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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We investigate the parameterized complexity of Vertex Cover parameterized above the optimum value of the linear programming (LP) relaxation of the integer linear programming formulation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that even the most straightforward branching algorithm (after some preprocessing) results in an O^*(2.6181^r) algorithm for the problem where r is the excess of the
vertex cover size over the LP optimum. We write O^*(f(k)) for a time complexity of the form O(f(k)n^{O(1)}), where f(k) grows exponentially with k.
Then, using known and new reductions, we give O^*(2.6181^k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an O^*(1.6181^k) algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The notable improvement is the bound for Odd Cycle Transversal for which this is the first major improvement after the first algorithm that showed it fixed-parameter tractable in 2003. We also observe that using our algorithm, one can obtain a simple kernel for the classical vertex cover problem with at most 2k-O(log k) vertices.

N.S. Narayanaswamy, Venkatesh Raman, M.S. Ramanujan, and Saket Saurabh. LP can be a cure for Parameterized Problems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 338-349, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{narayanaswamy_et_al:LIPIcs.STACS.2012.338, author = {Narayanaswamy, N.S. and Raman, Venkatesh and Ramanujan, M.S. and Saurabh, Saket}, title = {{LP can be a cure for Parameterized Problems}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {338--349}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.338}, URN = {urn:nbn:de:0030-drops-34291}, doi = {10.4230/LIPIcs.STACS.2012.338}, annote = {Keywords: Algorithms and data structures. Graph Algorithms, Parameterized Algorithms.} }

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

Given a (connected) undirected graph G, a set X ⊆ V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ⊇ X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness).
Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.

Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan. Finding a Highly Connected Steiner Subgraph and its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 45:1-45:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The study of the Knot-Free Vertex Deletion problem emerges from its application in the resolution of deadlocks called knots, detected in a classical distributed computation model, that is, the OR-model. A strongly connected subgraph Q of a digraph D with at least two vertices is said to be a knot if there is no arc (u,v) of D with u ∈ V(Q) and v ∉ V(Q) (no-out neighbors of the vertices in Q). Given a directed graph D, the Knot-Free Vertex Deletion (KFVD) problem asks to compute a minimum-size subset S ⊂ V(D) such that D[V⧵S] contains no knots. There is no exact algorithm known for the KFVD problem in the literature that is faster than the trivial O^⋆(2ⁿ) brute-force algorithm. In this paper, we obtain the first non-trivial upper bound for KFVD by designing an exact algorithm running in time 𝒪^⋆(1.576ⁿ), where n is the size of the vertex set in D.

M. S. Ramanujan, Abhishek Sahu, Saket Saurabh, and Shaily Verma. An Exact Algorithm for Knot-Free Vertex Deletion. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 78:1-78:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ramanujan_et_al:LIPIcs.MFCS.2022.78, author = {Ramanujan, M. S. and Sahu, Abhishek and Saurabh, Saket and Verma, Shaily}, title = {{An Exact Algorithm for Knot-Free Vertex Deletion}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {78:1--78:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.78}, URN = {urn:nbn:de:0030-drops-168769}, doi = {10.4230/LIPIcs.MFCS.2022.78}, annote = {Keywords: exact algorithm, knot-free graphs, branching algorithm} }

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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 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In the Connected ℱ-Deletion problem, ℱ is a fixed finite family of graphs and the objective is to compute a minimum set of vertices (or a vertex set of size at most k for some given k) such that (a) this set induces a connected subgraph of the given graph and (b) deleting this set results in a graph which excludes every F ∈ ℱ as a minor. In the area of kernelization, this problem is well known to exclude a polynomial kernel subject to standard complexity hypotheses even in very special cases such as ℱ = K₂, i.e., Connected Vertex Cover.
In this work, we give a (2+ε)-approximate polynomial compression for the Connected ℱ-Deletion problem when ℱ contains at least one planar graph. This is the first approximate polynomial compression result for this generic problem. As a corollary, we obtain the first approximate polynomial compression result for the special case of Connected η-Treewidth Deletion.

M. S. Ramanujan. On Approximate Compressions for Connected Minor-Hitting Sets. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 78:1-78:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ramanujan:LIPIcs.ESA.2021.78, author = {Ramanujan, M. S.}, title = {{On Approximate Compressions for Connected Minor-Hitting Sets}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {78:1--78:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.78}, URN = {urn:nbn:de:0030-drops-146590}, doi = {10.4230/LIPIcs.ESA.2021.78}, annote = {Keywords: Parameterized Complexity, Kernelization, Approximation Algorithms} }

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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 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Bulian and Dawar [Algorithmica, 2016] introduced the notion of elimination distance in an effort to define new tractable parameterizations for graph problems 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].
In this paper, we consider the problem of computing the elimination distance of a given graph to the class of cluster graphs and initiate the study of the parameterized complexity of a more general version - that of obtaining a modulator to such graphs. That is, we study the (η,Clq)-Elimination Deletion problem ((η,Clq)-ED Deletion) where, for a fixed η, one is given a graph G and k ∈ ℕ and the objective is to determine whether there is a set S ⊆ V(G) such that the graph G-S has elimination distance at most η to the class of cluster graphs.
Our main result is a polynomial kernelization (parameterized by k) for this problem. As components in the proof of our main result, we develop a k^𝒪(η k + η²)n^𝒪(1)-time fixed-parameter algorithm for (η,Clq)-ED Deletion and a polynomial-time factor-min{𝒪(η⋅ opt⋅ log² n),opt^𝒪(1)} approximation algorithm for the same problem.

Akanksha Agrawal and M. S. Ramanujan. On the Parameterized Complexity of Clique Elimination Distance. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 1:1-1:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2020.1, author = {Agrawal, Akanksha and Ramanujan, M. S.}, title = {{On the Parameterized Complexity of Clique Elimination Distance}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {1:1--1:13}, 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.2020.1}, URN = {urn:nbn:de:0030-drops-133043}, doi = {10.4230/LIPIcs.IPEC.2020.1}, annote = {Keywords: Elimination Distance, Cluster Graphs, Kernelization} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and co-graphic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition.
When L is of small rank, we show that the problem is NP-complete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upper-bound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs.

Fedor V. Fomin, Petr Golovach, Pranabendu Misra, and M. S. Ramanujan. On the Complexity of Recovering Incidence Matrices. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 50:1-50:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2020.50, author = {Fomin, Fedor V. and Golovach, Petr and Misra, Pranabendu and Ramanujan, M. S.}, title = {{On the Complexity of Recovering Incidence Matrices}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.50}, URN = {urn:nbn:de:0030-drops-129164}, doi = {10.4230/LIPIcs.ESA.2020.50}, annote = {Keywords: Graph Incidence Matrix, Matrix Recovery, Enumeration Algorithm} }

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

Directed Feedback Vertex Set (DFVS) is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the ℋ-SCC Deletion problem. Here, one is given a digraph D, an integer k and the objective is to decide whether there is a vertex set of size at most k whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ℋ as (not necessarily induced) subgraphs. When ℋ comprises only the digraph with a single arc, then this problem is precisely DFVS.
Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ℋ only contains rooted graphs or if ℋ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of Göke et al. [CIAC 2019] for the 1-Out-Regular Vertex Deletion and Bounded Size Strong Component Vertex Deletion problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for DFVS, without using the heavy machinery of shadow removal as is done by Göke et al. [CIAC 2019].

Rian Neogi, M. S. Ramanujan, Saket Saurabh, and Roohani Sharma. On the Parameterized Complexity of Deletion to ℋ-Free Strong Components. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 75:1-75:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{neogi_et_al:LIPIcs.MFCS.2020.75, author = {Neogi, Rian and Ramanujan, M. S. and Saurabh, Saket and Sharma, Roohani}, title = {{On the Parameterized Complexity of Deletion to ℋ-Free Strong Components}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {75:1--75:13}, 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.2020.75}, URN = {urn:nbn:de:0030-drops-127444}, doi = {10.4230/LIPIcs.MFCS.2020.75}, annote = {Keywords: Directed Cut Problems, Fixed-parameter Tractability, DFVS} }

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

The Feedback Vertex Set problem is a fundamental computational problem which has been the subject of intensive study in various domains of algorithmics. In this problem, one is given an undirected graph G and an integer k as input. The objective is to determine whether at most k vertices can be deleted from G such that the resulting graph is acyclic. The study of preprocessing algorithms for this problem has a long and rich history, culminating in the quadratic kernelization of Thomasse [SODA 2010].
However, it is known that when the solution is required to induce a connected subgraph (such a set is called a connected feedback vertex set), a polynomial kernelization is unlikely to exist and the problem is NP-hard to approximate below a factor of 2 (assuming the Unique Games Conjecture).
In this paper, we show that if one is interested in only preserving approximate solutions (even of quality arbitrarily close to the optimum), then there is a drastic improvement in our ability to preprocess this problem. Specifically, we prove that for every fixed 0<epsilon<1, graph G, and k in N, the following holds:
There is a polynomial time computable graph G' of size k^O(1) such that for every c >= 1, any c-approximate connected feedback vertex set of G' of size at most k is a c * (1+epsilon)-approximate connected feedback vertex set of G.
Our result adds to the set of approximate kernelization algorithms introduced by Lokshtanov et al. [STOC 2017]. As a consequence of our main result, we show that Connected Feedback Vertex Set can be approximated within a factor min{OPT^O(1),n^(1-delta)} in polynomial time for some delta>0.

M. S. Ramanujan. An Approximate Kernel for Connected Feedback Vertex Set. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 77:1-77:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ramanujan:LIPIcs.ESA.2019.77, author = {Ramanujan, M. S.}, title = {{An Approximate Kernel for Connected Feedback Vertex Set}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {77:1--77:14}, 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.2019.77}, URN = {urn:nbn:de:0030-drops-111989}, doi = {10.4230/LIPIcs.ESA.2019.77}, annote = {Keywords: Parameterized Complexity, Kernelization, Approximation Algorithms} }

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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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Brief Announcement

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

In the Directed Feedback Vertex Set (DFVS) problem, we are given as input a directed graph D and an integer k, and the objective is to check whether there exists a set S of at most k vertices such that F=D-S is a directed acyclic graph (DAG). Determining whether DFVS admits a polynomial kernel (parameterized by the solution size) is one of the most important open problems in parameterized complexity. In this article, we give a polynomial kernel for DFVS parameterized by the solution size plus the size of any treewidth-eta modulator, for any positive integer eta. We also give a polynomial kernel for the problem, which we call Vertex Deletion to treewidth-eta DAG, where given as input a directed graph D and a positive integer k, the objective is to decide whether there exists a set of at most k vertices, say S, such that D-S is a DAG and the treewidth of D-S is at most eta.

Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Brief Announcement: Treewidth Modulator: Emergency Exit for DFVS. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 110:1-110:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2018.110, author = {Lokshtanov, Daniel and Ramanujan, M. S. and Saurabh, Saket and Sharma, Roohani and Zehavi, Meirav}, title = {{Brief Announcement: Treewidth Modulator: Emergency Exit for DFVS}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {110:1--110:4}, 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.2018.110}, URN = {urn:nbn:de:0030-drops-91146}, doi = {10.4230/LIPIcs.ICALP.2018.110}, annote = {Keywords: Polynomial Kernel, Directed Feedback Vertex Set, Treewidth Modulator} }

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

Given a Counting Monadic Second Order (CMSO) sentence psi, the CMSO[psi] problem is defined as follows. The input to CMSO[psi] is a graph G, and the objective is to determine whether G |= psi. Our main theorem states that for every CMSO sentence psi, if CMSO[psi] is solvable in polynomial time on "globally highly connected graphs", then CMSO[psi] is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph G and the task is to find a connected induced subgraph of G such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property.

Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Reducing CMSO Model Checking to Highly Connected Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 135:1-135:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2018.135, author = {Lokshtanov, Daniel and Ramanujan, M. S. and Saurabh, Saket and Zehavi, Meirav}, title = {{Reducing CMSO Model Checking to Highly Connected Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {135:1--135:14}, 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.2018.135}, URN = {urn:nbn:de:0030-drops-91391}, doi = {10.4230/LIPIcs.ICALP.2018.135}, annote = {Keywords: Fixed Parameter Tractability Model Checking Recursive Understanding} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, \cup_{i=1}^{\alpha} E_{i}), where the edge set of G is partitioned into \alpha color classes, is called an \alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, \ldots, F_\alpha)-Deletion problem. In the latter problem, we are given an \alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 \leq i \leq \alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = \ldots = F_\alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and \alpha, and can be solved in O(2^{O(\alpha k)}n^{O(1)})
time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, \ldots, F_\alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i's is the class of bipartite graphs and the rest (if any) are forests.
We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = \ldots = F_\alpha is the family of all forests then the problem is fixed-parameter tractable
parameterized by k and \alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard.

Akanksha Agrawal, R. Krithika, Daniel Lokshtanov, Amer E. Mouawad, and M. S. Ramanujan. On the Parameterized Complexity of Simultaneous Deletion Problems. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2017.9, author = {Agrawal, Akanksha and Krithika, R. and Lokshtanov, Daniel and Mouawad, Amer E. and Ramanujan, M. S.}, title = {{On the Parameterized Complexity of Simultaneous Deletion Problems}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.9}, URN = {urn:nbn:de:0030-drops-84128}, doi = {10.4230/LIPIcs.FSTTCS.2017.9}, annote = {Keywords: parameterized complexity, feedback vertex set, odd cycle transversal, edge-colored graphs, simultaneous deletion} }

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

In the Directed Feedback Vertex Set (DFVS) problem, the input is
a directed graph D and an integer k. The objective is to determine
whether there exists a set of at most k vertices intersecting every
directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O'Sullivan and
Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics.
In this paper, we study DFVS parameterized by the feedback vertex
set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

Benjamin Bergougnoux, Eduard Eiben, Robert Ganian, Sebastian Ordyniak, and M. S. Ramanujan. Towards a Polynomial Kernel for Directed Feedback Vertex Set. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 36:1-36:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bergougnoux_et_al:LIPIcs.MFCS.2017.36, author = {Bergougnoux, Benjamin and Eiben, Eduard and Ganian, Robert and Ordyniak, Sebastian and Ramanujan, M. S.}, title = {{Towards a Polynomial Kernel for Directed Feedback Vertex Set}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {36:1--36:15}, 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.2017.36}, URN = {urn:nbn:de:0030-drops-81126}, doi = {10.4230/LIPIcs.MFCS.2017.36}, annote = {Keywords: parameterized algorithms, kernelization, (directed) feedback vertex set} }

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

In this paper, we study the Connected H-hitting Set and Dominating Set problems from the perspective of approximate kernelization, a framework recently introduced by Lokshtanov et al. [STOC 2017]. For the Connected H-hitting set problem, we obtain an \alpha-approximate kernel for every \alpha>1 and complement it with a lower bound for the natural weighted version. We then perform a refined analysis of the tradeoff between the approximation factor and kernel size for the Dominating Set problem on d-degenerate graphs and provide an interpolation of approximate kernels between the known d^2-approximate kernel of constant size and 1-approximate kernel of size k^{O(d^2)}.

Eduard Eiben, Danny Hermelin, and M. S. Ramanujan. Lossy Kernels for Hitting Subgraphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 67:1-67:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2017.67, author = {Eiben, Eduard and Hermelin, Danny and Ramanujan, M. S.}, title = {{Lossy Kernels for Hitting Subgraphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {67:1--67:14}, 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.2017.67}, URN = {urn:nbn:de:0030-drops-80955}, doi = {10.4230/LIPIcs.MFCS.2017.67}, annote = {Keywords: parameterized algorithms, lossy kernelization, graph theory} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: (Weighted) Biconnectivity Deletion, where we are tasked with deleting k edges while preserving biconnectivity in an undirected graph, Vertexdeletion Preserving Strong Connectivity, where we want to maintain strong connectivity of a digraph while deleting exactly k vertices, and Path-contraction Preserving Strong Connectivity, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed in [Bang-Jensen and Yeo, Discrete Applied Math 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are W[1]-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable (FPT) and we provide an FPT algorithm that solves Weighted Biconnectivity Deletion. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivitypreservation constraints in the parameterized setting.

Gregory Gutin, M. S. Ramanujan, Felix Reidl, and Magnus Wahlström. Path-Contractions, Edge Deletions and Connectivity Preservation. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 47:1-47:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gutin_et_al:LIPIcs.ESA.2017.47, author = {Gutin, Gregory and Ramanujan, M. S. and Reidl, Felix and Wahlstr\"{o}m, Magnus}, title = {{Path-Contractions, Edge Deletions and Connectivity Preservation}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {47:1--47:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.47}, URN = {urn:nbn:de:0030-drops-78270}, doi = {10.4230/LIPIcs.ESA.2017.47}, annote = {Keywords: connectivity, strong connectivity, vertex deletion, arc contraction} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Chitnis et al. [FOCS 2012, SICOMP 2016] proved that this problem is fixed-parameter tractable (FPT) and Wahlström [SODA 2014] gave an FPT algorithm with an improved parameter dependence. Subsequently, Iwata, Wahlström and Yoshida [SICOMP 2016] proved that the edge version of Unique Label Cover can be solved in linear FPT-time, and they left open the existence of such an algorithm for the node version of the problem. In this paper, we resolve this question by presenting the first linear-time FPT algorithm for Node Unique Label Cover.

Daniel Lokshtanov, M. S. Ramanujan, and Saket Saurabh. A Linear-Time Parameterized Algorithm for Node Unique Label Cover. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 57:1-57:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lokshtanov_et_al:LIPIcs.ESA.2017.57, author = {Lokshtanov, Daniel and Ramanujan, M. S. and Saurabh, Saket}, title = {{A Linear-Time Parameterized Algorithm for Node Unique Label Cover}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.57}, URN = {urn:nbn:de:0030-drops-78152}, doi = {10.4230/LIPIcs.ESA.2017.57}, annote = {Keywords: Algorithms and data structures, Fixed Parameter Tractability, Unique Label Cover, Linear Time FPT Algorithms.} }

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

We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters.
We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters.
We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between
the firefighting problem and the notions of important separators and tight separator sequences.
Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts our FPT result for the non-spreading model.

Jayesh Choudhari, Anirban Dasgupta, Neeldhara Misra, and M. S. Ramanujan. Saving Critical Nodes with Firefighters is FPT. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 135:1-135:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{choudhari_et_al:LIPIcs.ICALP.2017.135, author = {Choudhari, Jayesh and Dasgupta, Anirban and Misra, Neeldhara and Ramanujan, M. S.}, title = {{Saving Critical Nodes with Firefighters is FPT}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {135:1--135:13}, 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.2017.135}, URN = {urn:nbn:de:0030-drops-74968}, doi = {10.4230/LIPIcs.ICALP.2017.135}, annote = {Keywords: firefighting, cuts, FPT, kernelization} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth.

Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining Treewidth and Backdoors for CSP. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 36:1-36:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2017.36, author = {Ganian, Robert and Ramanujan, M. S. and Szeider, Stefan}, title = {{Combining Treewidth and Backdoors for CSP}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert 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.2017.36}, URN = {urn:nbn:de:0030-drops-69986}, doi = {10.4230/LIPIcs.STACS.2017.36}, annote = {Keywords: Algorithms and data structures, Fixed Parameter Tractability, Constraint Satisfaction} }

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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 purpose of this article is two fold: (a) to formally introduce a stronger version of graph deletion problems; and (b) to study this version in the context of bipartite graphs. Given a family of graphs F, a typical instance of parameterized graph deletion problem consists of an undirected graph G and a positive integer k and the objective is to check whether we can delete at most k vertices (or k edges) such that the resulting graph belongs to F. Another version that has been recently studied is the one where the input contains two integers k and l and the objective is to check whether we can delete at most k vertices and l edges such that the resulting graph belongs to F. In this paper, we propose and initiate the study of a more general version which we call strong deletion. In this problem, given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be transformed into a graph in F by deleting at most l edges. In this paper we study this stronger version of deletion problems for the class of bipartite graphs. In particular, we study Strong Bipartite Deletion, where given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be made bipartite by deleting at most l edges. While fixed-parameter tractability when parameterizing by k or l alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and l.

Ashutosh Rai and M. S. Ramanujan. Strong Parameterized Deletion: Bipartite Graphs. 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. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{rai_et_al:LIPIcs.FSTTCS.2016.21, author = {Rai, Ashutosh and Ramanujan, M. S.}, title = {{Strong Parameterized Deletion: Bipartite Graphs}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {21:1--21:14}, 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.2016.21}, URN = {urn:nbn:de:0030-drops-68561}, doi = {10.4230/LIPIcs.FSTTCS.2016.21}, annote = {Keywords: fixed parameter tractable, bipartite-editing, recursive understanding} }

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

Given an n-vertex graph G and a function f:V(G) -> {0, ..., n-1}, an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when f(v) is at least n^epsilon for each vertex v and epsilon<1; on the other side of the spectrum, the problem was known to be polynomial-time solvable when f(v) is at least n/3 for every vertex v.
In this paper, we extend this line of work and obtain new complexity results based on restricting the function f. In particular, we show that when f(v) is required to be at least n/(log n)^c, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c <= 1. We also show that when c>1, the problem is NP-intermediate.

Robert Ganian, N. S. Narayanaswamy, Sebastian Ordyniak, C. S. Rahul, and M. S. Ramanujan. On the Complexity Landscape of Connected f-Factor Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 41:1-41:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganian_et_al:LIPIcs.MFCS.2016.41, author = {Ganian, Robert and Narayanaswamy, N. S. and Ordyniak, Sebastian and Rahul, C. S. and Ramanujan, M. S.}, title = {{On the Complexity Landscape of Connected f-Factor Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {41:1--41:14}, 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.2016.41}, URN = {urn:nbn:de:0030-drops-65013}, doi = {10.4230/LIPIcs.MFCS.2016.41}, annote = {Keywords: f-factors, connected f-factors, quasi-polynomial time algorithms, randomized algorithms} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Given a graph G and a function f:V(G) -> [V(G)], an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. Tutte (1954) showed that one can check whether a given graph has a specified f-factor in polynomial time. However, detecting a connected f-factor is NP-complete, even when f is a constant function - a foremost example is the problem of checking whether a graph has a Hamiltonian cycle; here f is a function which maps every vertex to 2. The current best algorithm for this latter problem is due to Björklund (FOCS 2010), and runs in randomized O^*(1.657^n) time (the O^*() notation hides polynomial factors). This was the first superpolynomial improvement, in nearly fifty years, over the previous best algorithm of Bellman, Held and Karp (1962) which checks for a Hamiltonian cycle in deterministic O(2^n*n^2) time.
In this paper we present the first vertex-exponential algorithms for the more general problem of finding a connected f-factor. Our first result is a randomized algorithm which, given a graph G on n vertices and a function f:V(G) -> [n], checks whether G has a connected f-factor in O^*(2^n) time. We then extend our result to the case when f is a mapping from V(G) to {0,1} and the degree of every vertex v in the subgraph H is required to be f(v)(mod 2). This generalizes the problem of checking whether a graph has an Eulerian subgraph; this is a connected subgraph whose degrees are all even (f(v) equiv 0). Furthermore, we show that the min-cost editing and edge-weighted versions of these problems can be solved in randomized O^*(2^n) time as long as the costs/weights are bounded polynomially in n.

Geevarghese Philip and M. S. Ramanujan. Vertex Exponential Algorithms for Connected f-Factors. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 61-71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{philip_et_al:LIPIcs.FSTTCS.2014.61, author = {Philip, Geevarghese and Ramanujan, M. S.}, title = {{Vertex Exponential Algorithms for Connected f-Factors}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {61--71}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.61}, URN = {urn:nbn:de:0030-drops-48337}, doi = {10.4230/LIPIcs.FSTTCS.2014.61}, annote = {Keywords: Exact Exponential Time Algorithms, f-Factors} }

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

In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S' \subseteq S of size at most k satisfying some desired properties. If S' is required to contain all the elements of U then it corresponds to the classical Set Cover problem. On the other hand if we require S' to satisfy the property that for every pair of elements x,y \in U there exists a set S \in S' such that |S \cap {x,y}|=1 then it corresponds to the Test Cover problem. In this paper we consider a natural parameterization of Set Cover and Test Cover. More precisely, we study the (n-k)-Set Cover and (n-k)-Test Cover problems, where the objective is to find a subfamily S' of size at most n-k satisfying the respective properties, from the kernelization perspective. It is known in the literature that both (n-k)-Set Cover and (n-k)-Test Cover do not admit polynomial kernels (under some well known complexity theoretic assumptions). However, in this paper we show that they do admit "partially polynomial kernels". More precisely, we give polynomial time algorithms that take as input an instance (U,S,k) of (n-k)-Set Cover (n-k)-Test Cover) and return an equivalent instance (~U,~S,~k) of (n-k)-Set Cover (respectively (n-k)-Test Cover) with ~k <= k and |~U|= O(k^2) (|~U|=O(k^7)). These results allow us to generalize, improve and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain "sparsity properties". Using a part of our kernelization algorithm for (n-k)-Set Cover, we also get an improved FPT algorithm for this problem which runs in time O(4^k*k^{\O(1)}*(m+n)) improving over the previous best of O(8^{k+o(k)}*(m+n)^{O(1)}). On the other hand the partially polynomial kernel for (n-k)-Test Cover implies the first single exponential FPT algorithm, an algorithm with running time O(2^{O(k^2)}*(m+n)^{O(1)}). We believe such an approach will also be useful for other covering problems as well.

Manu Basavaraju, Mathew C. Francis, M. S. Ramanujan, and Saket Saurabh. Partially Polynomial Kernels for Set Cover and Test Cover. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 67-78, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{basavaraju_et_al:LIPIcs.FSTTCS.2013.67, author = {Basavaraju, Manu and Francis, Mathew C. and Ramanujan, M. S. and Saurabh, Saket}, title = {{Partially Polynomial Kernels for Set Cover and Test Cover}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {67--78}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.67}, URN = {urn:nbn:de:0030-drops-43621}, doi = {10.4230/LIPIcs.FSTTCS.2013.67}, annote = {Keywords: Set Cover, Test Cover, Kernelization, Parameterized Algorithms} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The class q-Horn, introduced by Boros, Crama and Hammer in 1990, is one of the largest known classes of propositional CNF formulas for which satisfiability can be decided in polynomial time. This class properly contains the fundamental classes of Horn and Krom formulas as well as the class of renamable (or disguised) Horn formulas.
In this paper we extend this class so that its favorable algorithmic properties can be made accessible to formulas that are outside but "close"' to this class. We show that deciding satisfiability is fixed-parameter tractable parameterized by the distance of the given formula from q-Horn. The distance is measured by the smallest number of variables that we need to delete from the formula in order to get a q-Horn formula, i.e., the size of a smallest deletion backdoor set into the class q-Horn.
This result generalizes known fixed-parameter tractability results for satisfiability decision with respect to the parameters distance from Horn, Krom, and renamable Horn.

Serge Gaspers, Sebastian Ordyniak, M. S. Ramanujan, Saket Saurabh, and Stefan Szeider. Backdoors to q-Horn. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 67-79, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{gaspers_et_al:LIPIcs.STACS.2013.67, author = {Gaspers, Serge and Ordyniak, Sebastian and Ramanujan, M. S. and Saurabh, Saket and Szeider, Stefan}, title = {{Backdoors to q-Horn}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {67--79}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.67}, URN = {urn:nbn:de:0030-drops-39236}, doi = {10.4230/LIPIcs.STACS.2013.67}, annote = {Keywords: Algorithms and data structures, Backdoor sets, Satisfiability, Fixed Parameter Tractability} }

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