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Keywords = k-clique search

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16 pages, 337 KB  
Article
Graph Coloring via Clique Search with Symmetry Breaking
by Sándor Szabó and Bogdán Zaválnij
Symmetry 2022, 14(8), 1574; https://doi.org/10.3390/sym14081574 - 30 Jul 2022
Cited by 3 | Viewed by 2319
Abstract
It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the [...] Read more.
It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the symmetries present in the auxiliary graph. The proposed method can also be used for efficient exact coloring of hyper graphs. We also precondition the auxiliary graph in order to further reduce the search space. We carry out numerical experiments to assess the practicality of these proposals. We solve some hard cases and prove a new lower limit of seven for the mycielski7 graph with the aid of the proposed technique. Full article
(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)
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22 pages, 447 KB  
Article
Clique Search in Graphs of Special Class and Job Shop Scheduling
by Sándor Szabó and Bogdán Zaválnij
Mathematics 2022, 10(5), 697; https://doi.org/10.3390/math10050697 - 23 Feb 2022
Cited by 5 | Viewed by 2187
Abstract
In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, [...] Read more.
In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructed graph. We carry out numerical experiments to assess the efficiency of the approach. It is a common practice that before one embarks on a large scale clique search typically one attempts to simplify and tidy up the given graph. This procedure is commonly referred as preconditioning or kernelization of the given graph. Of course, the preconditioning or kernelization is meant with respect to the given type of clique search problem. The other main topic of the paper is to describe a number of kernelization methods tailored particularly to the proposed special k-clique problem. Some of these techniques works in connection with the generic k-clique problem. In these situations, we will see that they are more efficient in the case of k-partite graphs. Some other preconditioning methods applicable only to k-partite graphs. We illustrate how expedient these preconditioning methods can be by solving non-trivial scheduling problems to optimality employing only kernelization techniques dispensing with exhaustive clique search algorithms altogether. Full article
(This article belongs to the Section E: Applied Mathematics)
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17 pages, 382 KB  
Article
A Restart Local Search for Solving Diversified Top-k Weight Clique Search Problem
by Jun Wu and Minghao Yin
Mathematics 2021, 9(21), 2674; https://doi.org/10.3390/math9212674 - 21 Oct 2021
Cited by 3 | Viewed by 2144
Abstract
Diversified top-k weight clique (DTKWC) search problem is an important generalization of the diversified top-k clique (DTKC) search problem with practical applications. The diversified top-k weight clique search problem aims to search k maximal cliques that can cover the maximum [...] Read more.
Diversified top-k weight clique (DTKWC) search problem is an important generalization of the diversified top-k clique (DTKC) search problem with practical applications. The diversified top-k weight clique search problem aims to search k maximal cliques that can cover the maximum weight in a vertex weighted graph. In this work, we propose a novel local search algorithm called TOPKWCLQ for the DTKWC search problem which mainly includes two strategies. First, a restart strategy is adopted, which repeated the construction and updating processes of the maximal weight clique set. Second, a scoring heuristic is designed by giving different priorities for maximal weight cliques in candidate set. Meanwhile, a constraint model of the DTKWC search problem is constructed such that the research concerns can be evaluated. Experimental results show that the proposed algorithm TOPKWCLQ outperforms than the comparison algorithm on large-scale real-world graphs. Full article
(This article belongs to the Special Issue Graphs, Metrics and Models)
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