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Solving the QAP by Two-Stage Graph Pointer Networks and Reinforcement Learning

Satoko Iida, Ryota Yasudo

TL;DR

The paper tackles solving the NP-hard Quadratic Assignment Problem (QAP) by framing it within neural combinatorial optimization. It extends Graph Pointer Networks (GPN) to handle matrix-input TSP and introduces a two-stage GPN architecture for QAP that uses a Distance-Flow Product (DFP) representation and a block-wise solving strategy. Empirical results show the approach yields semi-optimal solutions faster than traditional heuristics, though performance varies by instance type, especially in sparse or triangular cases. The work contributes a scalable, reinforcement-learning-based solver and releases code for broader use and benchmarking.

Abstract

Quadratic Assignment Problem (QAP) is a practical combinatorial optimization problems that has been studied for several years. Since it is NP-hard, solving large problem instances of QAP is challenging. Although heuristics can find semi-optimal solutions, the execution time significantly increases as the problem size increases. Recently, solving combinatorial optimization problems by deep learning has been attracting attention as a faster solver than heuristics. Even with deep learning, however, solving large QAP is still challenging. In this paper, we propose the deep reinforcement learning model called the two-stage graph pointer network (GPN) for solving QAP. Two-stage GPN relies on GPN, which has been proposed for Euclidean Traveling Salesman Problem (TSP). First, we extend GPN for general TSP, and then we add new algorithms to that model for solving QAP. Our experimental results show that our two-stage GPN provides semi-optimal solutions for benchmark problem instances from TSPlib and QAPLIB.

Solving the QAP by Two-Stage Graph Pointer Networks and Reinforcement Learning

TL;DR

The paper tackles solving the NP-hard Quadratic Assignment Problem (QAP) by framing it within neural combinatorial optimization. It extends Graph Pointer Networks (GPN) to handle matrix-input TSP and introduces a two-stage GPN architecture for QAP that uses a Distance-Flow Product (DFP) representation and a block-wise solving strategy. Empirical results show the approach yields semi-optimal solutions faster than traditional heuristics, though performance varies by instance type, especially in sparse or triangular cases. The work contributes a scalable, reinforcement-learning-based solver and releases code for broader use and benchmarking.

Abstract

Quadratic Assignment Problem (QAP) is a practical combinatorial optimization problems that has been studied for several years. Since it is NP-hard, solving large problem instances of QAP is challenging. Although heuristics can find semi-optimal solutions, the execution time significantly increases as the problem size increases. Recently, solving combinatorial optimization problems by deep learning has been attracting attention as a faster solver than heuristics. Even with deep learning, however, solving large QAP is still challenging. In this paper, we propose the deep reinforcement learning model called the two-stage graph pointer network (GPN) for solving QAP. Two-stage GPN relies on GPN, which has been proposed for Euclidean Traveling Salesman Problem (TSP). First, we extend GPN for general TSP, and then we add new algorithms to that model for solving QAP. Our experimental results show that our two-stage GPN provides semi-optimal solutions for benchmark problem instances from TSPlib and QAPLIB.
Paper Structure (18 sections, 9 equations, 7 figures, 4 tables)

This paper contains 18 sections, 9 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Traveling Salesman Problem and Quadratic Assignment Problem
  3. Euclidean TSP
  4. Matrix input TSP
  5. QAP
  6. Conventional Pointer Networks
  7. Pointer Network
  8. Graph Pointer Network
  9. Methods
  10. GPN for matrix input TSP
  11. GPN for QAP
  12. Distance-Flow Product Matrix
  13. Two-stage GPN for QAP
  14. Multiple Models for Various Sparsity
  15. Experiments
  16. ...and 3 more sections

Figures (7)

  • Figure 1: An example of QAP. (a) Problem and (b) Solution $[2,4,3,1]$.
  • Figure 2: Overview of a graph pointer network.
  • Figure 3: GPN for matrix input TSP.
  • Figure 4: Distance-Flow Product (DFP) matrix obtained from distance and flow matrices.
  • Figure 5: The added costs in the DFP matrix when solution is $[2,1,3]$.
  • ...and 2 more figures