Fast Enumeration of Large k-Plexes

Conte, Alessio; Firmani, Donatella; Mordente, Caterina; Patrignani, Maurizio; Torlone, Riccardo

doi:10.1145/3097983.3098031

Cited by 43 publications

(38 citation statements)

References 15 publications

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“…There has been extensive research on both clique enumeration in temporal graphs [21,39,40] and k-plex enumeration on static graphs [29,33,35,4,11,43,12]. To the best of our knowledge, we are the first to investigate the enumeration of k-plexes in temporal graphs.…”

Section: Related Workmentioning

confidence: 99%

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Listing All Maximal k -Plexes in Temporal Graphs

Bentert

Himmel

Molter

et al. 2019

ACM J. Exp. Algorithmics

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Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, Viard et al. [TCS 2016] introduced ∆-cliques as a natural temporal version of cliques. Himmel et al. [SNAM 2017] showed how to adapt the well-known Bron-Kerbosch algorithm to enumerate ∆-cliques. We continue this work and improve and extend the algorithm of Himmel et al. to enumerate temporal k-plexes (notably, cliques are the special case k = 1).We define a ∆-k-plex as a set of vertices with a lifetime, where during the lifetime each vertex has in each consecutive ∆ + 1 time steps edges to all but at most k − 1 vertices in the chosen set of vertices. We develop a recursive algorithm for enumerating all maximal ∆-k-plexes and perform experiments on real-world social networks that demonstrate the practical feasibility of our approach. In particular, for the special case of ∆-1-plexes (that is, ∆-cliques), we observe that our algorithm is on average significantly faster than the previous algorithms by Himmel et al. [SNAM 2017] and Viard et al. [IPL 2018] for enumerating ∆-cliques. * An extended abstract of this work appeared in the proceedings of the 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM '18) [3]. This version contains all proof details, extended experimental findings, and the analysis of a new version of our algorithm that fixed a small bug in the code (which has no large impact on the results). † Supported by the DFG, projects DAMM (NI 369/13) and FPTinP (NI 369/16). ‡ Supported by the DFG, project MATE (NI 369/17).

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Section: Related Workmentioning

confidence: 99%

“…To circumvent this fact, the clique concept has seen several relaxations. Our work focuses on a popular degree-based relaxation of cliques known as k-plexes [36,11,43,20,33]. In a k-plex, every vertex must be adjacent to all but at most k − 1 vertices in the k-plex (excluding itself).…”

Section: Introductionmentioning

confidence: 99%

Listing All Maximal k -Plexes in Temporal Graphs

Bentert

Himmel

Molter

et al. 2019

ACM J. Exp. Algorithmics

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“…Butenko et al [10] and Butenko and Trukhanov [11] were able to find exact maximum independent sets in graphs with thousands of vertices by first applying reductions. Further works have introduced reductions to more quickly solve the maximum clique problem [35,43] and enumerate kplexes [14]. Though these works apply reduction techniques as a preprocessing step, many works apply reductions as a natural step of the algorithm.…”

Section: Related Workmentioning

confidence: 99%

“…In practice, kernelization is used as a preprocessing step to other algorithms [10,14,15,35,40,43], where speeding up kernelization directly speeds up the algorithm. However, kernelization may also be applied repeatedly as part of an algorithm [2,12,31].…”

Section: Introductionmentioning

confidence: 99%

Scalable Kernelization for Maximum Independent Sets

Hespe¹,

Schulz²,

Strash³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms-or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching.We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.

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“…In practice, kernelization is used as a preprocessing step to other algorithms [10,14,15,34,39,42], where speeding up kernelization directly speeds up the algorithm. However, kernelization may also be applied repeatedly as part of an algorithm [2,12,30].…”

Section: Introductionmentioning

confidence: 99%

Scalable Kernelization for Maximum Independent Sets

Hespe

Schulz

Strash

2019

ACM J. Exp. Algorithmics

View full text Add to dashboard Cite

The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms-or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching. We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.

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Fast Enumeration of Large k-Plexes

Cited by 43 publications

References 15 publications

Listing All Maximal k -Plexes in Temporal Graphs

Listing All Maximal k -Plexes in Temporal Graphs

Scalable Kernelization for Maximum Independent Sets

Scalable Kernelization for Maximum Independent Sets

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