Numerical experiments with LP formulations of the maximum clique problem

Central European Journal of Operations Research ◽

10.1007/s10100-021-00776-z ◽

2021 ◽

Author(s):

Dóra Kardos ◽

Patrik Patassy ◽

Sándor Szabó ◽

Bogdán Zaválnij

Keyword(s):

Linear Programming ◽

Numerical Experiments ◽

Maximum Clique ◽

Maximum Clique Problem ◽

Graph Problem ◽

Clique Problem

AbstractThe maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one linear programming formulations. In this case study we deal with some of these formulations. We consider ways for tightening the formulations. We carry out numerical experiments to see the improvements the tightened formulations provide.

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Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

Mathematica Pannonica ◽

10.1556/314.2020.00005 ◽

2021 ◽

Vol 27_NS1 (1) ◽

pp. 32-47

Author(s):

Ákos Beke ◽

Sándor Szabó ◽

Bogdán Zavalnij

Keyword(s):

Linear Programming ◽

Optimization Problems ◽

Maximum Clique ◽

Independent Set ◽

Maximum Clique Problem ◽

Linear Programs ◽

Combinatorial Optimization Problems ◽

Edge Covering ◽

Covering Set ◽

Clique Problem

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

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Metric space method for constructing splitting partitions of graphs

Acta Universitatis Sapientiae Informatica ◽

10.2478/ausi-2019-0009 ◽

2019 ◽

Vol 11 (2) ◽

pp. 131-141

Author(s):

Sándor Szabó

Keyword(s):

Metric Space ◽

Numerical Experiments ◽

Search Algorithm ◽

Maximum Clique ◽

Clique Number ◽

Maximum Clique Problem ◽

Branching Rule ◽

The Given ◽

Space Method ◽

Clique Problem

Abstract In an earlier work [6] the concept of splitting partition of a graph was introduced in connection with the maximum clique problem. A splitting partition of a graph can be used to replace the graph by two smaller graphs in the course of a clique search algorithm. In other words splitting partitions can serve as a branching rule in an algorithm to compute the clique number of a given graph. In the paper we revisit this branching idea. We will describe a technique to construct not necessary optimal splitting partitions. The given graph can be viewed as a metric space and the geometry of this space plays a guiding role. In order to assess the performance of the procedure we carried out numerical experiments.

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