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Documents authored by Majumdar, Diptapriyo

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.19
Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

Authors: Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, and Prafullkumar Tale

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
Foucaud et al. [ICALP 2024] demonstrated that some problems in NP can admit (tight) double-exponential lower bounds when parameterized by treewidth or vertex cover number. They showed these first-of-their-kind results by proving conditional lower bounds for certain graph problems, in particular, the metric-based identification problems (Strong) Metric Dimension. We continue this line of research and highlight the usefulness of this type of problems, to prove relatively rare types of (tight) lower bounds. We investigate fine-grained algorithmic aspects of classical (non-metric based) identification problems in graphs, namely Locating-Dominating Set, and in set systems, namely Test Cover. In the first problem, an input is a graph G on n vertices and an integer k, and the objective is to decide whether there is a subset S of k vertices such that any two distinct vertices not in S are dominated by distinct subsets of S. In the second problem, an input is a set of items U, a collection of subsets ℱ of U called tests, and an integer k, and the objective is to select a set S of at most k tests such that any two distinct items are contained in a distinct subset of tests of S. For our first result, we adapt the techniques introduced by Foucaud et al. [ICALP 2024] to prove similar (tight) lower bounds for these two problems. - Locating-Dominating Set (respectively, Test Cover) parameterized by the treewidth of the input graph (respectively, the natural auxiliary graph) does not admit an algorithm running in time 2^{2^o(tw)} ⋅ poly(n) (respectively, 2^{2^o(tw)} ⋅ poly(|U| + |ℱ|))), unless the ETH fails. This augments the short list of NP-Complete problems that admit tight double-exponential lower bounds when parameterized by treewidth, and shows that "local" (non-metric-based) problems can also admit such bounds. We show that these lower bounds are tight by designing treewidth-based dynamic programming schemes with matching running times. Next, we prove that these two problems also admit "exotic" (and tight) lower bounds, when parameterized by the solution size k. We prove that unless the ETH fails, - Locating-Dominating Set does not admit an algorithm running in time 2^o(k²) ⋅ poly(n), nor a polynomial-time kernelization algorithm that reduces the solution size and outputs a kernel with 2^o(k) vertices, and - Test Cover does not admit an algorithm running in time 2^{2^o(k)} ⋅ poly(|U| + |ℱ|) nor a kernel with 2^{2^o(k)} vertices. Again, we show that these lower bounds are tight by designing (kernelization) algorithms with matching running times. To the best of our knowledge, Locating-Dominating Set is the first known problem which is FPT when parameterized by solution size k, where the optimal running time has a quadratic function in the exponent. These results also extend the (very) small list of problems that admit an ETH-based lower bound on the number of vertices in a kernel, and (for Test Cover) a double-exponential lower bound when parameterized by the solution size. Whereas it is the first example, to the best of our knowledge, that admit a double exponential lower bound for the number of vertices.
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Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, and Prafullkumar Tale. Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakraborty_et_al:LIPIcs.ISAAC.2024.19,
  author =	{Chakraborty, Dipayan and Foucaud, Florent and Majumdar, Diptapriyo and Tale, Prafullkumar},
  title =	{{Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.19},
  URN =		{urn:nbn:de:0030-drops-221469},
  doi =		{10.4230/LIPIcs.ISAAC.2024.19},
  annote =	{Keywords: Identification Problems, Locating-Dominating Set, Test Cover, Double Exponential Lower Bound, ETH, Kernelization Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.41
A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees

Authors: Ashwin Jacob, Diptapriyo Majumdar, and Meirav Zehavi

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes, where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. Here, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as dense as possible or as sparse as possible (while being connected). We develop a kernel of O(k⁵) vertices for the same.
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Ashwin Jacob, Diptapriyo Majumdar, and Meirav Zehavi. A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jacob_et_al:LIPIcs.ISAAC.2024.41,
  author =	{Jacob, Ashwin and Majumdar, Diptapriyo and Zehavi, Meirav},
  title =	{{A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.41},
  URN =		{urn:nbn:de:0030-drops-221687},
  doi =		{10.4230/LIPIcs.ISAAC.2024.41},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Scattered Graph Classes, New Expansion Lemma, Cliques or Trees Vertex Deletion}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.16
Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints

Authors: Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Given an undirected graph G and a set A ⊆ V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)⧵A. An A-path is called an (A,𝓁)-path if the length of the path is exactly 𝓁. In the (A, 𝓁)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, 𝓁)-paths in G or not. The problem is already known to be fixed-parmeter tractable when parameterized by k+𝓁 via color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or 𝓁 (P₃-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of G. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance between G and a path graph (distance to path). We use a randomized reduction to achieve the mentioned result. Toward this, we prove a lemma similar to the influential "isolation lemma": Given a set system (X,ℱ) if the elements of X are assigned a weight uniformly at random from a set of values fairly large, then each subset in ℱ will have a unique weight with high probability. We believe that this result will be useful beyond the scope of this paper. ALPP being hard even for structural parameters like distance to path+|A| rules out the possibility of any FPT algorithms for many well-known other structural parameters, including FVS+|A| and treewidth+|A|. There is a straightforward FPT algorithm for ALPP parameterized by vc, the vertex cover number of the input graph. Following this, we consider the parameters CVD(cluster vertex deletion)+|A| and CVD+|𝓁| and show the problem to be FPT with respect to these parameters. Note that CVD is incomparable to the treewidth of a graph and has been in vogue recently.
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Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu. Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandopadhyay_et_al:LIPIcs.MFCS.2024.16,
  author =	{Bandopadhyay, Susobhan and Banik, Aritra and Majumdar, Diptapriyo and Sahu, Abhishek},
  title =	{{Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.16},
  URN =		{urn:nbn:de:0030-drops-205725},
  doi =		{10.4230/LIPIcs.MFCS.2024.16},
  annote =	{Keywords: Parameterized complexity, (A,𝓁)-Path Packing, Kernelization, Randomized-Exponential Time Hypothesis, Graph Classes}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.45
Finding a Highly Connected Steiner Subgraph and its Applications

Authors: Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
Given a (connected) undirected graph G, a set X ⊆ V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ⊇ X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness). Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.
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Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan. Finding a Highly Connected Steiner Subgraph and its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2023.45,
  author =	{Eiben, Eduard and Majumdar, Diptapriyo and Ramanujan, M. S.},
  title =	{{Finding a Highly Connected Steiner Subgraph and its Applications}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{45:1--45:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.45},
  URN =		{urn:nbn:de:0030-drops-185793},
  doi =		{10.4230/LIPIcs.MFCS.2023.45},
  annote =	{Keywords: Parameterized Complexity, Steiner Subgraph Extension, p-edge-connected graphs, Matroids, Representative Families}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.18
Parameterized Complexity of Deletion to Scattered Graph Classes

Authors: Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most k vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including Vertex Cover, Feedback Vertex Set, Odd Cycle Transveral, Cluster Vertex Deletion, and Perfect Deletion. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is fixed-parameter tractable (FPT) is elusive, it has long been known that if the graph class is characterized by a finite set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to scattered graph classes where we need to delete at most k vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem (in the case where each graph class has a finite forbidden set) is fixed-parameter tractable by a O^*(2^(k^O(1))) algorithm, using a combination of the well-known techniques in parameterized complexity - iterative compression and important separators. Our approach follows closely that of a related problem in the context of satisfiability [Ganian, Ramanujan, Szeider, TAlg 2017], where one wants to find a small backdoor set so that the resulting CSP (constraint satisfaction problem) instance belongs to one of several easy instances of satisfiability. While we follow the main idea from this work, there are some challenges for our problem which we needed to overcome. When there are two graph classes with finite forbidden sets to get to, and if one of the forbidden sets has a path, then we show that the problem has a (better) singly exponential algorithm and a polynomial sized kernel. We also design an efficient FPT algorithm for a special case when one of the graph classes has an infinite forbidden set. Specifically, we give a O^*(4^k) algorithm to determine whether k vertices can be deleted from a given graph so that in the resulting graph, each connected component is a tree (the sparsest connected graph) or a clique (the densest connected graph).
Cite as

Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman. Parameterized Complexity of Deletion to Scattered Graph Classes. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jacob_et_al:LIPIcs.IPEC.2020.18,
  author =	{Jacob, Ashwin and Majumdar, Diptapriyo and Raman, Venkatesh},
  title =	{{Parameterized Complexity of Deletion to Scattered Graph Classes}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133210},
  doi =		{10.4230/LIPIcs.IPEC.2020.18},
  annote =	{Keywords: Parameterized Complexity, Scattered Graph Classes, Important Separators}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.19
Parameterized Pre-Coloring Extension and List Coloring Problems

Authors: Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors.
Cite as

Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström. Parameterized Pre-Coloring Extension and List Coloring Problems. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gutin_et_al:LIPIcs.STACS.2020.19,
  author =	{Gutin, Gregory and Majumdar, Diptapriyo and Ordyniak, Sebastian and Wahlstr\"{o}m, Magnus},
  title =	{{Parameterized Pre-Coloring Extension and List Coloring Problems}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.19},
  URN =		{urn:nbn:de:0030-drops-118801},
  doi =		{10.4230/LIPIcs.STACS.2020.19},
  annote =	{Keywords: Parameterized Algorithms, W-hardness, Kernelization, Graph Coloring, List Coloring}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2016.21
Structural Parameterizations of Feedback Vertex Set

Authors: Diptapriyo Majumdar

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

Abstract
A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. It is well-known that the problem of finding a minimum sized (or k sized in case of decision version of) feedback vertex set (FVS) is polynomial time solvable in (sub)-cubic graphs, in pseudo-forests (graphs where each component has at most one cycle) and mock-forests (graph where each vertex is part of at most one cycle). In general graphs, it is known that the problem is NP-Complete, and has an O*((3.619)^k) fixed-parameter algorithm and an O(k^2) kernel where k, the solution size is the parameter. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure of the input. In particular, we show that * FVS is fixed-parameter tractable, but is unlikely to have polynomial sized kernel when parameterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper. * When parameterized by k, the number of vertices, whose deletion results in a pseudo-forest, we give an O(k^6) vertices kernel improving from the previously known O(k^{10}) bound. * When parameterized by the number k of vertices, whose deletion results in a mock-d-forest, we give a kernel consisting of O(k^{3d+3}) vertices and prove a lower bound of Omega(k^{d+2}) vertices (under complexity theoretic assumptions). Mock-d-forest for a constant d is a mock-forest where each component has at most d cycles.
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Diptapriyo Majumdar. Structural Parameterizations of Feedback Vertex Set. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{majumdar:LIPIcs.IPEC.2016.21,
  author =	{Majumdar, Diptapriyo},
  title =	{{Structural Parameterizations of Feedback Vertex Set}},
  booktitle =	{11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
  pages =	{21:1--21:16},
  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.2016.21},
  URN =		{urn:nbn:de:0030-drops-69337},
  doi =		{10.4230/LIPIcs.IPEC.2016.21},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Feedback Vertex Set, Structural Parameterization}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.26
Kernelization of Cycle Packing with Relaxed Disjointness Constraints

Authors: Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, and Saket Saurabh

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1.
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Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, and Saket Saurabh. Kernelization of Cycle Packing with Relaxed Disjointness Constraints. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{agrawal_et_al:LIPIcs.ICALP.2016.26,
  author =	{Agrawal, Akanksha and Lokshtanov, Daniel and Majumdar, Diptapriyo and Mouawad, Amer E. and Saurabh, Saket},
  title =	{{Kernelization of Cycle Packing with Relaxed Disjointness Constraints}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.26},
  URN =		{urn:nbn:de:0030-drops-63053},
  doi =		{10.4230/LIPIcs.ICALP.2016.26},
  annote =	{Keywords: parameterized complexity, cycle packing, kernelization, relaxation}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2015.331
Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators

Authors: Diptapriyo Majumdar, Venkatesh Raman, and Saket Saurabh

Published in: LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Abstract
Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(l^2) edges and 2*l vertices. Here, l denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size O(f^3), where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two. While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial sized kernel. The input to our problem consists of an undirected graph G, S \subseteq V(G) such that |S| = k and G[V(G)\S] has maximum degree at most 2 and a positive integer l. Given (G,S,l), in polynomial time we output an instance (G',S',l') such that |V(G')|<= O(k^5), |E(G')|<= O(k^6) and G has a vertex cover of size at most l if and only if G' has a vertex cover of size at most l'. When G[V(G)\S] has maximum degree at most 1, we improve the known kernel bound from O(k^3) vertices to O(k^2) vertices (and O(k^3) edges). In general, if G[V(G)\S] is simply a collection of cliques of size at most d, then we transform the graph in polynomial time to an equivalent hypergraph with O(k^d) vertices and show that, for d >= 3, a kernel with O(k^{d-epsilon}) vertices is unlikely to exist for any epsilon >0 unless NP is a subset of coNO/poly.
Cite as

Diptapriyo Majumdar, Venkatesh Raman, and Saket Saurabh. Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 331-342, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

Copy BibTex To Clipboard

@InProceedings{majumdar_et_al:LIPIcs.IPEC.2015.331,
  author =	{Majumdar, Diptapriyo and Raman, Venkatesh and Saurabh, Saket},
  title =	{{Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{331--342},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Husfeldt, Thore and Kanj, Iyad},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.331},
  URN =		{urn:nbn:de:0030-drops-55943},
  doi =		{10.4230/LIPIcs.IPEC.2015.331},
  annote =	{Keywords: Parameterized Complexity, Kernelization, expansion lemma, vertex cover, structural parameterization}
}
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