Pointer Network Based Deep Learning Algorithm for the Maximum Clique Problem

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.

scholarly journals Randomized heuristic for the maximum clique problem

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Approximation Algorithm ◽  
Randomized Algorithm ◽  
Maximum Clique ◽  
Constant Factor ◽  
Maximum Clique Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Np Hard Problem ◽  
Java Code ◽  
Clique Problem

<p>A clique in a graph is a set of vertices that are all directly connected</p><p>to each other i.e. a complete sub-graph. A clique of the largest size is</p><p>called a maximum clique. Finding the maximum clique in a graph is an</p><p>NP-hard problem and it cannot be solved by an approximation algorithm</p><p>that returns a solution within a constant factor of the optimum. In this</p><p>work, we present a simple and very fast randomized algorithm for the</p><p>maximum clique problem. We also provide Java code of the algorithm</p><p>in our git repository. Results show that the algorithm is able to find</p><p>reasonably good solutions to some randomly chosen DIMACS benchmark</p><p>graphs. Rather than aiming for optimality, we aim to find good solutions</p><p>very fast.</p>

scholarly journals Randomized heuristic for the maximum clique problem

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Approximation Algorithm ◽  
Randomized Algorithm ◽  
Maximum Clique ◽  
Constant Factor ◽  
Maximum Clique Problem ◽  
Hard Problem ◽  
Np Hard ◽  
Np Hard Problem ◽  
Java Code ◽  
Clique Problem

<p>A clique in a graph is a set of vertices that are all directly connected</p><p>to each other i.e. a complete sub-graph. A clique of the largest size is</p><p>called a maximum clique. Finding the maximum clique in a graph is an</p><p>NP-hard problem and it cannot be solved by an approximation algorithm</p><p>that returns a solution within a constant factor of the optimum. In this</p><p>work, we present a simple and very fast randomized algorithm for the</p><p>maximum clique problem. We also provide Java code of the algorithm</p><p>in our git repository. Results show that the algorithm is able to find</p><p>reasonably good solutions to some randomly chosen DIMACS benchmark</p><p>graphs. Rather than aiming for optimality, we aim to find good solutions</p><p>very fast.</p>

hClique: An exact algorithm for maximum clique problem in uniform hypergraphs

2017 ◽  
Vol 09 (06) ◽  
pp. 1750078 ◽  
Author(s):  
Jose Torres-Jimenez ◽  
Jose Carlos Perez-Torres ◽  
Gildardo Maldonado-Martinez
Keyword(s):  
Maximum Clique ◽  
Exact Algorithm ◽  
Maximum Clique Problem ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
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.

scholarly journals GCLIQUE: An Open Source Genetic Algorithm for the Maximum Clique Problem

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Genetic Algorithm ◽  
Maximum Clique ◽  
Maximum Size ◽  
The Other ◽  
Maximum Clique Problem ◽  
Hard Problem ◽  
Maximum Cliques ◽  
Np Hard Problem ◽  
Optimum Solution ◽  
Clique Problem

<p>A clique in a graph is a set of vertices that are all connected to each</p><p>other. A maximum clique is a clique of maximum size. A graph may have</p><p>more than one maximum cliques. The problem of finding a maximum</p><p>clique is a strongly hard NP-hard problem. It is not possible to find an</p><p>approximation algorithm which finds a maximum clique that is a constant</p><p>factor of the optimum solution. In this work, we present a genetic algorithm</p><p>for the maximum clique problem that is able to find optimum or</p><p>close to optimum solutions to most DIMACS graphs. The genetic algorithm</p><p>uses new crossover mechanisms which are able to find reasonably</p><p>good cliques which can then be used in other applications downstream.</p><p>We also provide C++ code for our algorithm. Results show that our algorithm</p><p>is able to find maximum cliques for most DIMACS instances, and</p><p>if not, close to optimum solutions for the other instances.</p>

An exact algorithm for the maximum clique problem

1990 ◽  
Vol 9 (6) ◽  
pp. 375-382 ◽  
Author(s):  
Randy Carraghan ◽  
Panos M. Pardalos
Keyword(s):  
Maximum Clique ◽  
Exact Algorithm ◽  
Maximum Clique Problem ◽  
Clique Problem

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

2020 ◽  
Vol 32 (3) ◽  
pp. 747-762 ◽  
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.

scholarly journals GCLIQUE: An Open Source Genetic Algorithm for the Maximum Clique Problem

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Genetic Algorithm ◽  
Maximum Clique ◽  
Maximum Size ◽  
The Other ◽  
Maximum Clique Problem ◽  
Hard Problem ◽  
Maximum Cliques ◽  
Np Hard Problem ◽  
Optimum Solution ◽  
Clique Problem

<p>A clique in a graph is a set of vertices that are all connected to each</p><p>other. A maximum clique is a clique of maximum size. A graph may have</p><p>more than one maximum cliques. The problem of finding a maximum</p><p>clique is a strongly hard NP-hard problem. It is not possible to find an</p><p>approximation algorithm which finds a maximum clique that is a constant</p><p>factor of the optimum solution. In this work, we present a genetic algorithm</p><p>for the maximum clique problem that is able to find optimum or</p><p>close to optimum solutions to most DIMACS graphs. The genetic algorithm</p><p>uses new crossover mechanisms which are able to find reasonably</p><p>good cliques which can then be used in other applications downstream.</p><p>We also provide C++ code for our algorithm. Results show that our algorithm</p><p>is able to find maximum cliques for most DIMACS instances, and</p><p>if not, close to optimum solutions for the other instances.</p>

Sign in / Sign up

Export Citation Format

Share Document