The Shape of the Search Tree for the Maximum Clique Problem, and the Implications for Parallel Branch and Bound

TL;DR

This paper analyzes the search tree structure of the maximum clique problem and introduces a scalable parallel branch and bound method that leverages search order and solution location insights for improved performance.

Contribution

It provides a novel explanation for the success of parallel algorithms in maximum clique search and proposes a new work splitting mechanism that enhances scalability and load balancing.

Findings

Parallel search order and solution location influence algorithm success.

The proposed work splitting mechanism improves scalability.

Explicit early diversity prevents overcommitment to heuristics.

Abstract

Finding a maximum clique in a given graph is one of the fundamental NP-hard problems. We compare two multi-core thread-parallel adaptations of a state-of-the-art branch and bound algorithm for the maximum clique problem, and provide a novel explanation as to why they are successful. We show that load balance is sometimes a problem, but that the interaction of parallel search order and the most likely location of solutions within the search space is often the dominating consideration. We use this explanation to propose a new low-overhead, scalable work splitting mechanism. Our approach uses explicit early diversity to avoid strong commitment to the weakest heuristic advice, and late resplitting for balance. More generally, we argue that for branch and bound, parallel algorithm design should not be performed independently of the underlying sequential algorithm.