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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

For alpha > 1, an alpha-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I',k') (of a problem Q') of size bounded by a function of k such that, for every c >= 1, a c-approximate solution for the new instance can be turned into a (c alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every alpha > 1, Connected Dominating Set admits a polynomial-size alpha-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP \subseteq coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r \in N there cannot exist an alpha-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any alpha > 1 (assuming the Gap Exponential Time Hypothesis).

Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy Kernels for Connected Dominating Set on Sparse Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.29, author = {Eiben, Eduard and Kumar, Mithilesh and Mouawad, Amer E. and Panolan, Fahad and Siebertz, Sebastian}, title = {{Lossy Kernels for Connected Dominating Set on Sparse Graphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.29}, URN = {urn:nbn:de:0030-drops-85027}, doi = {10.4230/LIPIcs.STACS.2018.29}, annote = {Keywords: Lossy Kernelization, Connected Dominating Set, Sparse Graph Classes} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

In the l-Component Order Connectivity problem (l in N), we are given a graph G on n vertices, m edges and a non-negative integer k and asks whether there exists a set of vertices S subseteq V(G) such that |S| <= k and the size of the largest connected component in G-S is at most l. In this paper, we give a kernel for l-Component Order Connectivity with at most 2*l*k vertices that takes n^{O(l)} time for every constant l. On the way to obtaining our kernel, we prove a generalization of the q-Expansion Lemma to weighted graphs. This generalization may be of independent interest.

Mithilesh Kumar and Daniel Lokshtanov. A 2lk Kernel for l-Component Order Connectivity. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kumar_et_al:LIPIcs.IPEC.2016.20, author = {Kumar, Mithilesh and Lokshtanov, Daniel}, title = {{A 2lk Kernel for l-Component Order Connectivity}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.20}, URN = {urn:nbn:de:0030-drops-69345}, doi = {10.4230/LIPIcs.IPEC.2016.20}, annote = {Keywords: Parameterized algorithms, Kernel, Component Order Connectivity, Max-min allocation, Weighted expansion} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

A bipartite tournament is a directed graph T:=(A cup B, E) such that every pair of vertices (a,b), a in A, b in B are connected by an arc, and no arc connects two vertices of A or two vertices of B. A feedback vertex set is a set S of vertices in T such that T - S is acyclic. In this article we consider the Feedback Vertex Set problem in bipartite tournaments. Here the input is a bipartite tournament T on n vertices together with an integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for Feedback Vertex Set in Bipartite Tournaments. The running time of our algorithm is upper-bounded by O(1.6181^k + n^{O(1)}), improving over the previously best known algorithm with running time (2^k)k^{O(1)} + n^{O(1)} [Hsiao, ISAAC 2011]. As a by-product, we also obtain the fastest currently known exact exponential-time algorithm for the problem, with running time O(1.3820^n).

Mithilesh Kumar and Daniel Lokshtanov. Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.24, author = {Kumar, Mithilesh and Lokshtanov, Daniel}, title = {{Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.24}, URN = {urn:nbn:de:0030-drops-68591}, doi = {10.4230/LIPIcs.FSTTCS.2016.24}, annote = {Keywords: Parameterized algorithms, Exact algorithms, Feedback vertex set, Tour- naments, Bipartite tournaments} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T\S is acyclic. In this article we consider the FEEDBACK VERTEX SET problem in tournaments. Here the input is a tournament T and an integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for FEEDBACK VERTEX SET IN TOURNAMENTS. The running time of our algorithm is upper-bounded by O(1.6181^k + n^{O(1)}) and by O(1.466^n). Thus our algorithm simultaneously improves over the fastest known parameterized algorithm for the problem by Dom et al. running in time O(2^kk^{O(1)} + n^{O(1)}), and the fastest known exact exponential-time algorithm by Gaspers and Mnich with running time O(1.674^n). On the way to proving our main result we prove a strengthening of a special case of a graph partitioning theorem due to Bollobas and Scott. In particular we show that the vertices of any undirected m-edge graph of maximum degree d can be colored white or black in such a way that for each of the two colors, the number of edges with both endpoints of that color is between m/4-d/2 and m/4+d/2.

Mithilesh Kumar and Daniel Lokshtanov. Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.STACS.2016.49, author = {Kumar, Mithilesh and Lokshtanov, Daniel}, title = {{Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {49:1--49:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.49}, URN = {urn:nbn:de:0030-drops-57501}, doi = {10.4230/LIPIcs.STACS.2016.49}, annote = {Keywords: Parameterized algorithms, Exact algorithms, Feedback vertex set, Tour- naments, Graph partitions} }

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