3 Search Results for "Francis, Mathew C."


Document
On the Kernel and Related Problems in Interval Digraphs

Authors: Mathew C. Francis, Pavol Hell, and Dalu Jacob

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
Given a digraph G, a set X ⊆ V(G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S ⊆ V(G) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality kernel, absorbing set (or dominating set) and the problems of computing a maximum cardinality kernel, independent set are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals (S_u,T_u) can be assigned to each vertex u of G such that (u,v) ∈ E(G) if and only if S_u ∩ T_v ≠ ∅. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs - which arise when we require that the two intervals assigned to a vertex have to intersect. We see as our main contribution the identification of the class of reflexive interval digraphs as an important class of digraphs. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for reflexive interval digraphs. We also obtain some new results for undirected graphs along the way: (a) We get an O(n(n+m)) time algorithm for computing a minimum cardinality (undirected) independent dominating set in cocomparability graphs, which slightly improves the existing O(n³) time algorithm for the same problem by Kratsch and Stewart; and (b) We show that the Red Blue Dominating Set problem, which is NP-complete even for planar bipartite graphs, is linear-time solvable on interval bigraphs, which is a class of bipartite (undirected) graphs closely related to interval digraphs.

Cite as

Mathew C. Francis, Pavol Hell, and Dalu Jacob. On the Kernel and Related Problems in Interval Digraphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{francis_et_al:LIPIcs.ISAAC.2021.17,
  author =	{Francis, Mathew C. and Hell, Pavol and Jacob, Dalu},
  title =	{{On the Kernel and Related Problems in Interval Digraphs}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.17},
  URN =		{urn:nbn:de:0030-drops-154505},
  doi =		{10.4230/LIPIcs.ISAAC.2021.17},
  annote =	{Keywords: Interval digraphs, kernel, absorbing set, dominating set, independent set, algorithms, approximation hardness}
}
Document
Partially Polynomial Kernels for Set Cover and Test Cover

Authors: Manu Basavaraju, Mathew C. Francis, M. S. Ramanujan, and Saket Saurabh

Published in: LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)


Abstract
In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S' \subseteq S of size at most k satisfying some desired properties. If S' is required to contain all the elements of U then it corresponds to the classical Set Cover problem. On the other hand if we require S' to satisfy the property that for every pair of elements x,y \in U there exists a set S \in S' such that |S \cap {x,y}|=1 then it corresponds to the Test Cover problem. In this paper we consider a natural parameterization of Set Cover and Test Cover. More precisely, we study the (n-k)-Set Cover and (n-k)-Test Cover problems, where the objective is to find a subfamily S' of size at most n-k satisfying the respective properties, from the kernelization perspective. It is known in the literature that both (n-k)-Set Cover and (n-k)-Test Cover do not admit polynomial kernels (under some well known complexity theoretic assumptions). However, in this paper we show that they do admit "partially polynomial kernels". More precisely, we give polynomial time algorithms that take as input an instance (U,S,k) of (n-k)-Set Cover (n-k)-Test Cover) and return an equivalent instance (~U,~S,~k) of (n-k)-Set Cover (respectively (n-k)-Test Cover) with ~k <= k and |~U|= O(k^2) (|~U|=O(k^7)). These results allow us to generalize, improve and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain "sparsity properties". Using a part of our kernelization algorithm for (n-k)-Set Cover, we also get an improved FPT algorithm for this problem which runs in time O(4^k*k^{\O(1)}*(m+n)) improving over the previous best of O(8^{k+o(k)}*(m+n)^{O(1)}). On the other hand the partially polynomial kernel for (n-k)-Test Cover implies the first single exponential FPT algorithm, an algorithm with running time O(2^{O(k^2)}*(m+n)^{O(1)}). We believe such an approach will also be useful for other covering problems as well.

Cite as

Manu Basavaraju, Mathew C. Francis, M. S. Ramanujan, and Saket Saurabh. Partially Polynomial Kernels for Set Cover and Test Cover. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 67-78, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{basavaraju_et_al:LIPIcs.FSTTCS.2013.67,
  author =	{Basavaraju, Manu and Francis, Mathew C. and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{Partially Polynomial Kernels for Set Cover and Test Cover}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)},
  pages =	{67--78},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-64-4},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{24},
  editor =	{Seth, Anil and Vishnoi, Nisheeth K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.67},
  URN =		{urn:nbn:de:0030-drops-43621},
  doi =		{10.4230/LIPIcs.FSTTCS.2013.67},
  annote =	{Keywords: Set Cover, Test Cover, Kernelization, Parameterized Algorithms}
}
Document
Cubicity, Degeneracy, and Crossing Number

Authors: Abhijin Adiga, L. Sunil Chandran, and Rogers Mathew

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)


Abstract
A k-box B=(R_1,R_2,...,R_k), where each R_i is a closed interval on the real line, is defined to be the Cartesian product R_1 X R_2 X ... X R_k. If each R_i is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan. Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/0607092, 2006.] that, for a graph G with maximum degree \Delta, cub(G) <= \lceil 4(\Delta +1) ln n\rceil. In this paper we show that, for a k-degenerate graph G, cub(G) <= (k+2) \lceil 2e log n \rceil. Since k is at most \Delta and can be much lower, this clearly is a stronger result. We also give an efficient deterministic algorithm that runs in O(n^2k) time to output a 8k(\lceil 2.42 log n\rceil + 1) dimensional cube representation for G. The crossing number of a graph G, denoted as CR(G), is the minimum number of crossing pairs of edges, over all drawings of G in the plane. An important consequence of the above result is that if the crossing number of a graph G is t, then box(G) is O(t^{1/4}{\lceil log t\rceil}^{3/4}) . This bound is tight upto a factor of O((log t)^{3/4}). Let (P,\leq) be a partially ordered set and let G_{P} denote its underlying comparability graph. Let dim(P) denote the poset dimension of P. Another interesting consequence of our result is to show that dim(P) \leq 2(k+2) \lceil 2e \log n \rceil, where k denotes the degeneracy of G_{P}. Also, we get a deterministic algorithm that runs in O(n^2k) time to construct a 16k(\lceil 2.42 log n\rceil + 1) sized realizer for P. As far as we know, though very good upper bounds exist for poset dimension in terms of maximum degree of its underlying comparability graph, no upper bounds in terms of the degeneracy of the underlying comparability graph is seen in the literature.

Cite as

Abhijin Adiga, L. Sunil Chandran, and Rogers Mathew. Cubicity, Degeneracy, and Crossing Number. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 176-190, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{adiga_et_al:LIPIcs.FSTTCS.2011.176,
  author =	{Adiga, Abhijin and Chandran, L. Sunil and Mathew, Rogers},
  title =	{{Cubicity, Degeneracy, and Crossing Number}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{176--190},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.176},
  URN =		{urn:nbn:de:0030-drops-33428},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.176},
  annote =	{Keywords: Degeneracy, Cubicity, Boxicity, Crossing Number, Interval Graph, Intersection Graph, Poset Dimension, Comparability Graph}
}
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