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Relative-Interior Solution for the (Incomplete) Linear Assignment Problem with Applications to the Quadratic Assignment Problem

Tomáš Dlask, Bogdan Savchynskyy

TL;DR

This work develops a theory and practical toolkit for exploiting relative interiors of optimal solution sets in the linear assignment problem (LAP) and its incomplete variant (ILAP) to tighten bounds on the incomplete quadratic assignment problem (IQAP). It introduces a linear-time method to obtain a relative-interior solution from any dual-optimal LAP solution and an optimal assignment, and shows how to transfer these results to ILAP via a linear-time reduction that preserves optimality and ri. The IQAP framework is then decomposed into ILAP and WCSP subproblems, enabling efficient bound propagation through coordinate ascent updates and exact LAP/Hungarian steps. Empirical results on CV and QAPLIB benchmarks indicate that RI-based methods yield bounds close to the LP optimum and run substantially faster than a state-of-the-art LP solver, with robust performance across diverse instance types.

Abstract

We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since the LAP occurs as a subproblem in the linear programming (LP) relaxation of the quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of the QAP. To make our results applicable to the incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from the incomplete LAP to the complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution can frequently provide bounds near the optimum of the LP relaxation and its runtime is much lower when compared to a commercial LP solver.

Relative-Interior Solution for the (Incomplete) Linear Assignment Problem with Applications to the Quadratic Assignment Problem

TL;DR

This work develops a theory and practical toolkit for exploiting relative interiors of optimal solution sets in the linear assignment problem (LAP) and its incomplete variant (ILAP) to tighten bounds on the incomplete quadratic assignment problem (IQAP). It introduces a linear-time method to obtain a relative-interior solution from any dual-optimal LAP solution and an optimal assignment, and shows how to transfer these results to ILAP via a linear-time reduction that preserves optimality and ri. The IQAP framework is then decomposed into ILAP and WCSP subproblems, enabling efficient bound propagation through coordinate ascent updates and exact LAP/Hungarian steps. Empirical results on CV and QAPLIB benchmarks indicate that RI-based methods yield bounds close to the LP optimum and run substantially faster than a state-of-the-art LP solver, with robust performance across diverse instance types.

Abstract

We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since the LAP occurs as a subproblem in the linear programming (LP) relaxation of the quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of the QAP. To make our results applicable to the incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from the incomplete LAP to the complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution can frequently provide bounds near the optimum of the LP relaxation and its runtime is much lower when compared to a commercial LP solver.
Paper Structure (23 sections, 12 theorems, 42 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 12 theorems, 42 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Structure of the Paper
  5. The LAP and Relative-Interior Solution
  6. Relative Interior of the Set of Optimal Solutions
  7. Obtaining a Relative-Interior Solution from an Optimal Solution
  8. The Incomplete LAP: Reduction to the LAP and Relative-Interior Solution
  9. Reduction of the ILAP to the LAP
  10. Reduction
  11. Properties of Optimal Solutions
  12. The Incomplete QAP and Subproblems of Its LP Relaxation
  13. The ILAP Subproblem
  14. Coordinate Ascent
  15. The WCSP Subproblem
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mu$ and $(\alpha,\beta)$ be feasible for the primal and dual eq:LAP_LP_relax, respectively. The following are equivalent:

Figures (4)

  • Figure 1: Illustrations to Example \ref{['ex:lap_ri']}.
  • Figure 2: Illustrations to Example \ref{['ex:transformation']}.
  • Figure 3: Current bound and relative inter-iteration improvement of the bound (the latter in logarithmic scale) for the instance lipa20a. Only the first 100 iterations of $\texttt{Hung}$ are shown.
  • Figure 4: Cactus plots showing the distribution of the runtimes of the methods across all instances. See Section \ref{['sec:stopping-criteria']} on the runtime.

Theorems & Definitions (35)

  • Remark 1
  • Example 1: Interior vs. relative interior
  • Theorem 1: zhang1994strictly
  • Proposition 1: cf. Property 4 in claus2020analysis
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • ...and 25 more