On the Complexity of Maximum Clique Algorithms: usage of coloring heuristics leads to the 2^(n\5) algorithm running time lower bound

TL;DR

This paper establishes a theoretical lower bound of 2^(n/5) on the worst-case running time for a class of maximum clique algorithms that utilize coloring heuristics, highlighting fundamental complexity limits.

Contribution

It provides the first proven exponential lower bound for MCP algorithms based on coloring heuristics, advancing understanding of their computational complexity.

Findings

Proves a 2^(n/5) lower bound for certain MCP algorithms

Highlights limitations of current worst-case analysis methods

Emphasizes the need for alternative approaches to analyze algorithm complexity

Abstract

Maximum Clique Problem(MCP) is one of the 21 original NP--complete problems enumerated by Karp in 1972. In recent years a large number of exact methods to solve MCP have been appeared(Babel, Wood, Kumlander, Fahle, Li, Tomita and etc). Most of them are branch and bound algorithms that use branching rule introduced by Balas and Yu and based on coloring heuristics to establish an upper bound on the clique number. They differ from each other primarily in vertex preordering and vertex coloring methods. Current methods of worst case running time analysis for branch and bound algorithms do not allow to provide tight upper bounds. This motivates the study of lower bounds for such algorithms. We prove 2^(n\5) lower bound for group of MCP algorithms based on usage of coloring heuristics.