97 Search Results for "Zehavi, Meirav"


Volume

LIPIcs, Volume 214

16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

IPEC 2021, September 8-10, 2021, Lisbon, Portugal

Editors: Petr A. Golovach and Meirav Zehavi

Document
Kernelization of Counting Problems

Authors: Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time either explicitly or by implicitly reducing the counting problems to enumeration problems. Thus, our framework is the only one in the spirit of classic kernelization (as well as lossy kernelization). Specifically, we define a compression of a counting problem P into a counting problem Q as a pair of polynomial-time procedures: reduce and lift. Given an instance of P, reduce outputs an instance of Q whose size is bounded by a function f of the parameter, and given the number of solutions to the instance of Q, lift outputs the number of solutions to the instance of P. When P = Q, compression is termed kernelization, and when f is polynomial, compression is termed polynomial compression. Our technical (and other conceptual) contributions can be classified into two categories: Upper Bounds. We prove two theorems: (i) The #Vertex Cover problem parameterized by solution size admits a polynomial kernel; (ii) Every problem in the class of #Planar F-Deletion problems parameterized by solution size admits a polynomial compression. Lower Bounds. We introduce two new concepts of cross-compositions: EXACT-cross-composition and SUM-cross-composition. We prove that if a #P-hard counting problem P EXACT-cross-composes into a parameterized counting problem Q, then Q does not admit a polynomial compression unless the polynomial hierarchy collapses. We conjecture that the same statement holds for SUM-cross-compositions. Then, we prove that: (i) #Min (s,t)-Cut parameterized by treewidth does not admit a polynomial compression unless the polynomial hierarchy collapses; (ii) #Min (s,t)-Cut parameterized by minimum cut size, #Odd Cycle Transversal parameterized by solution size, and #Vertex Cover parameterized by solution size minus maximum matching size, do not admit polynomial compressions unless our conjecture is false.

Cite as

Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Kernelization of Counting Problems. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 77:1-77:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{lokshtanov_et_al:LIPIcs.ITCS.2024.77,
  author =	{Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Kernelization of Counting Problems}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{77:1--77:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.77},
  URN =		{urn:nbn:de:0030-drops-196059},
  doi =		{10.4230/LIPIcs.ITCS.2024.77},
  annote =	{Keywords: Kernelization, Counting Problems}
}
Document
Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs

Authors: Juhi Chaudhary, Harmender Gahlawat, Michal Włodarczyk, and Meirav Zehavi

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
Given an undirected graph G and a multiset of k terminal pairs 𝒳, the Vertex-Disjoint Paths (VDP) and Edge-Disjoint Paths (EDP) problems ask whether G has k pairwise internally vertex-disjoint paths and k pairwise edge-disjoint paths, respectively, connecting every terminal pair in 𝒳. In this paper, we study the kernelization complexity of VDP and EDP on subclasses of chordal graphs. For VDP, we design a 4k vertex kernel on split graphs and an 𝒪(k²) vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For EDP, we first prove that the problem is NP-complete on complete graphs. Then, we design an 𝒪(k^{2.75}) vertex kernel for EDP on split graphs, and improve it to a 7k+1 vertex kernel on threshold graphs. Lastly, we provide an 𝒪(k²) vertex kernel for EDP on block graphs and a 2k+1 vertex kernel for clique paths. Our contributions improve upon several results in the literature, as well as resolve an open question by Heggernes et al. [Theory Comput. Syst., 2015].

Cite as

Juhi Chaudhary, Harmender Gahlawat, Michal Włodarczyk, and Meirav Zehavi. Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 10:1-10:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{chaudhary_et_al:LIPIcs.IPEC.2023.10,
  author =	{Chaudhary, Juhi and Gahlawat, Harmender and W{\l}odarczyk, Michal and Zehavi, Meirav},
  title =	{{Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{10:1--10:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.10},
  URN =		{urn:nbn:de:0030-drops-194296},
  doi =		{10.4230/LIPIcs.IPEC.2023.10},
  annote =	{Keywords: Kernelization, Parameterized Complexity, Vertex-Disjoint Paths Problem, Edge-Disjoint Paths Problem}
}
Document
Collective Graph Exploration Parameterized by Vertex Cover

Authors: Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
We initiate the study of the parameterized complexity of the Collective Graph Exploration (CGE) problem. In CGE, the input consists of an undirected connected graph G and a collection of k robots, initially placed at the same vertex r of G, and each one of them has an energy budget of B. The objective is to decide whether G can be explored by the k robots in B time steps, i.e., there exist k closed walks in G, one corresponding to each robot, such that every edge is covered by at least one walk, every walk starts and ends at the vertex r, and the maximum length of any walk is at most B. Unfortunately, this problem is NP-hard even on trees [Fraigniaud et al., 2006]. Further, we prove that the problem remains W[1]-hard parameterized by k even for trees of treedepth 3. Due to the para-NP-hardness of the problem parameterized by treedepth, and motivated by real-world scenarios, we study the parameterized complexity of the problem parameterized by the vertex cover number (vc) of the graph, and prove that the problem is fixed-parameter tractable (FPT) parameterized by vc. Additionally, we study the optimization version of CGE, where we want to optimize B, and design an approximation algorithm with an additive approximation factor of O(vc).

Cite as

Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi. Collective Graph Exploration Parameterized by Vertex Cover. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gupta_et_al:LIPIcs.IPEC.2023.22,
  author =	{Gupta, Siddharth and Sa'ar, Guy and Zehavi, Meirav},
  title =	{{Collective Graph Exploration Parameterized by Vertex Cover}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.22},
  URN =		{urn:nbn:de:0030-drops-194413},
  doi =		{10.4230/LIPIcs.IPEC.2023.22},
  annote =	{Keywords: Collective Graph Exploration, Parameterized Complexity, Approximation Algorithm, Vertex Cover, Treedepth}
}
Document
Drawn Tree Decomposition: New Approach for Graph Drawing Problems

Authors: Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
Over the past decade, we witness an increasing amount of interest in the design of exact exponential-time and parameterized algorithms for problems in Graph Drawing. Unfortunately, we still lack knowledge of general methods to develop such algorithms. An even more serious issue is that, here, "standard" parameters very often yield intractability. In particular, for the most common structural parameter, namely, treewidth, we frequently observe NP-hardness already when the input graphs are restricted to have constant (often, being just 1 or 2) treewidth. Our work deals with both drawbacks simultaneously. We introduce a novel form of tree decomposition that, roughly speaking, does not decompose (only) a graph, but an entire drawing. As such, its bags and separators are of geometric (rather than only combinatorial) nature. While the corresponding parameter - like treewidth - can be arbitrarily smaller than the height (and width) of the drawing, we show that - unlike treewidth - it gives rise to efficient algorithms. Specifically, we get slice-wise polynomial (XP) time algorithms parameterized by our parameter. We present a general scheme for the design of such algorithms, and apply it to several central problems in Graph Drawing, including the recognition of grid graphs, minimization of crossings and bends, and compaction. Other than for the class of problems we discussed in the paper, we believe that our decomposition and scheme are of independent interest and can be further extended or generalized to suit even a wider class of problems. Additionally, we discuss classes of drawings where our parameter is bounded by 𝒪(√n) (where n is the number of vertices of the graph), yielding subexponential-time algorithms. Lastly, we prove which relations exist between drawn treewidth and other width measures, including treewidth, pathwidth, (dual) carving-width and embedded-width.

Cite as

Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi. Drawn Tree Decomposition: New Approach for Graph Drawing Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 23:1-23:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gupta_et_al:LIPIcs.IPEC.2023.23,
  author =	{Gupta, Siddharth and Sa'ar, Guy and Zehavi, Meirav},
  title =	{{Drawn Tree Decomposition: New Approach for Graph Drawing Problems}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{23:1--23:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.23},
  URN =		{urn:nbn:de:0030-drops-194424},
  doi =		{10.4230/LIPIcs.IPEC.2023.23},
  annote =	{Keywords: Graph Drawing, Parameterized Complexity, Tree decomposition}
}
Document
Sidestepping Barriers for Dominating Set in Parameterized Complexity

Authors: Ioannis Koutis, Michał Włodarczyk, and Meirav Zehavi

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
We study the classic Dominating Set problem with respect to several prominent parameters. Specifically, we present algorithmic results that sidestep time complexity barriers by the incorporation of either approximation or larger parameterization. Our results span several parameterization regimes, including: (i,ii,iii) time/ratio-tradeoff for the parameters treewidth, vertex modulator to constant treewidth and solution size; (iv,v) FPT-algorithms for the parameters vertex cover number and feedback edge set number; and (vi) compression for the parameter feedback edge set number.

Cite as

Ioannis Koutis, Michał Włodarczyk, and Meirav Zehavi. Sidestepping Barriers for Dominating Set in Parameterized Complexity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 31:1-31:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{koutis_et_al:LIPIcs.IPEC.2023.31,
  author =	{Koutis, Ioannis and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav},
  title =	{{Sidestepping Barriers for Dominating Set in Parameterized Complexity}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.31},
  URN =		{urn:nbn:de:0030-drops-194506},
  doi =		{10.4230/LIPIcs.IPEC.2023.31},
  annote =	{Keywords: Dominating Set, Parameterized Complexity, Approximation Algorithms}
}
Document
Parameterized Complexity of Incomplete Connected Fair Division

Authors: Harmender Gahlawat and Meirav Zehavi

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)


Abstract
Fair division of resources among competing agents is a fundamental problem in computational social choice and economic game theory. It has been intensively studied on various kinds of items (divisible and indivisible) and under various notions of fairness. We focus on Connected Fair Division (CFD), the variant of fair division on graphs, where the resources are modeled as an item graph. Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of CFD, termed Incomplete CFD (ICFD), where exactly p vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be "economical" as well as fair. We consider four well-known notions of fairness: PROP, EF, EF1, EFX. First, we prove that EF-ICFD, EF1-ICFD, and EFX-ICFD are W[1]-hard parameterized by p plus the number of agents, even for graphs having constant vertex cover number (vcn). In contrast, we present a randomized FPT algorithm for PROP-ICFD parameterized only by p. Additionally, we prove both positive and negative results concerning the kernelization complexity of ICFD under all four fairness notions, parameterized by p, vcn, and the total number of different valuations in the item graph (val).

Cite as

Harmender Gahlawat and Meirav Zehavi. Parameterized Complexity of Incomplete Connected Fair Division. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 14:1-14:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gahlawat_et_al:LIPIcs.FSTTCS.2023.14,
  author =	{Gahlawat, Harmender and Zehavi, Meirav},
  title =	{{Parameterized Complexity of Incomplete Connected Fair Division}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.14},
  URN =		{urn:nbn:de:0030-drops-193877},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.14},
  annote =	{Keywords: Fair Division, Kernelization, Connected Fair Allocation, Fixed parameter tractability}
}
Document
New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162)

Authors: Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, Meirav Zehavi, and Liana Khazaliya

Published in: Dagstuhl Reports, Volume 13, Issue 4 (2023)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 23162 "New Frontiers of Parameterized Complexity in Graph Drawing”. The seminar was held in-person from April 16 to April 21, 2023. It brought together 32 researchers from the Graph Drawing and the Parameterized Complexity research communities to discuss and explore new research frontiers on the interface between the two fields. The report collects the abstracts of talks and open problems presented in the seminar, as well as brief progress reports from the working groups.

Cite as

Robert Ganian, Fabrizio Montecchiani, Martin Nöllenburg, Meirav Zehavi, and Liana Khazaliya. New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162). In Dagstuhl Reports, Volume 13, Issue 4, pp. 58-97, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@Article{ganian_et_al:DagRep.13.4.58,
  author =	{Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana},
  title =	{{New Frontiers of Parameterized Complexity in Graph Drawing (Dagstuhl Seminar 23162)}},
  pages =	{58--97},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2023},
  volume =	{13},
  number =	{4},
  editor =	{Ganian, Robert and Montecchiani, Fabrizio and N\"{o}llenburg, Martin and Zehavi, Meirav and Khazaliya, Liana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.4.58},
  URN =		{urn:nbn:de:0030-drops-192393},
  doi =		{10.4230/DagRep.13.4.58},
  annote =	{Keywords: algorithm design, computational geometry, graph drawing, parameterized complexity}
}
Document
Kernelization for Spreading Points

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is "close" to each other. More precisely, for a family of n points, an integer k, and a real number d > 0, we ask whether at most k points could be relocated, each point at distance at most d from its original location, such that the distance between each pair of points is at least a fixed constant, say 1. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with 𝒪(d²k³) points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by k and d. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in k alone, unless NP ⊆ coNP/poly.

Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Kernelization for Spreading Points. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 48:1-48:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fomin_et_al:LIPIcs.ESA.2023.48,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Kernelization for Spreading Points}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{48:1--48:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.48},
  URN =		{urn:nbn:de:0030-drops-187017},
  doi =		{10.4230/LIPIcs.ESA.2023.48},
  annote =	{Keywords: parameterized algorithms, kernelization, spreading points, distant representatives, unit disk packing}
}
Document
Lossy Kernelization for (Implicit) Hitting Set Problems

Authors: Fedor V. Fomin, Tien-Nam Le, Daniel Lokshtanov, Saket Saurabh, Stéphan Thomassé, and Meirav Zehavi

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with 𝒪(k^d) bits (𝒪(k^d) sets and 𝒪(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization. In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (𝒪(k^{1+ε})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+ε)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments.

Cite as

Fedor V. Fomin, Tien-Nam Le, Daniel Lokshtanov, Saket Saurabh, Stéphan Thomassé, and Meirav Zehavi. Lossy Kernelization for (Implicit) Hitting Set Problems. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fomin_et_al:LIPIcs.ESA.2023.49,
  author =	{Fomin, Fedor V. and Le, Tien-Nam and Lokshtanov, Daniel and Saurabh, Saket and Thomass\'{e}, St\'{e}phan and Zehavi, Meirav},
  title =	{{Lossy Kernelization for (Implicit) Hitting Set Problems}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.49},
  URN =		{urn:nbn:de:0030-drops-187020},
  doi =		{10.4230/LIPIcs.ESA.2023.49},
  annote =	{Keywords: Hitting Set, Lossy Kernelization}
}
Document
Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large

Authors: Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP.

Cite as

Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi. Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 65:1-65:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{jacob_et_al:LIPIcs.ESA.2023.65,
  author =	{Jacob, Ashwin and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav},
  title =	{{Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.65},
  URN =		{urn:nbn:de:0030-drops-187184},
  doi =		{10.4230/LIPIcs.ESA.2023.65},
  annote =	{Keywords: Hamiltonian cycle, longest path, directed feedback vertex set, directed graphs, parameterized complexity}
}
Document
Parameterized Analysis of the Cops and Robber Game

Authors: Harmender Gahlawat and Meirav Zehavi

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


Abstract
Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber (CnR) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber. The aim is to compute the cop number of a graph, k, which is the minimum number of cops that ensures the capture of the robber. From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number (vcn). First, we establish that k ≤ vcn/3+1. Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number, and design a linear vertex kernel for neighborhood diversity. Additionally, we extend all of our results to several well-studied variations of CnR.

Cite as

Harmender Gahlawat and Meirav Zehavi. Parameterized Analysis of the Cops and Robber Game. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 49:1-49:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gahlawat_et_al:LIPIcs.MFCS.2023.49,
  author =	{Gahlawat, Harmender and Zehavi, Meirav},
  title =	{{Parameterized Analysis of the Cops and Robber Game}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{49:1--49:17},
  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.2023.49},
  URN =		{urn:nbn:de:0030-drops-185837},
  doi =		{10.4230/LIPIcs.MFCS.2023.49},
  annote =	{Keywords: Cops and Robber, Kernelization, Graph Searching, Fixed parameter tractability}
}
Document
On Computing Homological Hitting Sets

Authors: Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set 𝒮 of r-dimensional simplices of minimum cardinality so that 𝒮 meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+Δ, where Δ is the maximum degree of the Hasse graph of the complex 𝖪.

Cite as

Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi. On Computing Homological Hitting Sets. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 13:1-13:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bauer_et_al:LIPIcs.ITCS.2023.13,
  author =	{Bauer, Ulrich and Rathod, Abhishek and Zehavi, Meirav},
  title =	{{On Computing Homological Hitting Sets}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{13:1--13:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.13},
  URN =		{urn:nbn:de:0030-drops-175169},
  doi =		{10.4230/LIPIcs.ITCS.2023.13},
  annote =	{Keywords: Algorithmic topology, Cut problems, Surfaces, Parameterized complexity}
}
Document
A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

Authors: Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)


Abstract
The Delaunay graph of a point set P ⊆ ℝ² is the plane graph with the vertex-set P and the edge-set that contains {p,p'} if there exists a disc whose intersection with P is exactly {p,p'}. Accordingly, a triangulated graph G is Delaunay realizable if there exists a triangulation of the Delaunay graph of some P ⊆ ℝ², called a Delaunay triangulation of P, that is isomorphic to G. The objective of Delaunay Realization is to compute a point set P ⊆ ℝ² that realizes a given graph G (if such a P exists). Known algorithms do not solve Delaunay Realization as they are non-constructive. Obtaining a constructive algorithm for Delaunay Realization was mentioned as an open problem by Hiroshima et al. [Hiroshima et al., 2000]. We design an n^𝒪(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.

Cite as

Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi. A Finite Algorithm for the Realizabilty of a Delaunay Triangulation. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 1:1-1:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.1,
  author =	{Agrawal, Akanksha and Saurabh, Saket and Zehavi, Meirav},
  title =	{{A Finite Algorithm for the Realizabilty of a Delaunay Triangulation}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.1},
  URN =		{urn:nbn:de:0030-drops-173573},
  doi =		{10.4230/LIPIcs.IPEC.2022.1},
  annote =	{Keywords: Delaunay Triangulation, Delaunay Realization, Finite Algorithm, Integer Coordinate Realization}
}
Document
New Algorithms for Structure Informed Genome Rearrangement

Authors: Eden Ozery, Meirav Zehavi, and Michal Ziv-Ukelson

Published in: LIPIcs, Volume 242, 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)


Abstract
We define two new computational problems in the domain of perfect genome rearrangements, and propose three algorithms to solve them. The rearrangement scenarios modeled by the problems consider Reversal and Block Interchange operations, and a PQ-tree is utilized to guide the allowed operations and to compute their weights. In the first problem, Constrained TreeToString Divergence (CTTSD), we define the basic structure-informed rearrangement based divergence measure. Here, we assume that the gene order members of the gene cluster from which the PQ-tree is constructed are permutations. The PQ-tree representing the gene cluster is ordered such that the series of gene IDs spelled by its leaves is equivalent to the reference gene order. Then, a structure-informed gene rearrangement measure is computed between the ordered PQ-tree and the target gene order. The second problem, TreeToString Divergence (TTSD), generalizes CTTSD, where the gene order members are not necessarily permutations and the structure-informed rearrangement based divergence measure is extended to also consider up to d_S and d_T gene insertion and deletion operations, respectively, when modelling the PQ-tree informed divergence process from the reference order to the target order. The first algorithm solves CTTSD in O(n γ² ⋅ (m_p ⋅ 1.381^γ + m_q)) time and O(n²) space, where γ is the maximum number of children of a node, n is the length of the string and the number of leaves in the tree, and m_p and m_q are the number of P-nodes and Q-nodes in the tree, respectively. If one of the penalties of CTTSD is 0, then the algorithm runs in O(n m γ²) time and O(n²) space. The second algorithm solves TTSD in O(n² γ² {d_T}² {d_S}² m² (m_p ⋅ 5^γ γ + m_q)) time and O(d_T d_S m (m n + 5^γ)) space, where γ is the maximum number of children of a node, n is the length of the string, m is the number of leaves in the tree, m_p and m_q are the number of P-nodes and Q-nodes in the tree, respectively, and allowing d_T deletions from the tree and d_S deletions from the string. The third algorithm is intended to reduce the space complexity of the second algorithm. It solves a variant of the problem (where one of the penalties of TTSD is 0) in O(n γ² {d_T}² {d_S}² m² (m_p ⋅ 4^γ γ²n(d_T+d_S+m+n) + m_q)) time and O(γ² n m² d_T d_S (d_T+d_S+m+n)) space. The algorithm is implemented as a software tool, denoted MEM-Rearrange, and applied to the comparative and evolutionary analysis of 59 chromosomal gene clusters extracted from a dataset of 1,487 prokaryotic genomes.

Cite as

Eden Ozery, Meirav Zehavi, and Michal Ziv-Ukelson. New Algorithms for Structure Informed Genome Rearrangement. In 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 242, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ozery_et_al:LIPIcs.WABI.2022.11,
  author =	{Ozery, Eden and Zehavi, Meirav and Ziv-Ukelson, Michal},
  title =	{{New Algorithms for Structure Informed Genome Rearrangement}},
  booktitle =	{22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-243-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{242},
  editor =	{Boucher, Christina and Rahmann, Sven},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2022.11},
  URN =		{urn:nbn:de:0030-drops-170454},
  doi =		{10.4230/LIPIcs.WABI.2022.11},
  annote =	{Keywords: PQ-tree, Gene Cluster, Breakpoint Distance}
}
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