The maximum l-triangle k-club problem: Complexity, properties, and algorithms
•The computational complexity of the problem is established, for any l ≥ 1 and k ≥ 2.•Cohesiveness properties of graphs induced by l-triangle k-clubs are derived.•New valid inequalities are presented.•Solution approaches based on alternative formulations of the problem are proposed.•Numerical result...
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| Published in: | Computers & operations research Vol. 111; pp. 258 - 270 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Elsevier Ltd
01.11.2019
Pergamon Press Inc |
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| ISSN: | 0305-0548, 1873-765X, 0305-0548 |
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| Abstract | •The computational complexity of the problem is established, for any l ≥ 1 and k ≥ 2.•Cohesiveness properties of graphs induced by l-triangle k-clubs are derived.•New valid inequalities are presented.•Solution approaches based on alternative formulations of the problem are proposed.•Numerical results obtained on real-world networks are reported.
Given a graph G = (V, E) and two positive integers l and k, an l-triangle k-club is a subset of nodes that induces a subgraph with each node included in at least l triplets linked pairwise and maximum distance between each pair of nodes at most k. This structure aims to represent cohesive groups in social and other complex networks. The Maximum l-Triangle k-Club Problem (MlTkCP) consists of finding a maximum cardinality l-triangle k-club of a given graph. In this paper, we derive properties of l-triangle k-clubs and show that the decision version of the MlTkCP is NP-Complete, for any given integers l ≥ 1 and k ≥ 2. To solve the problem, polynomial and non-polynomial formulations designed in different variable spaces are considered. The computational performance of exact solution approaches based on them is tested on a set of real-world graphs. |
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| AbstractList | Given a graph G = (V, E) and two positive integers l and k, an l-triangle k-club is a subset of nodes that induces a subgraph with each node included in at least l triplets linked pairwise and maximum distance between each pair of nodes at most k. This structure aims to represent cohesive groups in social and other complex networks. The Maximum l-Triangle k-Club Problem (MlTkCP) consists of finding a maximum cardinality l-triangle k-club of a given graph. In this paper, we derive properties of l-triangle k-clubs and show that the decision version of the MlTkCP is NP-Complete, for any given integers l ≥ 1 and k ≥ 2. To solve the problem, polynomial and non-polynomial formulations designed in different variable spaces are considered. The computational performance of exact solution approaches based on them is tested on a set of real-world graphs. •The computational complexity of the problem is established, for any l ≥ 1 and k ≥ 2.•Cohesiveness properties of graphs induced by l-triangle k-clubs are derived.•New valid inequalities are presented.•Solution approaches based on alternative formulations of the problem are proposed.•Numerical results obtained on real-world networks are reported. Given a graph G = (V, E) and two positive integers l and k, an l-triangle k-club is a subset of nodes that induces a subgraph with each node included in at least l triplets linked pairwise and maximum distance between each pair of nodes at most k. This structure aims to represent cohesive groups in social and other complex networks. The Maximum l-Triangle k-Club Problem (MlTkCP) consists of finding a maximum cardinality l-triangle k-club of a given graph. In this paper, we derive properties of l-triangle k-clubs and show that the decision version of the MlTkCP is NP-Complete, for any given integers l ≥ 1 and k ≥ 2. To solve the problem, polynomial and non-polynomial formulations designed in different variable spaces are considered. The computational performance of exact solution approaches based on them is tested on a set of real-world graphs. |
| Author | Almeida, Maria Teresa Brás, Raúl |
| Author_xml | – sequence: 1 givenname: Maria Teresa orcidid: 0000-0002-9303-4407 surname: Almeida fullname: Almeida, Maria Teresa email: talmeida@iseg.ulisboa.pt organization: ISEG, Universidade de Lisboa, Lisboa, Portugal – sequence: 2 givenname: Raúl orcidid: 0000-0002-6088-6574 surname: Brás fullname: Brás, Raúl organization: ISEG, Universidade de Lisboa, Lisboa, Portugal |
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| Keywords | Integer programming Social network analysis Clique-relaxations Computational complexity k-club |
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| Snippet | •The computational complexity of the problem is established, for any l ≥ 1 and k ≥ 2.•Cohesiveness properties of graphs induced by l-triangle k-clubs are... Given a graph G = (V, E) and two positive integers l and k, an l-triangle k-club is a subset of nodes that induces a subgraph with each node included in at... |
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| SubjectTerms | Algorithms Clique-relaxations Complexity Computational complexity Formulations Graph theory Integer programming Integers k-club Nodes Operations research Polynomials Social network analysis |
| Title | The maximum l-triangle k-club problem: Complexity, properties, and algorithms |
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