550 research outputs found
Listing all maximal cliques in sparse graphs in near-optimal time
The degeneracy of an -vertex graph is the smallest number such that every subgraph of contains a vertex of degree at most . We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time . We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an -vertex graph with degeneracy (when is a multiple of 3 and ) is . Therefore, our algorithm matches the worst-case output size of the problem whenever
On combinatorial optimisation in analysis of protein-protein interaction and protein folding networks
Abstract: Protein-protein interaction networks and protein folding networks represent prominent research topics at the intersection of bioinformatics and network science. In this paper, we present a study of these networks from combinatorial optimisation point of view. Using a combination of classical heuristics and stochastic optimisation techniques, we were able to identify several interesting combinatorial properties of biological networks of the COSIN project. We obtained optimal or near-optimal solutions to maximum clique and chromatic number problems for these networks. We also explore patterns of both non-overlapping and overlapping cliques in these networks. Optimal or near-optimal solutions to partitioning of these networks into non-overlapping cliques and to maximum independent set problem were discovered. Maximal cliques are explored by enumerative techniques. Domination in these networks is briefly studied, too. Applications and extensions of our findings are discussed
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
Fine-grained Search Space Classification for Hard Enumeration Variants of Subset Problems
We propose a simple, powerful, and flexible machine learning framework for
(i) reducing the search space of computationally difficult enumeration variants
of subset problems and (ii) augmenting existing state-of-the-art solvers with
informative cues arising from the input distribution. We instantiate our
framework for the problem of listing all maximum cliques in a graph, a central
problem in network analysis, data mining, and computational biology. We
demonstrate the practicality of our approach on real-world networks with
millions of vertices and edges by not only retaining all optimal solutions, but
also aggressively pruning the input instance size resulting in several fold
speedups of state-of-the-art algorithms. Finally, we explore the limits of
scalability and robustness of our proposed framework, suggesting that
supervised learning is viable for tackling NP-hard problems in practice.Comment: AAAI 201
Shared-Memory Parallel Maximal Clique Enumeration
We present shared-memory parallel methods for Maximal Clique Enumeration
(MCE) from a graph. MCE is a fundamental and well-studied graph analytics task,
and is a widely used primitive for identifying dense structures in a graph. Due
to its computationally intensive nature, parallel methods are imperative for
dealing with large graphs. However, surprisingly, there do not yet exist
scalable and parallel methods for MCE on a shared-memory parallel machine. In
this work, we present efficient shared-memory parallel algorithms for MCE, with
the following properties: (1) the parallel algorithms are provably
work-efficient relative to a state-of-the-art sequential algorithm (2) the
algorithms have a provably small parallel depth, showing that they can scale to
a large number of processors, and (3) our implementations on a multicore
machine shows a good speedup and scaling behavior with increasing number of
cores, and are substantially faster than prior shared-memory parallel
algorithms for MCE.Comment: 10 pages, 3 figures, proceedings of the 25th IEEE International
Conference on. High Performance Computing, Data, and Analytics (HiPC), 201
Listing k-cliques in Sparse Real-World Graphs
International audienceMotivated by recent studies in the data mining community which require to efficiently list all k-cliques, we revisit the iconic algorithm of Chiba and Nishizeki and develop the most efficient parallel algorithm for such a problem. Our theoretical analysis provides the best asymptotic upper bound on the running time of our algorithm for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs shows that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any k) in graphs containing up to tens of millions of edges as well as all 10-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. Finally, we show how our algorithm can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs
Mining Maximal Cliques from an Uncertain Graph
We consider mining dense substructures (maximal cliques) from an uncertain
graph, which is a probability distribution on a set of deterministic graphs.
For parameter 0 < {\alpha} < 1, we present a precise definition of an
{\alpha}-maximal clique in an uncertain graph. We present matching upper and
lower bounds on the number of {\alpha}-maximal cliques possible within an
uncertain graph. We present an algorithm to enumerate {\alpha}-maximal cliques
in an uncertain graph whose worst-case runtime is near-optimal, and an
experimental evaluation showing the practical utility of the algorithm.Comment: ICDE 201
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