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# Algorithm Selection for the Maximum Clique Problem

Name
Alexey
Surname
Nikolaev
Scientific organization
Laboratory of Algorithms and Technologies for Network Analysis
Academic degree
Master
Position
Probationer
Scientific discipline
Information technologies
Topic
Algorithm Selection for the Maximum Clique Problem
Abstract
In this talk a new approach for solving the maximum clique problem is presented. For every instance the suggested approach predicts if a heuristic should be used before starting an exact algorithm and predicts the fastest algorithm from several algorithms for the maximum clique problem. Then the chosen algorithm is applied for solving the maximum clique problem in a given graph. According to the computational results the proposed approach is effective and it overcomes state-of-the-art exact algorithms.
Keywords
maximum clique problem, exact algorithms, machine learning
Summary

A simple, undirected graph G = (V ,E) is defined by a set of vertices V = {v1, v2,...,vn} and a set of edges E made up of pairs of distinct vertices (E ⊆ V × V ). A clique in graph G is a complete subgraph, that is, a subgraph in which vertices are pairwise adjacent. In this work, we consider the maximum clique problem (MCP), which asks for a clique of the largest cardinality in the graph. The size of a maximum clique is called the clique number of the graph and is usually denoted as ω(G).

The MCP is a well known and deeply studied NP-hard problem in graph theory. Moreover, it has found applications in many different fields, such as data association problems in bioinformatics and computational biology [1–3], computer vision , and robotics . Such association problems may be reduced to the MCP in a correspondence graph, which subsumes the matching criteria between the two entities involved. With the upsurge of Web technologies, cliques have also been applied to capture the structure of massive networks. For example, in social networks a clique can identify a group of cooperating agents (e.g. a terrorist cell); in the World Wide Web, cliques or quasi-cliques can help detect frequently visited pages concerning a certain topic. Clique kernels can also help to identify clusters.

In the literature, there are many different approaches to solving the MCP exactly. Most successful exact solvers belong to the family of branch-and-bound algorithms that employ approximate-colour bounds [6–12]. But theoretical analysis of these algorithms appears to be difficult. Instead of it empirical analysis of the algorithm properties is made using DIMACS benchmarks. Many papers on this topic conclude that a new algorithm is better than other algorithms, if it spends less time in total on solving benchmarks than other algorithms. However the fact that some specific algorithms turn out to show better results for some specific graphs is usually ignored.

The ideal solution for this problem is to create a kind of "oracle" algorithm that would predict how much time each of the existing algorithms would spend on solving the MCP for a given graph. After that the oracle would choose the fastest algorithm. Unfortunately there is no such thing as the ideal oracle thus in practice we can only develop a heuristic for predicting the fastest algorithm from a predifined set of exact algorithms.

This work presents an approach which is based on machine learning. It selects the fastest algorithm from the set of state-of-the-art solvers. In order to do it some features were extracted and are used for the proposed approach. The computational results show that the suggested aproach is effective.

References
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2. Eblen J, Phillips C, Rogers G, Langston M (2012) The maximum clique enumeration problem: algorithms, applications, and implementations. BMC Bioinforma 13:S5

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12. San Segundo P, Nikolaev A, Batsyn M (2015) Infra-chromatic bound for exact maximum clique search. Comput Oper Res 64:293–303