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Exploitation Strategies in Conditional Markov Chain Search: A case study on the three-index assignment problem

Sahil Patel, Daniel Karapetyan

TL;DR

The paper investigates exploitation in Conditional Markov Chain Search (CMCS) for discrete combinatorial optimization, focusing on improving exploitation without losing exploration. It introduces two exploitation-enhancing extensions: Strategy B, which integrates Variable Neighborhood Descent (VND) triggered by new best-found solutions, and Strategy C, which splits the time budget into exploration and exploitation via two sub-configurations. Applied to the NP-hard Three-Index Assignment Problem (AP3), Strategy C consistently outperforms the baseline Strategy A and the similar Strategy B, though Strategy C incurs higher configurator complexity. The contributions include a CMCS configurator with mutation-driven transition-matrix optimisation and a rich AP3 component pool, enabling automated design of high-performing two- and multi-phase CMCS configurations for complex combinatorial problems.

Abstract

The Conditional Markov Chain Search (CMCS) is a framework for automated design of metaheuristics for discrete combinatorial optimisation problems. Given a set of algorithmic components such as hill climbers and mutations, CMCS decides in which order to apply those components. The decisions are dictated by the CMCS configuration that can be learnt offline. CMCS does not have an acceptance criterion; any moves are accepted by the framework. As a result, it is particularly good in exploration but is not as good at exploitation. In this study, we explore several extensions of the framework to improve its exploitation abilities. To perform a computational study, we applied the framework to the three-index assignment problem. The results of our experiments showed that a two-stage CMCS is indeed superior to a single-stage CMCS.

Exploitation Strategies in Conditional Markov Chain Search: A case study on the three-index assignment problem

TL;DR

The paper investigates exploitation in Conditional Markov Chain Search (CMCS) for discrete combinatorial optimization, focusing on improving exploitation without losing exploration. It introduces two exploitation-enhancing extensions: Strategy B, which integrates Variable Neighborhood Descent (VND) triggered by new best-found solutions, and Strategy C, which splits the time budget into exploration and exploitation via two sub-configurations. Applied to the NP-hard Three-Index Assignment Problem (AP3), Strategy C consistently outperforms the baseline Strategy A and the similar Strategy B, though Strategy C incurs higher configurator complexity. The contributions include a CMCS configurator with mutation-driven transition-matrix optimisation and a rich AP3 component pool, enabling automated design of high-performing two- and multi-phase CMCS configurations for complex combinatorial problems.

Abstract

The Conditional Markov Chain Search (CMCS) is a framework for automated design of metaheuristics for discrete combinatorial optimisation problems. Given a set of algorithmic components such as hill climbers and mutations, CMCS decides in which order to apply those components. The decisions are dictated by the CMCS configuration that can be learnt offline. CMCS does not have an acceptance criterion; any moves are accepted by the framework. As a result, it is particularly good in exploration but is not as good at exploitation. In this study, we explore several extensions of the framework to improve its exploitation abilities. To perform a computational study, we applied the framework to the three-index assignment problem. The results of our experiments showed that a two-stage CMCS is indeed superior to a single-stage CMCS.
Paper Structure (35 sections, 2 figures, 1 table, 4 algorithms)

This paper contains 35 sections, 2 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. CMCS Exploitation Strategies
  3. $\text{CMCS}^\text{A}$: Strategy A
  4. $\text{CMCS}^\text{B}$: Strategy B
  5. $\text{CMCS}^\text{C}$: Strategy C
  6. CMCS Configurator
  7. Configuration mutations
  8. Swap Rows.
  9. Shuffle Row.
  10. Minimum change.
  11. Ruin and Recreate.
  12. Void.
  13. Case study: The three-index assignment problem (AP3)
  14. Data structures
  15. AP3 components
  16. ...and 20 more sections

Figures (2)

  • Figure 1: Evaluation of the generated configurations. The graph shows how the solution error changes throughout the run of CMCS. The solution error is averaged across all the instances in the test set.
  • Figure 2: Generated 3-component Strategy C configuration (the best configuration found in this study). The blue arcs correspond to transitions following improvement of the solution, while the red arcs correspond to the transitions following the solution not being improved). The thickness of each arc is proportionate to the frequency of the corresponding transition.