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A Unified Column Generation and Elimination Method for Solving Large-Scale Set Partitioning Problems

Yasuyuki Ihara

TL;DR

The paper tackles large-scale Set Partitioning Problems by integrating a relaxation-based Column Generation and a systematic elimination of redundant columns, forming the Column Generation and Elimination Method. This approach tightens the traditional CG framework to increase the likelihood of attaining exact solutions and to improve the accuracy of approximate solutions, without requiring branch-and-price. Experimental results on real-world GTFS bus data show a dramatic rise in exact-solution rate for uniform-cost crew scheduling (from ≈3% to ≈99%) and notable accuracy gains for non-uniform costs, while controlling problem size growth. The work offers a practical, scalable alternative to branching methods for exactness and highlights directions for theoretical analysis and acceleration via hybrid technologies.

Abstract

The Set Partitioning Problem is a combinatorial optimization problem with wide-ranging applicability, used to model various real-world tasks such as facility location and crew scheduling. However, real-world applications often require solving large-scale instances that involve hundreds of thousands of variables. Although the conventional Column Generation method is popular for its computational efficiency, it lacks a guarantee for exact solutions. This paper proposes a novel solution method integrating relaxation of Column Generation conditions and automatic elimination of redundant columns, aimed at overcoming the limitations of conventional Column Generation methods in guaranteeing exact optimal solutions. Numerical experiments using actual bus route data reveal that while the traditional method achieves an exact solution rate of only about 3%, the proposed method attains a rate of approximately 99% and remarkably improves solution accuracy.

A Unified Column Generation and Elimination Method for Solving Large-Scale Set Partitioning Problems

TL;DR

The paper tackles large-scale Set Partitioning Problems by integrating a relaxation-based Column Generation and a systematic elimination of redundant columns, forming the Column Generation and Elimination Method. This approach tightens the traditional CG framework to increase the likelihood of attaining exact solutions and to improve the accuracy of approximate solutions, without requiring branch-and-price. Experimental results on real-world GTFS bus data show a dramatic rise in exact-solution rate for uniform-cost crew scheduling (from ≈3% to ≈99%) and notable accuracy gains for non-uniform costs, while controlling problem size growth. The work offers a practical, scalable alternative to branching methods for exactness and highlights directions for theoretical analysis and acceleration via hybrid technologies.

Abstract

The Set Partitioning Problem is a combinatorial optimization problem with wide-ranging applicability, used to model various real-world tasks such as facility location and crew scheduling. However, real-world applications often require solving large-scale instances that involve hundreds of thousands of variables. Although the conventional Column Generation method is popular for its computational efficiency, it lacks a guarantee for exact solutions. This paper proposes a novel solution method integrating relaxation of Column Generation conditions and automatic elimination of redundant columns, aimed at overcoming the limitations of conventional Column Generation methods in guaranteeing exact optimal solutions. Numerical experiments using actual bus route data reveal that while the traditional method achieves an exact solution rate of only about 3%, the proposed method attains a rate of approximately 99% and remarkably improves solution accuracy.

Paper Structure

This paper contains 12 sections, 3 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Set Partitioning Problem
  4. Column Generation Method
  5. Proposed Method: Column Generation and Elimination Method
  6. Evaluation Experiments
  7. Experimental Data Configuration
  8. Optimization Problem Formulation
  9. Evaluation Criteria
  10. Experimental Results
  11. Conclusions
  12. Greedy Algorithm

Figures (1)

  • Figure 1: Boxplots showing the number of bus runs (a) and bus duty candidates (b) summarized by route count. In (a), the number of trips increases proportionally with the number of routes. In (b), the number of duty candidates grows exponentially with the route count, accompanied by a larger variation range.