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Analysis of Evolutionary Diversity Optimisation for the Maximum Matching Problem

Jonathan Gadea Harder, Aneta Neumann, Frank Neumann

TL;DR

This work investigates evolutionary diversity optimization (EDO) for the maximum matching problem on complete bipartite graphs and path graphs, using binary string encoding and Hamming distance to quantify diversity. It analyzes two population-based algorithms, the (μ+1)-EA_D and the Two-Phase Matching EA_D (2P-EA_D), providing drift-theorem–based runtime bounds and supporting empirical evidence. The theoretical results show polynomial-time convergence to a maximally diverse population of high-quality matchings, with distinct bounds depending on the graph structure and gap between partitions; the 2P-EA_D generally yields tighter runtimes. Empirical experiments corroborate the theory and indicate potential refinements to the bounds, while demonstrating the practical relevance of maintaining diverse, high-quality matchings for decision-making in combinatorial settings.

Abstract

This paper explores the enhancement of solution diversity in evolutionary algorithms (EAs) for the maximum matching problem, concentrating on complete bipartite graphs and paths. We adopt binary string encoding for matchings and use Hamming distance to measure diversity, aiming for its maximization. Our study centers on the $(μ+1)$-EA and $2P-EA_D$, which are applied to optimize diversity. We provide a rigorous theoretical and empirical analysis of these algorithms. For complete bipartite graphs, our runtime analysis shows that, with a reasonably small $μ$, the $(μ+1)$-EA achieves maximal diversity with an expected runtime of $O(μ^2 m^4 \log(m))$ for the small gap case (where the population size $μ$ is less than the difference in the sizes of the bipartite partitions) and $O(μ^2 m^2 \log(m))$ otherwise. For paths, we establish an upper runtime bound of $O(μ^3 m^3)$. The $2P-EA_D$ displays stronger performance, with bounds of $O(μ^2 m^2 \log(m))$ for the small gap case, $O(μ^2 n^2 \log(n))$ otherwise, and $O(μ^3 m^2)$ for paths. Here, $n$ represents the total number of vertices and $m$ the number of edges. Our empirical studies, which examine the scaling behavior with respect to $m$ and $μ$, complement these theoretical insights and suggest potential for further refinement of the runtime bounds.

Analysis of Evolutionary Diversity Optimisation for the Maximum Matching Problem

TL;DR

This work investigates evolutionary diversity optimization (EDO) for the maximum matching problem on complete bipartite graphs and path graphs, using binary string encoding and Hamming distance to quantify diversity. It analyzes two population-based algorithms, the (μ+1)-EA_D and the Two-Phase Matching EA_D (2P-EA_D), providing drift-theorem–based runtime bounds and supporting empirical evidence. The theoretical results show polynomial-time convergence to a maximally diverse population of high-quality matchings, with distinct bounds depending on the graph structure and gap between partitions; the 2P-EA_D generally yields tighter runtimes. Empirical experiments corroborate the theory and indicate potential refinements to the bounds, while demonstrating the practical relevance of maintaining diverse, high-quality matchings for decision-making in combinatorial settings.

Abstract

This paper explores the enhancement of solution diversity in evolutionary algorithms (EAs) for the maximum matching problem, concentrating on complete bipartite graphs and paths. We adopt binary string encoding for matchings and use Hamming distance to measure diversity, aiming for its maximization. Our study centers on the -EA and , which are applied to optimize diversity. We provide a rigorous theoretical and empirical analysis of these algorithms. For complete bipartite graphs, our runtime analysis shows that, with a reasonably small , the -EA achieves maximal diversity with an expected runtime of for the small gap case (where the population size is less than the difference in the sizes of the bipartite partitions) and otherwise. For paths, we establish an upper runtime bound of . The displays stronger performance, with bounds of for the small gap case, otherwise, and for paths. Here, represents the total number of vertices and the number of edges. Our empirical studies, which examine the scaling behavior with respect to and , complement these theoretical insights and suggest potential for further refinement of the runtime bounds.
Paper Structure (27 sections, 12 theorems, 22 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 27 sections, 12 theorems, 22 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Preliminaries
  5. Maximum matching problem and diversity optimization
  6. Diversity measure
  7. Algorithms
  8. Drift theorems
  9. Runtime Analysis for complete bipartite graphs
  10. Runtime Analysis for paths
  11. Matching $E^3O^0$:
  12. Matching $E^2O^1$:
  13. Matching $E^1O^2$:
  14. Matching $E^0O^3$:
  15. Matching $E^3O^0$:
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $S \subseteq \mathbb{R}$ be a finite set of positive numbers and let $({X^t})_{t\in \mathbb{N}}$ be a sequence of random variables over $S \cup \{0\}$. Let $T$ be the random variable that denotes the first point in time $t \in \mathbb{N}$ for which $X^t \leq 0$. Suppose that there exists a const holds. Then If there exists a constant $\delta_2 > 0$ such that holds. Then

Figures (2)

  • Figure 1: Experimental results on complete bipartite graphs
  • Figure 2: Experimental results on paths

Theorems & Definitions (22)

  • Theorem 2.1: Additive Drift Theoremdriftthm
  • Theorem 2.2: Multiplicative Drift Theoremmultdrift
  • Lemma 3.1: Diversity of a Population
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 12 more