On the parameterized complexity of BicliqueCover and partition

Chandran, S and Issac, D and Karrenbauer, A (2017) On the parameterized complexity of BicliqueCover and partition. In: 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, 24 - 26 August 2016, Aarhus.

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Abstract

Given a bipartite graph G, we consider the decision problem called BicliqueCover for a fixed positive integer parameter k where we are asked whether the edges of G can be covered with at most k complete bipartite subgraphs (a.k.a. bicliques). In the BicliquePartition problem, we have the additional constraint that each edge should appear in exactly one of the k bicliques. These problems are both known to be NP-complete but fixed parameter tractable. However, the known FPT algorithms have a running time that is doubly exponential in k, and the best known kernel for both problems is exponential in k. We build on this kernel and improve the running time for BicliquePartition to O(22k2+k log k+k) by exploiting a linear algebraic view on this problem. On the other hand, we show that no such improvement is possible for BicliqueCover unless the Exponential Time Hypothesis (ETH) is false by proving a doubly exponential lower bound on the running time. We achieve this by giving a reduction from 3SAT on n variables to an instance of BicliqueCover with k = O(log n). As a further consequence of this reduction, we show that there is no subexponential kernel for BicliqueCover unless P = NP. Finally, we point out the significance of the exponential kernel mentioned above for the design of polynomialtime approximation algorithms for the optimization versions of both problems. That is, we show that it is possible to obtain approximation factors of n/log n for both problems, whereas the previous best approximation factor was n/√plog n.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Keywords:	Algebra; Approximation algorithms; Graph theory; Linear algebra; Optimization; Parameterization, Approximation factor; Biclique; Exponential time hypothesis; Finite fields; Fixed positive integers; Lower bounds; Parameterized complexity; Polynomial-time approximation algorithms, Parameter estimation
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	12 Jul 2022 12:20
Last Modified:	12 Jul 2022 12:20
URI:	https://eprints.iisc.ac.in/id/eprint/74587

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