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APPROX

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

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.

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} }

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

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).

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} }

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

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.

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} }

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

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.

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} }

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Track A: Algorithms, Complexity and Games

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

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.

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} }

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

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.

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} }

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

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.

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} }

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

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).

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} }

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

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.

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} }

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

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}).

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} }

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

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].

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} }

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

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)).

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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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.

Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, and Kirill Simonov. Low-Rank Binary Matrix Approximation in Column-Sum Norm. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2020.32, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Simonov, Kirill}, title = {{Low-Rank Binary Matrix Approximation in Column-Sum Norm}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.32}, URN = {urn:nbn:de:0030-drops-126355}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.32}, annote = {Keywords: Binary Matrix Factorization, PTAS, Column-sum norm} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the Split Vertex Deletion (SVD) problem, the input is an n-vertex undirected graph G and a weight function w: V(G) → ℕ, and the objective is to find a minimum weight subset S of vertices such that G-S is a split graph (i.e., there is bipartition of V(G-S) = C ⊎ I such that C is a clique and I is an independent set in G-S). This problem is a special case of 5-Hitting Set and consequently, there is a simple factor 5-approximation algorithm for this. On the negative side, it is easy to show that the problem does not admit a polynomial time (2-δ)-approximation algorithm, for any fixed δ > 0, unless the Unique Games Conjecture fails.
We start by giving a simple quasipolynomial time (n^O(log n)) factor 2-approximation algorithm for SVD using the notion of clique-independent set separating collection. Thus, on the one hand SVD admits a factor 2-approximation in quasipolynomial time, and on the other hand this approximation factor cannot be improved assuming UGC. It naturally leads to the following question: Can SVD be 2-approximated in polynomial time? In this work we almost close this gap and prove that for any ε > 0, there is a n^O(log 1/(ε))-time 2(1+ε)-approximation algorithm.

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. A (2 + ε)-Factor Approximation Algorithm for Split Vertex Deletion. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2020.80, author = {Lokshtanov, Daniel and Misra, Pranabendu and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{A (2 + \epsilon)-Factor Approximation Algorithm for Split Vertex Deletion}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {80:1--80:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.80}, URN = {urn:nbn:de:0030-drops-124879}, doi = {10.4230/LIPIcs.ICALP.2020.80}, annote = {Keywords: Approximation Algorithms, Graph Algorithms, Split Vertex Deletion} }

Document

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2020.44, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {44:1--44:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.44}, URN = {urn:nbn:de:0030-drops-122024}, doi = {10.4230/LIPIcs.SoCG.2020.44}, annote = {Keywords: Optimality Program, ETH, Unit Disk Graphs, Parameterized Complexity, Long Path, Long Cycle} }

Document

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ⋅ g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε>0, multiplicative parameterization above g(I)^(1+ε) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Parameterization Above a Multiplicative Guarantee. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 39:1-39:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ITCS.2020.39, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Parameterization Above a Multiplicative Guarantee}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {39:1--39:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.39}, URN = {urn:nbn:de:0030-drops-117248}, doi = {10.4230/LIPIcs.ITCS.2020.39}, annote = {Keywords: Parameterized Complexity, Above-Guarantee Parameterization, Girth} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^O(1). In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log{n} can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^O(1). We complement these results by showing that the choice of degeneracy as the "above guarantee parameterization" is optimal in the following sense: For any epsilon>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+epsilon)d.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Going Far From Degeneracy. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2019.47, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Going Far From Degeneracy}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.47}, URN = {urn:nbn:de:0030-drops-111688}, doi = {10.4230/LIPIcs.ESA.2019.47}, annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a sqrt{k} x sqrt{k}-grid as a minor, or its treewidth is O(sqrt{k}). However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs like unit disk or map graphs. This is mainly due to the presence of large cliques in these classes of graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs, the intersection graphs of finitely many simply-connected and interior-disjoint regions of the Euclidean plane. Informally, our lemma states the following. For any map graph G, there exists a collection (U_1,...,U_t) of cliques of G with the following property: G either contains a sqrt{k} x sqrt{k}-grid as a minor, or it admits a tree decomposition where every bag is the union of O(sqrt{k}) cliques in the above collection.
The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time 2^{O({sqrt{k}log{k}})} * n^{O(1)} for Connected Planar F-Deletion (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions.
For Longest Cycle/Path, these are the first subexponential-time parameterized algorithm on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known 2^{O({k^{0.75}log{k}})} * n^{O(1)}-time algorithms on map graphs.

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of Map Graphs with Applications. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2019.60, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Decomposition of Map Graphs with Applications}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {60:1--60:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.60}, URN = {urn:nbn:de:0030-drops-106366}, doi = {10.4230/LIPIcs.ICALP.2019.60}, annote = {Keywords: Longest Cycle, Cycle Packing, Feedback Vertex Set, Map Graphs, FPT} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T subseteq V(G) with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H subseteq G of the minimum cost such that there is a directed path from s to t in H for all st in A(R).
It is known that the problem can be solved in time |V(G)|^{O(|A(R)|)} [Feldman and Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^{o(|A(R)|)} even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^{o(q)}, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent.
We show that Directed Steiner Network is solvable in time f(q)* |V(G)|^{O(c_g * q)}, where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q)* |V(G)|^{o(q^2/ log q)} algorithm for any function f for the problem on general graphs, unless ETH fails.

Eduard Eiben, Dušan Knop, Fahad Panolan, and Ondřej Suchý. Complexity of the Steiner Network Problem with Respect to the Number of Terminals. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2019.25, author = {Eiben, Eduard and Knop, Du\v{s}an and Panolan, Fahad and Such\'{y}, Ond\v{r}ej}, title = {{Complexity of the Steiner Network Problem with Respect to the Number of Terminals}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.25}, URN = {urn:nbn:de:0030-drops-102642}, doi = {10.4230/LIPIcs.STACS.2019.25}, annote = {Keywords: Directed Steiner Network, Planar Graphs, Parameterized Algorithms, Bounded Genus, Exponential Time Hypothesis} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

For a graph G, a set D subseteq V(G) is called a [1,j]-dominating set if every vertex in V(G) setminus D has at least one and at most j neighbors in D. A set D subseteq V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study [1,j]-Dominating Set on sparse graph classes from the perspective of parameterized complexity and prove the following results when the problem is parameterized by solution size:
- [1,j]-Dominating Set is W[1]-hard on d-degenerate graphs for d = j + 1;
- [1,j]-Dominating Set is FPT on nowhere dense graphs.
We also prove that the known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH). Finally, assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.

Mohsen Alambardar Meybodi, Fedor Fomin, Amer E. Mouawad, and Fahad Panolan. On the Parameterized Complexity of [1,j]-Domination Problems. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alambardarmeybodi_et_al:LIPIcs.FSTTCS.2018.34, author = {Alambardar Meybodi, Mohsen and Fomin, Fedor and Mouawad, Amer E. and Panolan, Fahad}, title = {{On the Parameterized Complexity of \lbrack1,j\rbrack-Domination Problems}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.34}, URN = {urn:nbn:de:0030-drops-99330}, doi = {10.4230/LIPIcs.FSTTCS.2018.34}, annote = {Keywords: \lbrack1, j\rbrack-dominating set, parameterized complexity, sparse graphs} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH.
This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.

Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. On the Optimality of Pseudo-polynomial Algorithms for Integer Programming. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 31:1-31:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2018.31, author = {Fomin, Fedor V. and Panolan, Fahad and Ramanujan, M. S. and Saurabh, Saket}, title = {{On the Optimality of Pseudo-polynomial Algorithms for Integer Programming}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {31:1--31:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.31}, URN = {urn:nbn:de:0030-drops-94949}, doi = {10.4230/LIPIcs.ESA.2018.31}, annote = {Keywords: Integer Programming, Strong Exponential Time Hypothesis, Branch-width of a matrix, Fine-grained Complexity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We provide a number of algorithmic results for the following family of problems: For a given binary m x n matrix A and a nonnegative integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an integer r, the "simplicity" of B is characterized as follows.
- Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r=2. We show that the problem is solvable in time 2^{O(k log k)}*(nm)^O(1) and thus is fixed-parameter tractable parameterized by k. We also complement this result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}).
- Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r=1. It is also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}).
- Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k=0 as well as for r=1. We show that it is solvable in subexponential in k time 2^{O(r2^r * sqrt{k log k})}(nm)^O(1).

Fedor V. Fomin, Petr A. Golovach, and Fahad Panolan. Parameterized Low-Rank Binary Matrix Approximation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2018.53, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad}, title = {{Parameterized Low-Rank Binary Matrix Approximation}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {53:1--53:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.53}, URN = {urn:nbn:de:0030-drops-90571}, doi = {10.4230/LIPIcs.ICALP.2018.53}, annote = {Keywords: Binary matrices, clustering, low-rank approximation, fixed-parameter tractability} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

For alpha > 1, an alpha-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I',k') (of a problem Q') of size bounded by a function of k such that, for every c >= 1, a c-approximate solution for the new instance can be turned into a (c alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every alpha > 1, Connected Dominating Set admits a polynomial-size alpha-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP \subseteq coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r \in N there cannot exist an alpha-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any alpha > 1 (assuming the Gap Exponential Time Hypothesis).

Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy Kernels for Connected Dominating Set on Sparse Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eiben_et_al:LIPIcs.STACS.2018.29, author = {Eiben, Eduard and Kumar, Mithilesh and Mouawad, Amer E. and Panolan, Fahad and Siebertz, Sebastian}, title = {{Lossy Kernels for Connected Dominating Set on Sparse Graphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.29}, URN = {urn:nbn:de:0030-drops-85027}, doi = {10.4230/LIPIcs.STACS.2018.29}, annote = {Keywords: Lossy Kernelization, Connected Dominating Set, Sparse Graph Classes} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union representation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the transversal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r\in\mathbb{N}, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union~representation.
Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element.

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lokshtanov_et_al:LIPIcs.ITCS.2018.32, author = {Lokshtanov, Daniel and Misra, Pranabendu and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Quasipolynomial Representation of Transversal Matroids with Applications in Parameterized Complexity}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {32:1--32:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.32}, URN = {urn:nbn:de:0030-drops-83144}, doi = {10.4230/LIPIcs.ITCS.2018.32}, annote = {Keywords: travserval matroid, matroid representation, union representation, representative set} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Given a graph G and a pair (\mathcal{F}_1,\mathcal{F}_2) of graph families, the function {\sf GDISJ}_{G,{\cal F}_1,{\cal F}_2} takes as input, two induced subgraphs G_1 and G_2 of G, such that G_1 \in \mathcal{F}_1 and G_2 \in \mathcal{F}_2 and returns 1 if V(G_1)\cap V(G_2)=\emptyset and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we
obtain nuanced upper bounds on the communication complexity of GDISJ_G,\cal F_1,\cal F_2. A concept related to communication protocols is that of a (\mathcal{F}_1,\mathcal{F}_2)-separating family of a graph G. A collection \mathcal{F} of subsets of V(G) is
called a (\mathcal{F}_1,\mathcal{F}_2)-separating family} for G, if for any two vertex disjoint induced subgraphs G_1\in \mathcal{F}_1,G_2\in \mathcal{F}_2, there is a set F \in \mathcal{F} with V(G_1) \subseteq F and V(G_2) \cap F = \emptyset.
Given a graph G on n vertices, for any pair (\mathcal{F}_1,\mathcal{F}_2) of hereditary graph families with sublinear communication complexity for GDISJ_G,\cal F_1,\cal F_2, we give an enumeration algorithm that finds a subexponential sized (\mathcal{F}_1,\mathcal{F}_2)-separating
family. In fact, we give an enumeration algorithm that finds a 2^{o(k)}n^{\Oh(1)} sized (\mathcal{F}_1,\mathcal{F}_2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V(G)\setminus S has a bipartition (V_1,V_2) with G[V_1] \in {\cal F}_1 and G[V_2]\in {\cal F}_2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.

Sudeshna Kolay, Fahad Panolan, and Saket Saurabh. Communication Complexity of Pairs of Graph Families with Applications. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kolay_et_al:LIPIcs.MFCS.2017.13, author = {Kolay, Sudeshna and Panolan, Fahad and Saurabh, Saket}, title = {{Communication Complexity of Pairs of Graph Families with Applications}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.13}, URN = {urn:nbn:de:0030-drops-80849}, doi = {10.4230/LIPIcs.MFCS.2017.13}, annote = {Keywords: Communication Complexity, Separating Family, FPT algorithms} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We give algorithms with running time 2^{O({\sqrt{k}\log{k}})} n^{O(1)} for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains (i) a path on exactly/at least k vertices, (ii) a cycle on exactly k vertices, (iii) a cycle on at least k vertices, (iv) a feedback vertex set of size at most k, and (v) a set of k pairwise vertex disjoint cycles.
For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time 2^{O(k^{0.75}\log{k})} n^{O(1)}. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to k^{O(1)} and there exists a solution that crosses every separator at most O(\sqrt{k}) times. The running times of our algorithms are optimal up to the log{k} factor in the exponent, assuming the Exponential Time Hypothesis.

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 65:1-65:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2017.65, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {65:1--65:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.65}, URN = {urn:nbn:de:0030-drops-73937}, doi = {10.4230/LIPIcs.ICALP.2017.65}, annote = {Keywords: Longest Cycle, Cycle Packing, Feedback Vertex Set, Unit Disk Graph, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

The classic K-Cycle problem asks if a graph G, with vertex set V(G), has a simple cycle containing all vertices of a given set K subseteq V(G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K subset of V(G) and an injective coloring c from K to {1,2,...,|K|}, decide if G has a simple cycle containing each color in {1,2,...,|K|} (once). Another problem widely known since the introduction of color coding is {Colorful Cycle}. Given a graph G and a coloring c from V(G) to {1,2,...,k} for some natural number k, it asks if G has a simple cycle of length k containing each color in {1,2,...,k} (once). We study a generalization of these problems: Given a graph G, a set K subset of V(G), a list-coloring L from K to 2^{{1,2,...,k^*}} for some natural number k^* and a parameter k, List K-Cycle asks if one can assign a color to each vertex in K so that G would have a simple cycle (of arbitrary length) containing exactly k vertices from K with distinct colors.
We design a randomized algorithm for List K-Cycle running in time 2^kn^{O(1)} on an -vertex graph, matching the best known running times of algorithms for both K-Cycle and Colorful Cycle. Moreover, unless the Set Cover Conjecture is false, our algorithm is essentially optimal. We also study a variant of List K-Cycle that generalizes the classic Hamiltonicity problem, where one specifies the size of a solution. Our results integrate three related algebraic approaches, introduced by Bjorklund, Husfeldt and Taslaman (SODA'12), Bjorklund, Kaski and Kowalik (STACS'13), and Bjorklund (FOCS'10).

Fahad Panolan and Meirav Zehavi. Parameterized Algorithms for List K-Cycle. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{panolan_et_al:LIPIcs.FSTTCS.2016.22, author = {Panolan, Fahad and Zehavi, Meirav}, title = {{Parameterized Algorithms for List K-Cycle}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.22}, URN = {urn:nbn:de:0030-drops-68571}, doi = {10.4230/LIPIcs.FSTTCS.2016.22}, annote = {Keywords: Parameterized Complexity, K-Cycle, Colorful Path, k-Path} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes Q={Q_1,Q_2,...,Q_n}, where each class Q_i consists of a pair of points Q_i = {q_i, bar{q_i}}. We call a subset A subset {q_i , bar{q_i}:1 <= i <= n} conflict free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict free subset that optimizes a given function.
Given a line segment l and set Q of a pair of points in R^2, our objective is to find a conflict free subset that minimizes the Frechet distance between l and the point set, where the minimum is taken over all possible conflict free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm which is later derandomized using universal family of sets. We believe that this technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in O(2^k * n * log^2(n)) time, and the derandomized deterministic algorithm runs in O(2^k * k^{O(log(k))} * n * log^2(n)) time, where k, the parameter, is the number of elements in the conflict free subset solution. Finally we present a simple branching algorithm for the problem running in O(2^k * n^{2} *log(n)) time. We also show that the problem is unlikely to have a polynomial sized kernel under standard complexity theoretic assumption.

Aritra Banik, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot. Fréchet Distance Between a Line and Avatar Point Set. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{banik_et_al:LIPIcs.FSTTCS.2016.32, author = {Banik, Aritra and Panolan, Fahad and Raman, Venkatesh and Sahlot, Vibha}, title = {{Fr\'{e}chet Distance Between a Line and Avatar Point Set}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.32}, URN = {urn:nbn:de:0030-drops-68676}, doi = {10.4230/LIPIcs.FSTTCS.2016.32}, annote = {Keywords: Frechet Distance, Avatar Problems, Multiple Choice, FPT} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) -> 2^[alpha] , and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Here, G_i = (V (G), {e in E(G) | i in col(e)}) and [alpha] = {1,...,alpha}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Unlike the vertex variant of the problem, when alpha = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for alpha = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2^o(k) n^O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2^((omega k alpha) + (alpha log k)) n^O(1)), where omega is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when alpha = 2. We also give a kernel for Sim-FES with (k alpha)^O(alpha) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) -> 2^[alpha] . The question is whether there is a edge subset F of cardinality at least q in G such that for all i in [alpha], G[F_i] is acyclic. Here, F_i = {e in F | i in col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2^(omega q alpha) n^O(1) ). All our algorithms are based on parameterized version of the Matroid Parity problem.

Akanksha Agrawal, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Simultaneous Feedback Edge Set: A Parameterized Perspective. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{agrawal_et_al:LIPIcs.ISAAC.2016.5, author = {Agrawal, Akanksha and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Simultaneous Feedback Edge Set: A Parameterized Perspective}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {5:1--5:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.5}, URN = {urn:nbn:de:0030-drops-67767}, doi = {10.4230/LIPIcs.ISAAC.2016.5}, annote = {Keywords: parameterized complexity, feedback edge set, alpha-matroid parity} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs.
When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l.
In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l).
1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l.
2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0.
3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

Sudeshna Kolay, Fahad Panolan, Venkatesh Raman, and Saket Saurabh. Parameterized Algorithms on Perfect Graphs for Deletion to (r,l)-Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 75:1-75:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kolay_et_al:LIPIcs.MFCS.2016.75, author = {Kolay, Sudeshna and Panolan, Fahad and Raman, Venkatesh and Saurabh, Saket}, title = {{Parameterized Algorithms on Perfect Graphs for Deletion to (r,l)-Graphs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {75:1--75:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.75}, URN = {urn:nbn:de:0030-drops-64846}, doi = {10.4230/LIPIcs.MFCS.2016.75}, annote = {Keywords: graph deletion, FPT algorithms, polynomial kernels} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

Fedor Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2016.39, author = {Fomin, Fedor and Kolay, Sudeshna and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket}, title = {{Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {39:1--39:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.39}, URN = {urn:nbn:de:0030-drops-59310}, doi = {10.4230/LIPIcs.SoCG.2016.39}, annote = {Keywords: Rectilinear graphs, Steiner arborescence, parameterized algorithms} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In the EDGE EDITING TO CONNECTED f-DEGREE GRAPH problem we are given a graph G, an integer k and a function f assigning integers to vertices of G. The task is to decide whether there is a connected graph F on the same vertex set as G, such that for every vertex v, its degree in F is f(v) and the number of edges inthe symmetric difference of E(G) and E(F), is at most k. We show that EDGE EDITING TO CONNECTED f-DEGREE GRAPH is fixed-parameter tractable (FPT) by providing an algorithm solving the problem on an n-vertex graph in time 2^{O(k)}n^{O(1)}. Our FPT algorithm is based on a non-trivial combination of color-coding and fast computations of representative families over direct sum matroid of l-elongation of co-graphic matroid associated with G and uniform matroid over the set of non-edges of G. We believe that this combination could be useful in designing parameterized algorithms for other edge editing problems.

Fedor V. Fomin, Petr Golovach, Fahad Panolan, and Saket Saurabh. Editing to Connected f-Degree Graph. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2016.36, author = {Fomin, Fedor V. and Golovach, Petr and Panolan, Fahad and Saurabh, Saket}, title = {{Editing to Connected f-Degree Graph}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {36:1--36:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.36}, URN = {urn:nbn:de:0030-drops-57370}, doi = {10.4230/LIPIcs.STACS.2016.36}, annote = {Keywords: Connected f-factor, FPT, Representative Family, Color Coding} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results:
- We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems.
- Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open.
For our approximation algorithms and thus for Turing kernels we use
an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Sudeshna Kolay and Fahad Panolan. Parameterized Algorithms for Deletion to (r,ell)-Graphs. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 420-433, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kolay_et_al:LIPIcs.FSTTCS.2015.420, author = {Kolay, Sudeshna and Panolan, Fahad}, title = {{Parameterized Algorithms for Deletion to (r,ell)-Graphs}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {420--433}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.420}, URN = {urn:nbn:de:0030-drops-56456}, doi = {10.4230/LIPIcs.FSTTCS.2015.420}, annote = {Keywords: FPT, Turing kernels, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on n vertices and a positive integer parameter k, find if there exist k edges(arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the
vertices have even degrees--where the resulting graph is Eulerian,the problem is called Undirected Eulerian Edge Deletion. The corresponding
problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its
out-degree is called Directed Eulerian Edge Deletion. Cygan et al.[Algorithmica, 2014] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time
2^O(k log k)n^O(1). They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time
2^O(k)n^O(1). It was also posed as an open problem at the School on Parameterized Algorithms and Complexity 2014, Bedlewo, Poland.
In this paper we answer their question in the affirmative: using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2^O(k)n^O(1). The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity.

Prachi Goyal, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Finding Even Subgraphs Even Faster. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 434-447, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2015.434, author = {Goyal, Prachi and Misra, Pranabendu and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{Finding Even Subgraphs Even Faster}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {434--447}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.434}, URN = {urn:nbn:de:0030-drops-56336}, doi = {10.4230/LIPIcs.FSTTCS.2015.434}, annote = {Keywords: Eulerian Edge Deletion, FPT, Representative Family.} }

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

Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G\S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C_{odd}, families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C_{odd} are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12^{k}*n^{O(1)} for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.

Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Quick but Odd Growth of Cacti. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 258-269, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kolay_et_al:LIPIcs.IPEC.2015.258, author = {Kolay, Sudeshna and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket}, title = {{Quick but Odd Growth of Cacti}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {258--269}, 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.2015.258}, URN = {urn:nbn:de:0030-drops-55883}, doi = {10.4230/LIPIcs.IPEC.2015.258}, annote = {Keywords: Even Cycle Transversal, Diamond Hitting Set, Randomized Algorithms} }

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

The b-chromatic number of a graph G, chi_b(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the B-Chromatic Number problem, the objective is to decide whether chi_b(G) >= k. Testing whether chi_b(G)=Delta(G)+1, where Delta(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvil, Tuza and Voigt, WG 2002). In this paper we study B-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that B-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k=Delta(G)+1, we design an algorithm for B-Chromatic Number running in time 2^{O(k^2 * log(k))}*n^{O(1)}. Finally, we show that B-Chromatic Number for an n-vertex graph can be solved in time O(3^n * n^{4} * log(n)).

Fahad Panolan, Geevarghese Philip, and Saket Saurabh. B-Chromatic Number: Beyond NP-Hardness. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 389-401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{panolan_et_al:LIPIcs.IPEC.2015.389, author = {Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{B-Chromatic Number: Beyond NP-Hardness}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {389--401}, 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.2015.389}, URN = {urn:nbn:de:0030-drops-55997}, doi = {10.4230/LIPIcs.IPEC.2015.389}, annote = {Keywords: b-chromatic number, exact algorithm, parameterized complexity} }

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

Given a universe U := U_1 + .... + U_r and a r-uniform family F
which is a subset of U_1 x .... x U_r, the r-dimensional matching
problem asks if F admits a collection of k mutually disjoint sets. The
special case when r=3 is the classic 3-Dimensional Matching problem.
Recently, several improvements have been suggested for these (and closely related) problems in the setting of randomized parameterized algorithms. Also, many approaches have evolved for deterministic parameterized algorithms. For instance, for the 3-Dimensional Matching problem, a combination of color coding and iterative expansion yields a running time of O^*(2.80^{(3k)}), and for the r-dimensional matching problem, a recently developed derandomization for known algebraic techniques leads to a running time of O^*(5.44^{(r-1)k}).
In this work, we employ techniques based on dynamic programming and
representative families, leading to a deterministic algorithm with running time O^*(2.85^{(r-1)k}) for the r-Dimensional Matching problem. Further, we incorporate the principles of iterative expansion used in the literature [TALG 2012] to obtain a better algorithm for 3D-matching, with a running time of O^*(2.003^{(3k)}). Apart from the significantly improved running times, we believe that these algorithms demonstrate an interesting application of representative families in conjunction with more traditional techniques.

Prachi Goyal, Neeldhara Misra, and Fahad Panolan. Faster Deterministic Algorithms for r-Dimensional Matching Using Representative Sets. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 237-248, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2013.237, author = {Goyal, Prachi and Misra, Neeldhara and Panolan, Fahad}, title = {{Faster Deterministic Algorithms for r-Dimensional Matching Using Representative Sets}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {237--248}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.237}, URN = {urn:nbn:de:0030-drops-43761}, doi = {10.4230/LIPIcs.FSTTCS.2013.237}, annote = {Keywords: 3-Dimensional Matching, Fixed-Parameter Algorithms, Iterative Expansion} }

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