Sublinear-Space and Bounded-Delay Algorithms for Maximal Clique Enumeration in Graphs

Conte, Alessio; Grossi, Roberto; Marino, Andrea; Versari, Luca

doi:10.1007/s00453-019-00656-8

Cited by 14 publications

(9 citation statements)

References 29 publications

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“…Such an algorithm is bound to use an amount of memory that is exponential in the size of r. This is due to the fact that some databases have exponentially many minimal solutions and the decision subroutine at least has to read all of H. Note that such a large memory consumption is not at all necessary as there are algorithms known for Transversal Hypergraph whose space complexity is only linear in the input size [27,39]. In fact, enumeration algorithms are often analyzed not only with respect to their running time, but also in terms of space consumption, see [9,14]. For data profiling problems like Enumerate Minimal UCCs on the other hand, space-efficient algorithms have only recently started to received some attention [5,6,36].…”

Section: Approximation and Discoverymentioning

confidence: 99%

The Complexity of Dependency Detection and Discovery in Relational Databases

Bläsius¹,

Friedrich²,

Schirneck³

2021

Preprint

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Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs).We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs.We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.

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Section: Approximation and Discoverymentioning

confidence: 99%

The Complexity of Dependency Detection and Discovery in Relational Databases

Bläsius¹,

Friedrich²,

Schirneck³

2021

Preprint

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“…It defines a "successor" function to traverse from one solution to the others, which corresponds to a DFS on an implicit solution graph where the nodes represent desired subgraphs and the links capture the traversals from desired subgraphs to other desired subgraphs. Recently, this framework has been adopted to solve various enumeration problems, e.g., independent sets [11,40], cliques [7,12,32], 𝑘-plexes [4], and general structures that satisfy the hereditary property [6,10,14]. The algorithms following this framework achieve an output-sensitive time complexity that is proportional to the number of desired subgraphs (within the input graph).…”

Section: Related Workmentioning

confidence: 99%

Efficient Algorithms for Maximal k-Biplex Enumeration

Yu¹,

Cheng²,

Liu³

et al. 2021

Preprint

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Mining maximal subgraphs with cohesive structures from a bipartite graph has been widely studied. One important cohesive structure on bipartite graphs is 𝑘-biplex, where each vertex on one side disconnects at most 𝑘 vertices on the other side. In this paper, we study the maximal 𝑘-biplex enumeration problem which enumerates all maximal 𝑘-biplexes. Existing methods suffer from efficiency and/or scalability issues and have the time of waiting for the next output exponential w.r.t. the size of the input bipartite graph (i.e., an exponential delay). In this paper, we adopt a reverse search framework called bTraversal, which corresponds to a depth-first search (DFS) procedure on an implicit solution graph on top of all maximal 𝑘-biplexes. We then develop a series of techniques for improving and implementing this framework including (1) carefully selecting an initial solution to start DFS, (2) pruning the vast majority of links from the solution graph of bTraversal, and (3) implementing abstract procedures of the framework. The resulting algorithm is called iTraversal, which has its underlying solution graph significantly sparser than (around 0.1% of) that of bTraversal. Besides, iTraversal provides a guarantee of polynomial delay. Our experimental results on real and synthetic graphs, where the largest one contains one billion edges, show that our algorithm is up to four orders of magnitude faster than existing algorithms. CCS CONCEPTS• Mathematics of computing → Graph algorithms.

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“…Thus, it seems relevant to find nice parameters of the graph to get tractable complexities. There has been a long line of work following this approach and such results can be found in [2,3,4,5,8,10,11,14] for instance. The parameters used in these papers often try to capture the sparseness of the input graph, since many real words graphs have, in fact, few edges.…”

Section: Introductionmentioning

confidence: 99%

Efficient maximal cliques enumeration in weakly closed graphs

Manoussakis¹

2023

Preprint

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We show that the algorithm presented in [J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein. Finding cliques in social networks: A new distribution-free model. SIAM journal on computing, 49(2):448-464, 2020.] can be modified to have enumeration time complexity αO(npoly(c)). Here parameter c is the weakly closure of the graph and α its number of maximal cliques. This result improves on their complexity which was not output sensitive and exponential in the closure of the graph.

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Sublinear-Space and Bounded-Delay Algorithms for Maximal Clique Enumeration in Graphs

Cited by 14 publications

References 29 publications

The Complexity of Dependency Detection and Discovery in Relational Databases

The Complexity of Dependency Detection and Discovery in Relational Databases

Efficient Algorithms for Maximal k-Biplex Enumeration

Efficient maximal cliques enumeration in weakly closed graphs

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