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Summary
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This study explores finding the Maximum Clique in random graphs. Local algorithms can find larger cliques than previously thought, even in challenging "hidden clique" problems.

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Area of Science:

  • Graph Theory
  • Computational Complexity
  • Random Graph Theory

Background:

  • The Maximum Clique problem is a fundamental challenge in graph theory, involving the identification of the largest complete subgraph within a given graph.
  • Erdös-Rényi G(N, p) random graphs provide a standard model for studying graph properties as a function of size (N) and edge probability (p).

Purpose of the Study:

  • To investigate the structure of the Maximum Clique problem in Erdös-Rényi random graphs.
  • To analyze the performance of local algorithms in finding cliques, particularly in finite-sized systems and challenging scenarios like the hidden clique problem.

Main Methods:

  • Analysis of the Maximum Clique problem as a function of graph size (N) and clique size (K).
  • Exploration of phase boundaries and their finite widths in random graphs.
  • Evaluation of extended traditional fast local algorithms for clique finding.
  • Comparison of local search algorithms with message passing and spectral algorithms for the hidden clique problem.

Main Results:

  • A complex phase boundary was observed, characterized by a staircase structure where the maximum findable clique size increases incrementally.
  • The finite width of these phase boundaries enables local algorithms to identify cliques beyond theoretical limits established for infinite systems.
  • Extended local algorithms demonstrate accessibility to much of the 'hard' problem space even at finite graph sizes (N).
  • Early-stopping local searches show potential to outperform advanced algorithms (message passing, spectral) in the unique hidden clique problem.

Conclusions:

  • Local algorithms are effective for finding Maximum Cliques in random graphs, even in challenging configurations.
  • The finite nature of phase boundaries in random graphs offers new opportunities for clique-finding algorithms.
  • For the specific case of the hidden clique problem, tailored local search strategies can be highly efficient.