41 Search Results for "Panolan, Fahad"

Document
DOI: 10.4230/LIPIcs.STACS.2025.5
Parameterized Saga of First-Fit and Last-Fit Coloring

Authors: Akanksha Agrawal, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Shaily Verma

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
The classic greedy coloring algorithm considers the vertices of an input graph G in a given order and assigns the first available color to each vertex v in G. In the Grundy Coloring problem, the task is to find an ordering of the vertices that will force the greedy algorithm to use as many colors as possible. In the Partial Grundy Coloring, the task is also to color the graph using as many colors as possible. This time, however, we may select both the ordering in which the vertices are considered and which color to assign the vertex. The only constraint is that the color assigned to a vertex v is a color previously used for another vertex if such a color is available. Whether Grundy Coloring and Partial Grundy Coloring admit fixed-parameter tractable (FPT) algorithms, algorithms with running time f(k)n^O(1), where k is the number of colors, was posed as an open problem by Zaker and by Effantin et al., respectively. Recently, Aboulker et al. (STACS 2020 and Algorithmica 2022) resolved the question for Grundy Coloring in the negative by showing that the problem is W[1]-hard. For Partial Grundy Coloring, they obtain an FPT algorithm on graphs that do not contain K_{i,j} as a subgraph (a.k.a. K_{i,j}-free graphs). Aboulker et al. re-iterate the question of whether there exists an FPT algorithm for Partial Grundy Coloring on general graphs and also asks whether Grundy Coloring admits an FPT algorithm on K_{i,j}-free graphs. We give FPT algorithms for Partial Grundy Coloring on general graphs and for Grundy Coloring on K_{i,j}-free graphs, resolving both the questions in the affirmative. We believe that our new structural theorems for partial Grundy coloring and "representative-family" like sets for K_{i,j}-free graphs that we use in obtaining our results may have wider algorithmic applications.
Cite as

Akanksha Agrawal, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Shaily Verma. Parameterized Saga of First-Fit and Last-Fit Coloring. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{agrawal_et_al:LIPIcs.STACS.2025.5,
  author =	{Agrawal, Akanksha and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Verma, Shaily},
  title =	{{Parameterized Saga of First-Fit and Last-Fit Coloring}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{5:1--5:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.5},
  URN =		{urn:nbn:de:0030-drops-228304},
  doi =		{10.4230/LIPIcs.STACS.2025.5},
  annote =	{Keywords: Grundy Coloring, Partial Grundy Coloring, FPT Algorithm, K\underline\{i,j\}-free graphs}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.24
When Far Is Better: The Chamberlin-Courant Approach to Obnoxious Committee Selection

Authors: Sushmita Gupta, Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
Classical work on metric space based committee selection problem interprets distance as "near is better". In this work, motivated by real-life situations, we interpret distance as "far is better". Formally stated, we initiate the study of "obnoxious" committee scoring rules when the voters' preferences are expressed via a metric space. To accomplish this, we propose a model where large distances imply high satisfaction (in contrast to the classical setting where shorter distances imply high satisfaction) and study the egalitarian avatar of the well-known Chamberlin-Courant voting rule and some of its generalizations. For a given integer value λ between 1 and k, the committee size, a voter derives satisfaction from only the λth favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of λ = 1, this yields the egalitarian Chamberlin-Courant rule. In this paper, we consider general metric space and the special case of a d-dimensional Euclidean space. We show that when λ is 1 and k, the problem is polynomial-time solvable in ℝ² and general metric space, respectively. However, for λ = k-1, it is NP-hard even in ℝ². Thus, we have "double-dichotomy" in ℝ² with respect to the value of λ, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be "tight" for ℝ² because the problem is NP-hard for general metric space, even for λ = 1. Consequently, we are motivated to explore the problem in the realm of (parameterized) approximation algorithms and obtain positive results. Interestingly, we note that this generalization of Chamberlin-Courant rules encodes practical constraints that are relevant to solutions for certain facility locations.
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Sushmita Gupta, Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. When Far Is Better: The Chamberlin-Courant Approach to Obnoxious Committee Selection. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 24:1-24:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2024.24,
  author =	{Gupta, Sushmita and Inamdar, Tanmay and Jain, Pallavi and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket},
  title =	{{When Far Is Better: The Chamberlin-Courant Approach to Obnoxious Committee Selection}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{24:1--24:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.24},
  URN =		{urn:nbn:de:0030-drops-222135},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.24},
  annote =	{Keywords: Metric Space, Parameterized Complexity, Approximation, Obnoxious Facility Location}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.31
Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem

Authors: Rogers Mathew, Fahad Panolan, and Seshikanth

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
An edge q-coloring of a graph G is a coloring of its edges such that every vertex sees at most q colors on the edges incident on it. The size of an edge q-coloring is the total number of colors used in the coloring. Given a graph G and a positive integer t, the Maximum Edge q-Coloring problem is about whether G has an edge q-coloring of size t. Studies on this coloring problem were motivated by its application in the channel assignment problem in wireless networks. Goyal, Kamat, and Misra (MFCS 2013) studied Maximum Edge 2-Coloring from the perspective of parameterized complexity. Given a graph on n vertices, they considered the standard parameter t, the number of colors in an optimal edge 2-coloring, and the dual parameter 𝓁, where n-𝓁 is the number of colors in an optimal edge 2-coloring. They designed FPT algorithms for Maximum Edge 2-Coloring parameterized by t and 𝓁. In this paper, we revisit and study Maximum Edge 2-Coloring from the perspective of parameterized complexity and show the following results. 1) Let γ(G) denote the maximum matching size in a given graph G. It is easy to see that a maximum edge 2-coloring of G is of size at least γ(G). Goyal, Kamat, and Misra (MFCS 2013) had asked if there exists an FPT algorithm for Maximum Edge 2-Coloring parameterized by k, where k: = (size of a maximum edge 2-coloring of G) - γ(G). We show that Maximum Edge 2-Coloring parameterized by k is W[1] hard. 2) On the positive side, we show that there is an algorithm that, given a graph G on n vertices and a tree decomposition of width tw, runs in time 2^{O(qtw log {q tw})}n and outputs a maximum edge q-coloring of G.
Cite as

Rogers Mathew, Fahad Panolan, and Seshikanth. Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mathew_et_al:LIPIcs.FSTTCS.2024.31,
  author =	{Mathew, Rogers and Panolan, Fahad and Seshikanth},
  title =	{{Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.31},
  URN =		{urn:nbn:de:0030-drops-222202},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.31},
  annote =	{Keywords: FPT algorithm, Edge coloring, Treewidth, W\lbrack1\rbrack-hardness}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.6
Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3

Authors: Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
In a disk graph, every vertex corresponds to a disk in ℝ² and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) 3-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a (3-α)-approximation algorithm for Bipartization on disk graphs, for some constant α > 0. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.
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Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lokshtanov_et_al:LIPIcs.APPROX/RANDOM.2024.6,
  author =	{Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Xue, Jie and Zehavi, Meirav},
  title =	{{Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.6},
  URN =		{urn:nbn:de:0030-drops-209990},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.6},
  annote =	{Keywords: bipartization, geometric intersection graphs, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.39
Covering and Partitioning of Split, Chain and Cographs with Isometric Paths

Authors: Dibyayan Chakraborty, Haiko Müller, Sebastian Ordyniak, Fahad Panolan, and Mateusz Rychlicki

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

Abstract
Given a graph G, an isometric path cover of a graph is a set of isometric paths that collectively contain all vertices of G. An isometric path cover 𝒞 of a graph G is also an isometric path partition if no vertex lies in two paths in 𝒞. Given a graph G, and an integer k, the objective of Isometric Path Cover (resp. Isometric Path Partition) is to decide whether G has an isometric path cover (resp. partition) of cardinality k. In this paper, we show that Isometric Path Partition is NP-complete even on split graphs, i.e. graphs whose vertex set can be partitioned into a clique and an independent set. In contrast, we show that both Isometric Path Cover and Isometric Path Partition admit polynomial time algorithms on cographs (graphs with no induced P₄) and chain graphs (bipartite graphs with no induced 2K₂).
Cite as

Dibyayan Chakraborty, Haiko Müller, Sebastian Ordyniak, Fahad Panolan, and Mateusz Rychlicki. Covering and Partitioning of Split, Chain and Cographs with Isometric Paths. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2024.39,
  author =	{Chakraborty, Dibyayan and M\"{u}ller, Haiko and Ordyniak, Sebastian and Panolan, Fahad and Rychlicki, Mateusz},
  title =	{{Covering and Partitioning of Split, Chain and Cographs with Isometric Paths}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{39:1--39:14},
  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.2024.39},
  URN =		{urn:nbn:de:0030-drops-205959},
  doi =		{10.4230/LIPIcs.MFCS.2024.39},
  annote =	{Keywords: Isometric path partition (cover), chordal graphs, chain graphs, split graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.72
A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs

Authors: Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Vertex Cover is a fundamental optimization problem, and is among Karp’s 21 NP-complete problems. The problem aims to compute, for a given graph G, a minimum-size set S of vertices of G such that G - S contains no edge. Vertex Cover admits a simple polynomial-time 2-approximation algorithm, which is the best approximation ratio one can achieve in polynomial time, assuming the Unique Game Conjecture. However, on many restrictive graph classes, it is possible to obtain better-than-2 approximation in polynomial time (or even PTASes) for Vertex Cover. In the club of geometric intersection graphs, examples of such graph classes include unit-disk graphs, disk graphs, pseudo-disk graphs, rectangle graphs, etc. In this paper, we study Vertex Cover on the broadest class of geometric intersection graphs in the plane, known as string graphs, which are intersection graphs of any connected geometric objects in the plane. Our main result is a polynomial-time 1.9999-approximation algorithm for Vertex Cover on string graphs, breaking the natural 2 barrier. Prior to this work, no better-than-2 approximation (in polynomial time) was known even for special cases of string graphs, such as intersection graphs of segments. Our algorithm is simple, robust (in the sense that it does not require the geometric realization of the input string graph to be given), and also works for the weighted version of Vertex Cover. Due to a connection between approximation for Independent Set and approximation for Vertex Cover observed by Har-Peled, our result can be viewed as a first step towards obtaining constant-approximation algorithms for Independent Set on string graphs.
Cite as

Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 72:1-72:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lokshtanov_et_al:LIPIcs.SoCG.2024.72,
  author =	{Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Xue, Jie and Zehavi, Meirav},
  title =	{{A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{72:1--72:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.72},
  URN =		{urn:nbn:de:0030-drops-200174},
  doi =		{10.4230/LIPIcs.SoCG.2024.72},
  annote =	{Keywords: vertex cover, geometric intersection graphs, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.44
Decremental Sensitivity Oracles for Covering and Packing Minors

Authors: Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
In this paper, we present the first decremental fixed-parameter sensitivity oracles for a number of basic covering and packing problems on graphs. In particular, we obtain the first decremental sensitivity oracles for Vertex Planarization (delete k vertices to make the graph planar) and Cycle Packing (pack k vertex-disjoint cycles in the given graph). That is, we give a sensitivity oracle that preprocesses the given graph in time f(k,𝓁)n^{{O}(1)} such that, when given a set of 𝓁 edge deletions, the data structure decides in time f(k,𝓁) whether the updated graph is a positive instance of the problem. These results are obtained as a corollary of our central result, which is the first decremental sensitivity oracle for Topological Minor Deletion (cover all topological minors in the input graph that belong to a specified set, using k vertices). Though our methodology closely follows the literature, we are able to produce the first explicit bounds on the preprocessing and query times for several problems. We also initiate the study of fixed-parameter sensitivity oracles with so-called structural parameterizations and give sufficient conditions for the existence of fixed-parameter sensitivity oracles where the parameter is just the treewidth of the graph. In contrast, all existing literature on this topic and the aforementioned results in this paper assume a bound on the solution size (a weaker parameter than treewidth for many problems). As corollaries, we obtain decremental sensitivity oracles for well-studied problems such as Vertex Cover and Dominating Set when only the treewidth of the input graph is bounded. A feature of our methodology behind these results is that we are able to obtain query times independent of treewidth.
Cite as

Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo. Decremental Sensitivity Oracles for Covering and Packing Minors. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kanesh_et_al:LIPIcs.STACS.2024.44,
  author =	{Kanesh, Lawqueen and Panolan, Fahad and Ramanujan, M. S. and Strulo, Peter},
  title =	{{Decremental Sensitivity Oracles for Covering and Packing Minors}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{44:1--44:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.44},
  URN =		{urn:nbn:de:0030-drops-197544},
  doi =		{10.4230/LIPIcs.STACS.2024.44},
  annote =	{Keywords: Sensitivity oracles, Data Structures, FPT algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.91
Backdoor Sets on Nowhere Dense SAT

Authors: Daniel Lokshtanov, Fahad Panolan, and M. S. Ramanujan

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
For a satisfiable CNF formula ϕ and an integer t, a weak backdoor set to treewidth-t is a set of variables such that there is an assignment to this set that reduces ϕ to a satisfiable formula that has an incidence graph of treewidth at most t. A natural research program in the work on fixed-parameter algorithms (FPT algorithms) for SAT is to delineate the tractability borders for the problem of detecting a small weak backdoor set to treewidth-t formulas. In this line of research, Gaspers and Szeider (ICALP 2012) showed that detecting a weak backdoor set of size at most k to treewidth-1 is W[2]-hard parameterized by k if the input is an arbitrary CNF formula. Fomin, Lokshtanov, Misra, Ramanujan and Saurabh (SODA 2015), showed that if the input is d-CNF, then detecting a weak backdoor set of size at most k to treewidth-t is fixed-parameter tractable (parameterized by k,t,d). These two results indicate that sparsity of the input plays a role in determining the parameterized complexity of detecting weak backdoor sets to treewidth-t. In this work, we take a major step towards characterizing the precise impact of sparsity on the parameterized complexity of this problem by obtaining algorithmic results for detecting small weak backdoor sets to treewidth-t for input formulas whose incidence graphs belong to a nowhere-dense graph class. Nowhere density provides a robust and well-understood notion of sparsity that is at the heart of several advances on model checking and structural graph theory. Moreover, nowhere-dense graph classes contain many well-studied graph classes such as bounded treewidth graphs, graphs that exclude a fixed (topological) minor and graphs of bounded expansion. Our main contribution is an algorithm that, given a formula ϕ whose incidence graph belongs to a fixed nowhere-dense graph class and an integer k, in time f(t,k)|ϕ|^O(1), either finds a satisfying assignment of ϕ, or concludes correctly that ϕ has no weak backdoor set of size at most k to treewidth-t. To obtain this algorithm, we develop a strategy that only relies on the fact that nowhere-dense graph classes are biclique-free. That is, for every nowhere-dense graph class, there is a p such that it is contained in the class of graphs that exclude K_{p,p} as a subgraph. This is a significant feature of our techniques since the class of biclique-free graphs also generalizes the class of graphs of bounded degeneracy, which are incomparable with nowhere-dense graph classes. As a result, our algorithm also generalizes the results of Fomin, Lokshtanov, Misra, Ramanujan and Saurabh (SODA 2015) for the special case of d-CNF formulas as input when d is fixed. This is because the incidence graphs of such formulas exclude K_{d+1,d+1} as a subgraph.
Cite as

Daniel Lokshtanov, Fahad Panolan, and M. S. Ramanujan. Backdoor Sets on Nowhere Dense SAT. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 91:1-91:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2022.91,
  author =	{Lokshtanov, Daniel and Panolan, Fahad and Ramanujan, M. S.},
  title =	{{Backdoor Sets on Nowhere Dense SAT}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{91:1--91:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.91},
  URN =		{urn:nbn:de:0030-drops-164323},
  doi =		{10.4230/LIPIcs.ICALP.2022.91},
  annote =	{Keywords: Fixed-parameter Tractability, Satisfiability, Backdoors, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.5
An FPT Algorithm for Elimination Distance to Bounded Degree Graphs

Authors: Akanksha Agrawal, Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most k to any minor-closed class of graphs is fixed-parameter tractable parameterized by k [Algorithmica, 2017]. They showed that Graph Isomorphism parameterized by the elimination distance to bounded degree graphs is fixed-parameter tractable and asked whether determining the elimination distance to the class of bounded degree graphs is fixed-parameter tractable. Recently, Lindermayr et al. [MFCS 2020] obtained a fixed-parameter algorithm for this problem in the special case where the input is restricted to K₅-minor free graphs. In this paper, we answer the question of Bulian and Dawar in the affirmative for general graphs. In fact, we give a more general result capturing elimination distance to any graph class characterized by a finite set of graphs as forbidden induced subgraphs.
Cite as

Akanksha Agrawal, Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. An FPT Algorithm for Elimination Distance to Bounded Degree Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 5:1-5:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{agrawal_et_al:LIPIcs.STACS.2021.5,
  author =	{Agrawal, Akanksha and Kanesh, Lawqueen and Panolan, Fahad and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{An FPT Algorithm for Elimination Distance to Bounded Degree Graphs}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{5:1--5:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.5},
  URN =		{urn:nbn:de:0030-drops-136507},
  doi =		{10.4230/LIPIcs.STACS.2021.5},
  annote =	{Keywords: Elimination Distance, Fixed-parameter Tractability, Graph Modification}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.31
Diverse Collections in Matroids and Graphs

Authors: Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, Geevarghese Philip, and Saket Saurabh

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid M, a weight function ω:E(M)→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k bases B_1, ..., B_k of M such that the weight of the symmetric difference of any pair of these bases is at least d. This is a diverse variant of the classical matroid base packing problem. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids M₁,M₂ defined on the same ground set E, a weight function ω:E→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k common independent sets I_1, ..., I_k of M₁ and M₂ such that the weight of the symmetric difference of any pair of these sets is at least d. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph G and integers k ≥ 1, d ≥ 0. The task is to decide if G contains k perfect matchings M_1, ..., M_k such that the symmetric difference of any two of these matchings is at least d. The underlying problem of finding one solution (basis, common independent set, or perfect matching) is known to be doable in polynomial time for each of these problems, and Diverse Perfect Matchings is known to be NP-hard for k = 2. We show that Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard. We show also that Diverse Perfect Matchings cannot be solved in polynomial time (unless P=NP) even for the case d = 1. We derive fixed-parameter tractable (FPT) algorithms for all three problems with (k,d) as the parameter. The above results on matroids are derived under the assumption that the input matroids are given as independence oracles. For Weighted Diverse Bases we present a polynomial-time algorithm that takes a representation of the input matroid over a finite field and computes a poly(k,d)-sized kernel for the problem.
Cite as

Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Diverse Collections in Matroids and Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2021.31,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket},
  title =	{{Diverse Collections in Matroids and Graphs}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.31},
  URN =		{urn:nbn:de:0030-drops-136769},
  doi =		{10.4230/LIPIcs.STACS.2021.31},
  annote =	{Keywords: Matroids, Diverse solutions, Fixed-parameter tractable algorithms}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.19
Structural Parameterizations with Modulator Oblivion

Authors: Ashwin Jacob, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot

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

Abstract
It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph, when parameterized by k. While this investigation fits naturally into the recent trend of what are called "structural parameterizations", here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k^O(k)n^O(1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2^O(k)n^O(1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in time 2^O(k)n^O(1) where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the Strong Exponential Time Hypothesis (SETH).
Cite as

Ashwin Jacob, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot. Structural Parameterizations with Modulator Oblivion. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jacob_et_al:LIPIcs.IPEC.2020.19,
  author =	{Jacob, Ashwin and Panolan, Fahad and Raman, Venkatesh and Sahlot, Vibha},
  title =	{{Structural Parameterizations with Modulator Oblivion}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{19:1--19:18},
  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.2020.19},
  URN =		{urn:nbn:de:0030-drops-133222},
  doi =		{10.4230/LIPIcs.IPEC.2020.19},
  annote =	{Keywords: Parameterized Complexity, Chordal Graph, Tree Decomposition, Strong Exponential Time Hypothesis}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.24
On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets

Authors: Daniel Lokshtanov, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz

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

Abstract
In a reconfiguration version of a decision problem 𝒬 the input is an instance of 𝒬 and two feasible solutions S and T. The objective is to determine whether there exists a step-by-step transformation between S and T such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the Connected Dominating Set Reconfiguration problem (CDS-R). It was shown in previous work that the Dominating Set Reconfiguration problem (DS-R) parameterized by k, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique K_{d,d} as a subgraph, for some constant d ≥ 1. We show that the additional connectivity constraint makes the problem much harder, namely, that CDS-R is W[1]-hard parameterized by k+𝓁, the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on 5-degenerate graphs. On the positive side, we show that CDS-R parameterized by k is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs.
Cite as

Daniel Lokshtanov, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lokshtanov_et_al:LIPIcs.IPEC.2020.24,
  author =	{Lokshtanov, Daniel and Mouawad, Amer E. and Panolan, Fahad and Siebertz, Sebastian},
  title =	{{On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{24:1--24:15},
  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.2020.24},
  URN =		{urn:nbn:de:0030-drops-133276},
  doi =		{10.4230/LIPIcs.IPEC.2020.24},
  annote =	{Keywords: reconfiguration, parameterized complexity, connected dominating set, graph structure theory}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.34
Improved FPT Algorithms for Deletion to Forest-Like Structures

Authors: Kishen N. Gowda, Aditya Lonkar, Fahad Panolan, Vraj Patel, and Saket Saurabh

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset S ⊆ V(G) of size at most k such that G-S is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized algorithm for the problem running in time 𝒪^⋆(2.7^k). In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set S should be an independent set in G), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in G-S has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph G and non-negative integers k,𝓁 ∈ ℕ, and the objective is to test whether there exists a vertex subset S of size at most k, such that G-S is 𝓁 edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms. 1) Independent Feedback Vertex Set can be solved in time 𝒪^⋆(2.7^k). 2) Pseudo Forest Deletion can be solved in time 𝒪^⋆(2.85^k). 3) Almost Forest Deletion can be solved in 𝒪^⋆(min{2.85^k ⋅ 8.54^𝓁, 2.7^k ⋅ 36.61^𝓁, 3^k ⋅ 1.78^𝓁}).
Cite as

Kishen N. Gowda, Aditya Lonkar, Fahad Panolan, Vraj Patel, and Saket Saurabh. Improved FPT Algorithms for Deletion to Forest-Like Structures. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gowda_et_al:LIPIcs.ISAAC.2020.34,
  author =	{Gowda, Kishen N. and Lonkar, Aditya and Panolan, Fahad and Patel, Vraj and Saurabh, Saket},
  title =	{{Improved FPT Algorithms for Deletion to Forest-Like Structures}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.34},
  URN =		{urn:nbn:de:0030-drops-133781},
  doi =		{10.4230/LIPIcs.ISAAC.2020.34},
  annote =	{Keywords: Parameterized Complexity, Independent Feedback Vertex Set, PseudoForest, Almost Forest, Cut and Count, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.18
Parameterized Complexity of Feedback Vertex Sets on Hypergraphs

Authors: Pratibha Choudhary, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
A feedback vertex set in a hypergraph H is a set of vertices S such that deleting S from H results in an acyclic hypergraph. Here, deleting a vertex means removing the vertex and all incident hyperedges, and a hypergraph is acyclic if its vertex-edge incidence graph is acyclic. We study the (parameterized complexity of) the Hypergraph Feedback Vertex Set (HFVS) problem: given as input a hypergraph H and an integer k, determine whether H has a feedback vertex set of size at most k. It is easy to see that this problem generalizes the classic Feedback Vertex Set (FVS) problem on graphs. Remarkably, despite the central role of FVS in parameterized algorithms and complexity, the parameterized complexity of a generalization of FVS to hypergraphs has not been studied previously. In this paper, we fill this void. Our main results are as follows - HFVS is W[2]-hard (as opposed to FVS, which is fixed parameter tractable). - If the input hypergraph is restricted to a linear hypergraph (no two hyperedges intersect in more than one vertex), HFVS admits a randomized algorithm with running time 2^{𝒪(k³log k)}n^{𝒪(1)}. - If the input hypergraph is restricted to a d-hypergraph (hyperedges have cardinality at most d), then HFVS admits a deterministic algorithm with running time d^{𝒪(k)}n^{𝒪(1)}. The algorithm for linear hypergraphs combines ideas from the randomized algorithm for FVS by Becker et al. [J. Artif. Intell. Res., 2000] with the branching algorithm for Point Line Cover by Langerman and Morin [Discrete & Computational Geometry, 2005].
Cite as

Pratibha Choudhary, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Parameterized Complexity of Feedback Vertex Sets on Hypergraphs. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{choudhary_et_al:LIPIcs.FSTTCS.2020.18,
  author =	{Choudhary, Pratibha and Kanesh, Lawqueen and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket},
  title =	{{Parameterized Complexity of Feedback Vertex Sets on Hypergraphs}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.18},
  URN =		{urn:nbn:de:0030-drops-132596},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.18},
  annote =	{Keywords: feedback vertex sets, hypergraphs, FPT, randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.70
Quick Separation in Chordal and Split Graphs

Authors: Pranabendu Misra, Fahad Panolan, Ashutosh Rai, Saket Saurabh, and Roohani Sharma

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of 𝓁 vertex pairs (s_i, t_i), i ∈ [𝓁], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V(G)⧵ {s_i,t_i : i ∈ [𝓁]} of size at most k such that for every vertex pair (s_i,t_i), s_i and t_i are in two different connected components of G-S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(s_i,t_i): i ∈ [𝓁]}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T× T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2^{{𝒪}(k³)}n^{{𝒪}(1)} and 2^k n^{{𝒪}(1)}, respectively. However, the kernelization complexity of both these problems is not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results. 1) Multicut on chordal graphs admits a polynomial kernel with {𝒪}(k³ 𝓁⁷) vertices. Multiway Cut on chordal graphs admits a polynomial kernel with {𝒪}(k^{13}) vertices. 2) Multicut on chordal graphs can be solved in time min {𝒪(2^{k} ⋅ (k³+𝓁) ⋅ (n+m)), 2^{𝒪(𝓁 log k)} ⋅ (n+m) + 𝓁 (n+m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP. 3) Multicut on split graphs can be solved in time min {𝒪(1.2738^k + kn+𝓁(n+m), 𝒪(2^{𝓁} ⋅ 𝓁 ⋅ (n+m))}. Unrestricted Multicut on split graphs can be solved in time 𝒪(4^{𝓁}⋅ 𝓁 ⋅ (n+m)).
Cite as

Pranabendu Misra, Fahad Panolan, Ashutosh Rai, Saket Saurabh, and Roohani Sharma. Quick Separation in Chordal and Split Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{misra_et_al:LIPIcs.MFCS.2020.70,
  author =	{Misra, Pranabendu and Panolan, Fahad and Rai, Ashutosh and Saurabh, Saket and Sharma, Roohani},
  title =	{{Quick Separation in Chordal and Split Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{70:1--70:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.70},
  URN =		{urn:nbn:de:0030-drops-127391},
  doi =		{10.4230/LIPIcs.MFCS.2020.70},
  annote =	{Keywords: chordal graphs, multicut, multiway cut, FPT, kernel}
}
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