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DOI: 10.4230/LIPIcs.FSTTCS.2025.31

Token Sliding Independent Set Reconfiguration on Block Graphs

Authors: Mathew C. Francis and Veena Prabhakaran

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

Let S be an independent set of a simple undirected graph G. Suppose that each vertex of S has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of G while maintaining the property that after each move, the vertices having tokens always form an independent set of G. We would like to determine whether the tokens can be eventually brought to stay on the vertices of another independent set S' of G in this manner. In other words, we would like to decide if we can transform S into S' through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph "is reachable" from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, and bipartite permutation graphs. We present a polynomial time algorithm for the problem on block graphs, which are the graphs in which every maximal 2-connected subgraph is a clique. Our algorithm is the first generalization of the known polynomial time algorithm for trees to a larger class of graphs.

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Mathew C. Francis and Veena Prabhakaran. Token Sliding Independent Set Reconfiguration on Block Graphs. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{francis_et_al:LIPIcs.FSTTCS.2025.31,
  author =	{Francis, Mathew C. and Prabhakaran, Veena},
  title =	{{Token Sliding Independent Set Reconfiguration on Block Graphs}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.31},
  URN =		{urn:nbn:de:0030-drops-251120},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.31},
  annote =	{Keywords: Token sliding independent set reconfiguration, block graphs, polynomial time algorithm}
}

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DOI: 10.4230/LIPIcs.ISAAC.2021.17

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.

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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

DOI: 10.4230/LIPIcs.FSTTCS.2013.67

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.

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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}
}

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Documents authored by Francis, Mathew C.

Token Sliding Independent Set Reconfiguration on Block Graphs

Abstract

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On the Kernel and Related Problems in Interval Digraphs

Abstract

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Partially Polynomial Kernels for Set Cover and Test Cover

Abstract

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