scholarly journals Alternative Branching Strategies in the Branch and Bound Algorithm by Using a K-Clique Covering Vertex Set for Maximum Clique Problems

2020 ◽  
Vol 1 (4) ◽  
pp. 208-216
Author(s):  
Mochamad Suyudi ◽  
Asep K. Supriatna ◽  
Sukono Sukono
Keyword(s):  
Lower Bound ◽  
Branch And Bound ◽  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Theory Problem ◽  
Maximum Cardinality ◽  
Arbitrary Graph ◽  
Complete Subgraph ◽  
Vertex Set ◽  
The Difference

The maximum clique problem (MCP) is graph theory problem that demand complete subgraph with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm. In this paper, we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarantee finding maximum clique.

scholarly journals Alternative Branching Strategies in the Branch and Bound Algorithm by Using a k-clique covering vertex set for Maximum Clique Problems.

2020 ◽  
Vol 1 (4) ◽  
pp. 208-216
Author(s):  
Mochamad Suyudi ◽  
Asep K Supriatna ◽  
Sukono Sukono
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Branch And Bound ◽  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Theory Problem ◽  
Maximum Cardinality ◽  
Arbitrary Graph ◽  
Vertex Set ◽  
The Difference

The Maximum clique problem (MCP) is graph theory problem that demand complete subgraf with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm, in this paper we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarenteed finnding maximum clique.

scholarly journals A Convex Relaxation Bound for Subgraph Isomorphism

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Christian Schellewald
Keyword(s):  
Lower Bound ◽  
Structural Information ◽  
Convex Relaxation ◽  
Maximum Clique ◽  
Combinatorial Problem ◽  
Isomorphism Problem ◽  
Maximum Clique Problem ◽  
Subgraph Isomorphism ◽  
Semidefinite Programming Relaxation ◽  
Problem Instances

In this work a convex relaxation of a subgraph isomorphism problem is proposed, which leads to a new lower bound that can provide a proof that a subgraph isomorphism between two graphs can not be found. The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail. We consider subgraph isomorphism problem instances of simple graphs which means that only the structural information of the two graphs is exploited and other information that might be available (e.g., node positions) is ignored. The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation. Therefore, for problem instances with a lower bound that is larger than zero this represents a proof that a subgraph isomorphism can not exist. But note that conversely, a negative lower bound does not imply that a subgraph isomorphism must be present and only indicates that a subgraph isomorphism can not be excluded. In addition, the relation of our approach and the reformulation of the largest common subgraph problem into a maximum clique problem is discussed.

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 A New Lower Bound on the Independence Number of a Graph and Applications

10.37236/3601 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Christian Löwenstein ◽  
Justin Southey ◽  
Anders Yeo
Keyword(s):  
Lower Bound ◽  
Maximum Degree ◽  
Independence Number ◽  
Maximum Clique ◽  
Independent Set ◽  
Uniform Hypergraph ◽  
Maximum Cardinality ◽  
Uniform Hypergraphs ◽  
Transversal Number ◽  
Graph Parameters

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

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