Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations

Trukhanov, Svyatoslav; Balasubramaniam, Chitra; Balasundaram, Balabhaskar; Butenko, Sergiy

doi:10.1007/s10589-013-9548-5

Cited by 41 publications

(51 citation statements)

References 36 publications

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“…Other approaches based on branch and cut [30] and on combinatorial algorithms [31] have roughly comparable performance. The current-best algorithm for s-PLEX is also based on a combinatorial search tree algorithm, which exploits that the s-plex property is hereditary [32].…”

Section: S-plexmentioning

confidence: 99%

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Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.

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Section: S-plexmentioning

confidence: 99%

“…Furthermore, given that s-PLEX can be solved quite efficiently in practice [29][30][31][32], an implementation of efficient fixed-parameter algorithms for s-BUNDLE seems clearly within reach.…”

Section: Open Topic 6 Perform a Multivariate Complexity Analysis Formentioning

confidence: 99%

Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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“…These models may be more practical on denser networks, however, the proposed scale reduction technique will not be applicable in this case, calling for alternative solution methods, such as a combinatorial branch-and-bound method similar to RDS [29,44]. As noted above, RDS is not directly applicable due to the lack of heredity property, however, it can be modified to address the cases when heredity is violated.…”

Section: Discussionmentioning

confidence: 99%

Scale Reduction Techniques for Computing Maximum Induced Bicliques

et al. 2017

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Given a simple, undirected graph G, a biclique is a subset of vertices inducing a complete bipartite subgraph in G. In this paper, we consider two associated optimization problems, the maximum biclique problem, which asks for a biclique of the maximum cardinality in the graph, and the maximum edge biclique problem, aiming to find a biclique with the maximum number of edges in the graph. These NP-hard problems find applications in biclustering-type tasks arising in complex network analysis. Real-life instances of these problems often involve massive, but sparse networks. We develop exact approaches for detecting optimal bicliques in large-scale graphs that combine effective scale reduction techniques with integer programming methodology. Results of computational experiments with numerous real-life network instances demonstrate the performance of the proposed approach.

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“…Once a vertex u is chosen from the RCL and the current solution is updated to

S^{0} \cup {u}

, the candidate list C must be updated to correspond to

S^{0} \cup {u}

. To construct the new candidate list, explicitly verifying if

S^{0} \cup {u, v}

is a k ‐plex for every

v \in C ∖ {u}

is very inefficient as reported in . However, this update can be done more efficiently based on the following properties.…”

Section: A New Grasp For K‐plex Detectionmentioning

confidence: 99%

“…However, this update can be done more efficiently based on the following properties. Vertex

v \in C ∖ {u}

is no longer a candidate with respect to the solution

S^{0} \cup {u}

if one of the following conditions is true :

v \notin N (u)

and

| S^{0} ∖ N (v) | = k - 1

, There exists

w \in S^{0} ∖ N (v)

such that

| S^{0} \cup {u} ∖ N (w) | = k - 1

. Under the first condition,

v \in C ∖ {u}

has exactly k −1 non‐neighbors in S 0 . As v is not adjacent to u , it must have k non‐neighbors in

S^{0} \cup {u}

.…”

Section: A New Grasp For K‐plex Detectionmentioning

confidence: 99%

Approaches for finding cohesive subgroups in large‐scale social networks via maximum k‐plex detection

Miao

Balasundaram

2017

Networks

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A k‐plex is a clique relaxation introduced in social network analysis to model cohesive social subgroups that allows for a limited number of nonadjacent vertices (strangers) inside the cohesive subgroup. Several exact algorithms and heuristic approaches to find a maximum‐size k‐plex in the graph have been developed recently for this NP‐hard problem. This article develops a greedy randomized adaptive search procedure (GRASP) for the maximum k‐plex problem. We offer a key improvement in the design of the construction procedure that alleviates a drawback observed in multiple past studies. In existing construction heuristics, k‐plexes found for smaller values of parameter k are sometimes not found for larger k even though they are feasible; instead inferior solutions are found. We identify the reasons behind this behavior and address these in our new construction procedure. We then show that an existing exact algorithm for solving this problem on power‐law graphs can be considerably enhanced by using GRASP. The overall approach is able to solve the problem to optimality on massive social networks, including some with several million vertices and edges. These are orders of magnitude larger than the largest real‐life social networks on which this problem has been solved to optimality in the current literature. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 388–407 2017

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Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations

Cited by 41 publications

References 36 publications

Multivariate Algorithmics for Finding Cohesive Subnetworks

Multivariate Algorithmics for Finding Cohesive Subnetworks

Scale Reduction Techniques for Computing Maximum Induced Bicliques

Approaches for finding cohesive subgroups in large‐scale social networks via maximum k‐plex detection

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