD can be time-consuming. We explore how restricted BDDs, which have already been shown to be effective as heuristics for single-objective problems, can be adapted to multiobjective optimization through the use of machine learning (ML). MORBDD, our ML-based BDD sparsifier, first trains a binary classifier to eliminate BDD nodes that are unlikely to contribute to Pareto solutions, then post-processes the sparse BDD to ensure its connectivity via optimization. Experimental results on multiobjective knapsack problems show that MORBDD is highly effective at producing very small restricted BDDs with excellent approximation quality, outperforming width-limited restricted BDDs and the well-known evolutionary algorithm NSGA-II.

MORBDD: Multiobjective Restricted Binary Decision Diagrams by Learning to Sparsify

TL;DR

This work tackles the challenge of efficiently approximating the Pareto frontier in multiobjective ILPs by leveraging Binary Decision Diagrams (BDDs). It introduces MORBDD, a learning-based sparsification pipeline that prunes nodes in an exact BDD and then stitches the remaining subgraph to preserve connectivity, yielding small restricted BDDs with strong Pareto-quality trade-offs. The approach is demonstrated on multiobjective knapsack problems with up to seven objectives, showing that MORBDD substantially speeds frontier enumeration while recovering a large fraction of Pareto-optimal solutions and achieving competitive hypervolume compared to NSGA-II and width-restricted BDDs. The results suggest a practical pathway for integrating ML with combinatorial optimization representations, enabling fast, scalable decision-support for many-objective problems.

Abstract

In multicriteria decision-making, a user seeks a set of non-dominated solutions to a (constrained) multiobjective optimization problem, the so-called Pareto frontier. In this work, we seek to bring a state-of-the-art method for exact multiobjective integer linear programming into the heuristic realm. We focus on binary decision diagrams (BDDs) which first construct a graph that represents all feasible solutions to the problem and then traverse the graph to extract the Pareto frontier. Because the Pareto frontier may be exponentially large, enumerating it over the BDD can be time-consuming. We explore how restricted BDDs, which have already been shown to be effective as heuristics for single-objective problems, can be adapted to multiobjective optimization through the use of machine learning (ML). MORBDD, our ML-based BDD sparsifier, first trains a binary classifier to eliminate BDD nodes that are unlikely to contribute to Pareto solutions, then post-processes the sparse BDD to ensure its connectivity via optimization. Experimental results on multiobjective knapsack problems show that MORBDD is highly effective at producing very small restricted BDDs with excellent approximation quality, outperforming width-limited restricted BDDs and the well-known evolutionary algorithm NSGA-II.
Paper Structure (31 sections, 2 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 31 sections, 2 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Exact approaches
  4. Heuristic approaches
  5. ML for MOO
  6. Problem Description
  7. Multiobjective Integer Linear Programming
  8. Binary Decision Diagrams
  9. Exact BDD
  10. Restricted BDD
  11. Compiling BDDs
  12. Solving MOILPs with BDDs
  13. The MORBDD approach
  14. ML formulation
  15. Sparsifiers
  16. ...and 16 more sections

Figures (1)

  • Figure 1: A binary decision diagram for a biobjective knapsack problem with two Pareto optimal solutions. The nodes used by the Pareto optimal solutions pass through only a subset of the nodes, red ones in the Figure. Each arc is associated with a pair of values in parentheses representing the objective function contributions of an arc. Each node is associated with one or more pairs of values (in red) representing the set of (unique) cumulative objective function values of each path from the root node.