Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved Dynamics for the Maximum Common Subgraph Problem

Davide Guidobene, Guido Cera

TL;DR

New heuristics aimed at mitigating challenges of the Maximum Common Subgraph problem through the reformulation of the MCS problem as the Maximum Clique and its complement, the Maximum Independent Set are introduced.

Abstract

The Maximum Common Subgraph (MCS) problem plays a crucial role across various domains, bridging theoretical exploration and practical applications in fields like bioinformatics and social network analysis. Despite its wide applicability, MCS is notoriously challenging and is classified as an NP-Complete (NPC) problem. This study introduces new heuristics aimed at mitigating these challenges through the reformulation of the MCS problem as the Maximum Clique and its complement, the Maximum Independent Set. Our first heuristic leverages the Motzkin-Straus theorem to reformulate the Maximum Clique Problem as a constrained optimization problem, continuing the work of Pelillo in Replicator Equations, Maximal Cliques, and Graph Isomorphism (1999) with replicator dynamics and introducing annealed imitation heuristics as in Dominant Sets and Hierarchical Clustering (Pavan and Pelillo, 2003) to improve chances of convergence to better local optima. The second technique applies heuristics drawn upon strategies for the Maximum Independent Set problem to efficiently reduce graph sizes as used by Akiwa and Iwata in 2014. This enables faster computation and, in many instances, yields near-optimal solutions. Furthermore we look at the implementation of both techniques in a single algorithm and find that it is a promising approach. Our techniques were tested on randomly generated Erdős-Rényi graph pairs. Results indicate the potential for application and substantial impact on future research directions.

Improved Dynamics for the Maximum Common Subgraph Problem

TL;DR

New heuristics aimed at mitigating challenges of the Maximum Common Subgraph problem through the reformulation of the MCS problem as the Maximum Clique and its complement, the Maximum Independent Set are introduced.

Abstract

The Maximum Common Subgraph (MCS) problem plays a crucial role across various domains, bridging theoretical exploration and practical applications in fields like bioinformatics and social network analysis. Despite its wide applicability, MCS is notoriously challenging and is classified as an NP-Complete (NPC) problem. This study introduces new heuristics aimed at mitigating these challenges through the reformulation of the MCS problem as the Maximum Clique and its complement, the Maximum Independent Set. Our first heuristic leverages the Motzkin-Straus theorem to reformulate the Maximum Clique Problem as a constrained optimization problem, continuing the work of Pelillo in Replicator Equations, Maximal Cliques, and Graph Isomorphism (1999) with replicator dynamics and introducing annealed imitation heuristics as in Dominant Sets and Hierarchical Clustering (Pavan and Pelillo, 2003) to improve chances of convergence to better local optima. The second technique applies heuristics drawn upon strategies for the Maximum Independent Set problem to efficiently reduce graph sizes as used by Akiwa and Iwata in 2014. This enables faster computation and, in many instances, yields near-optimal solutions. Furthermore we look at the implementation of both techniques in a single algorithm and find that it is a promising approach. Our techniques were tested on randomly generated Erdős-Rényi graph pairs. Results indicate the potential for application and substantial impact on future research directions.
Paper Structure (14 sections, 9 theorems, 14 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 9 theorems, 14 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Maximum Clique Problem
  3. Maximum Independent Set Problem
  4. Aim of this Paper
  5. Techniques
  6. Replicator Dynamics (RD)
  7. Annealed Imitation Heuristics (AIH)
  8. Kernelization techniques
  9. Kernelization-AIH Pipeline
  10. Experimental results
  11. AIH vs RD
  12. Kernelization Techniques
  13. Kernelization-AIH Pipeline
  14. Conclusion

Key Result

Theorem 2.1

Let $G'=(V', E'), G"=(V", E")$ two graphs of order n and m respectively (i.e$|V'| = n, |V"|=m$) and let $G=(V, E)$ be their association graphs. Then $C' \subseteq V'$ and $C" \subseteq V"$ are a solution of the maximum common subgraph iff $C=\{(i, h) \in C'\times C"| h=\phi(i)\}$ is a maximum clique

Figures (5)

  • Figure 1:
  • Figure 2: Size of maximum common subgraph found with RD and AIH heuristics comparison
  • Figure 3: Average accuracy for graphs whose final kernel is empty. Numbers inside brackets refer to standard deviation.
  • Figure 4: Size of maximum common subgraph found with kernelization + RD and kernelization + AIH heuristics comparison
  • Figure 5: Average time to apply reduction techniques. Numbers inside brackets refer to standard deviation.

Theorems & Definitions (12)

  • Theorem 2.1
  • Definition 2.1.1
  • Definition 2.1.2
  • Theorem 2.2: Motzkin-Straus
  • Theorem 2.3: BomzePelillo1999neuripsBomze1997
  • Theorem 2.4: Fundamental Theorem of Natural Selection Pelillo1999neurips
  • Definition 2.4.1
  • Theorem 2.5: PavanPelillo2003
  • Theorem 2.6: PavanPelillo2003
  • Theorem 2.7: PelilloTorsello2006
  • ...and 2 more