An exact algorithm for the maximum clique problem

A hypergraph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text] differs from a graph in that an edge can connect more than two vertices. An r-uniform hypergraph [Formula: see text] is a hypergraph with hyperedges of size [Formula: see text]. For an r-uniform hypergraph [Formula: see text], an r-uniform clique is a subset [Formula: see text] of [Formula: see text] such as every subset of [Formula: see text] elements of [Formula: see text] belongs to [Formula: see text]. We present hClique, an exact algorithm to find a maximum r-uniform clique for [Formula: see text]-uniform graphs. In order to evidence the performance of hClique, 32 random [Formula: see text]-graphs were solved.

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A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem

INFORMS Journal on Computing ◽

10.1287/ijoc.2019.0898 ◽

2020 ◽

Vol 32 (3) ◽

pp. 747-762 ◽

Cited By ~ 1

Author(s):

Seyedmohammadhossein Hosseinian ◽

Dalila B. M. M. Fontes ◽

Sergiy Butenko

Keyword(s):

Maximum Clique ◽

Exact Algorithm ◽

Clique Number ◽

Superior Performance ◽

Maximum Clique Problem ◽

Edge Weight ◽

Linear Programming Formulation ◽

Lagrangian Bound ◽

Integer Linear Programming Formulation ◽

Clique Problem

This paper explores the connections between the classical maximum clique problem and its edge-weighted generalization, the maximum edge weight clique (MEWC) problem. As a result, a new analytic upper bound on the clique number of a graph is obtained and an exact algorithm for solving the MEWC problem is developed. The bound on the clique number is derived using a Lagrangian relaxation of an integer (linear) programming formulation of the MEWC problem. Furthermore, coloring-based bounds on the clique number are used in a novel upper-bounding scheme for the MEWC problem. This scheme is employed within a combinatorial branch-and-bound framework, yielding an exact algorithm for the MEWC problem. Results of computational experiments demonstrate a superior performance of the proposed algorithm compared with existing approaches.

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Pointer Network Based Deep Learning Algorithm for the Maximum Clique Problem

International Journal of Artificial Intelligence Tools ◽

10.1142/s0218213021400042 ◽

2021 ◽

Vol 30 (01) ◽

pp. 2140004

Author(s):

Shenshen Gu ◽

Hanmei Yao

Keyword(s):

Reinforcement Learning ◽

Supervised Learning ◽

Learning Strategy ◽

Learning Algorithm ◽

Maximum Clique ◽

Exact Algorithm ◽

Maximum Clique Problem ◽

Deep Learning Algorithm ◽

Np Hard Problem ◽

Clique Problem

The maximum clique problem (MCP) is a famous NP-hard problem, which is difficult for the exact algorithm to solve when the dimension is large. In this paper, we applied the pointer network based method to solve this problem. First, we illustrated how to train the network with supervised learning strategy to obtain the solution to the maximum clique problem. We then further trained the pointer network with reinforcement learning strategy to obtain the vertices from the graph. For both strategies, backtracking algorithm is used to reselect the vertices. From the experimental results, we can see that both supervised learning and reinforcement learning work well. Promising results can be obtained up to 100 dimensions. This illustrates that the deep neural network based algorithms have great potentials for solving the maximum clique problem effectively and efficiently.

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