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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number.
We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2024.1, author = {Agrawal, Akanksha and Cabello, Sergio and Kaufmann, Michael and Saurabh, Saket and Sharma, Roohani and Uno, Yushi and Wolff, Alexander}, title = {{Eliminating Crossings in Ordered Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.1}, URN = {urn:nbn:de:0030-drops-200417}, doi = {10.4230/LIPIcs.SWAT.2024.1}, annote = {Keywords: Ordered graphs, book embedding, edge deletion, d-planar, hitting set} }

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**Published in:** LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)

We study the SUPPORTED model of distributed computing introduced by Schmid and Suomela [Schmid and Suomela, 2013], which generalizes the LOCAL and CONGEST models. In this framework, multiple instances of the same problem, differing from each other by the subnetwork to which they apply. recur over time, and need to be solved efficiently online. To do that, one may rely on an initial preprocessing phase for computing some useful information. This preprocessing phase makes it possible, in some cases, to obtain improved distributed algorithms, overcoming locality-based time lower bounds.
Our main contribution is to expand the class of problems to which the SUPPORTED model applies, by handling also multiple recurring instances of the same problem that differ from each other by some problem specific input, and not only the subnetwork to which they apply. We illustrate this by considering two extended problem classes. The first class, denoted PCS, concerns problems where client nodes of the network need to be served, and each recurring instance applies to some Partial Client Set. The second class, denoted PFO, concerns situations where each recurrent instance of the problem includes a partially fixed output, which needs to be completed to a full consistent solution.
Specifically, we propose some natural recurrent variants of the dominating set problem and the coloring problem that are of interest particularly in the distributed setting. For these problems, we show that information about the topology can be used to overcome locality-based lower bounds. We also categorize the round complexity of Locally Checkable Labellings in the SUPPORTED model for the simple case of paths. Finally we present some interesting open problems and some partial results towards resolving them.

Akanksha Agrawal, John Augustine, David Peleg, and Srikkanth Ramachandran. Local Recurrent Problems in the SUPPORTED Model. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.OPODIS.2023.22, author = {Agrawal, Akanksha and Augustine, John and Peleg, David and Ramachandran, Srikkanth}, title = {{Local Recurrent Problems in the SUPPORTED Model}}, booktitle = {27th International Conference on Principles of Distributed Systems (OPODIS 2023)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-308-9}, ISSN = {1868-8969}, year = {2024}, volume = {286}, editor = {Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.22}, URN = {urn:nbn:de:0030-drops-195124}, doi = {10.4230/LIPIcs.OPODIS.2023.22}, annote = {Keywords: Distributed Algorithms, LOCAL Model, SUPPORTED Model} }

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

The problem of computing a minimum set of vertices intersecting a finite set of forbidden minors in a given graph is a fundamental graph problem in the area of kernelization with numerous well-studied special cases.
A major breakthrough in this line of research was made by Fomin et al. [FOCS 2012], who showed that the ρ-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most ρ) has a polynomial kernel of size k^g(ρ) for some function g. A second standout result in this line is that of Giannapoulou et al. [ACM TALG 2017], who obtained an f(η)k^𝒪(1)-size kernel (for some function f) for the η-Treedepth Modulator problem (delete fewest number of vertices to make treedepth at most η) and showed that some dependence of the exponent of k on ρ in the result of Fomin et al. for the ρ-Treewidth Modulator problem is unavoidable under reasonable complexity hypotheses.
In this work, we provide an approximate interpolation between these two results by giving, for every ε > 0, a (1+ε)-approximate kernel of size f'(η,ρ,1/ε)⋅ k^g'(ρ) (for some functions f' and g') for the problem of deciding whether k vertices can be deleted from a given graph to obtain a graph that has elimination distance at most η to the class of graphs that have treewidth at most ρ.
Graphs of treedepth η are precisely the graphs with elimination distance at most η-1 to the graphs of treewidth 0 and graphs of treewidth ρ are simply graphs with elimination distance 0 to graphs of treewidth ρ. Consequently, our result "approximately" interpolates between these two major results in this active line of research.

Akanksha Agrawal and M. S. Ramanujan. Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2023.36, author = {Agrawal, Akanksha and Ramanujan, M. S.}, title = {{Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.36}, URN = {urn:nbn:de:0030-drops-194096}, doi = {10.4230/LIPIcs.FSTTCS.2023.36}, annote = {Keywords: Lossy Kernelization, Treewidth Modulator, Vertex Deletion Problems} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 22481 "Vertex Partitioning in Graphs: From Structure to Algorithms", which was held from 27 November to 2 December 2023. The report contains abstracts for presentations about recent structural and algorithmic developments for a variety of vertex partitioning problems. It also contains a collection of open problems which were posed during the seminar.

Maria Chudnovsky, Neeldhara Misra, Daniel Paulusma, Oliver Schaudt, and Akanksha Agrawal. Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481). In Dagstuhl Reports, Volume 12, Issue 11, pp. 109-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{chudnovsky_et_al:DagRep.12.11.109, author = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, title = {{Vertex Partitioning in Graphs: From Structure to Algorithms (Dagstuhl Seminar 22481)}}, pages = {109--123}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Chudnovsky, Maria and Misra, Neeldhara and Paulusma, Daniel and Schaudt, Oliver and Agrawal, Akanksha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.11.109}, URN = {urn:nbn:de:0030-drops-178384}, doi = {10.4230/DagRep.12.11.109}, annote = {Keywords: computational complexity, hereditary graph classes, parameterized algorithms, polynomial-time algorithms, vertex partitioning} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Assume we are given a graph G, two independent sets S and T in G of size k ≥ 1, and a positive integer 𝓁 ≥ 1. The goal is to decide whether there exists a sequence ⟨ I₀, I₁, ..., I_𝓁 ⟩ of independent sets such that for all j ∈ {0,…,𝓁-1} the set I_j is an independent set of size k, I₀ = S, I_𝓁 = T, and I_{j+1} is obtained from I_j by a predetermined reconfiguration rule. We consider two reconfiguration rules, namely token sliding and token jumping. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most 𝓁 steps that transforms S into T, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex (while maintaining independence). In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph (as long as we maintain independence). Both TSO and TJO are known to be fixed-parameter tractable when parameterized by 𝓁 on nowhere dense classes of graphs. In this work, we investigate the boundary of tractability for sparse classes of graphs. We show that both problems are fixed-parameter tractable for parameter k + 𝓁 + d on d-degenerate graphs as well as for parameter |M| + 𝓁 + Δ on graphs having a modulator M whose deletion leaves a graph of maximum degree Δ. We complement these result by showing that for parameter 𝓁 alone both problems become W[1]-hard already on 2-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. [Daniel Lokshtanov et al., 2020]. Finally, we show as a side result that using such families we can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem (a.k.a. Token Jumping) parameterized by k on both degenerate and nowhere dense classes of graphs.

Akanksha Agrawal, Soumita Hait, and Amer E. Mouawad. On Finding Short Reconfiguration Sequences Between Independent Sets. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agrawal_et_al:LIPIcs.ISAAC.2022.39, author = {Agrawal, Akanksha and Hait, Soumita and Mouawad, Amer E.}, title = {{On Finding Short Reconfiguration Sequences Between Independent Sets}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.39}, URN = {urn:nbn:de:0030-drops-173244}, doi = {10.4230/LIPIcs.ISAAC.2022.39}, annote = {Keywords: Token sliding, token jumping, fixed-parameter tractability, combinatorial reconfiguration, shortest reconfiguration sequence} }

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

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

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

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

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

For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(√k)⋅ n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^𝒪(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu. Parameterized Complexity of Perfectly Matched Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.2, author = {Agrawal, Akanksha and Bhattacharjee, Sutanay and Jana, Satyabrata and Sahu, Abhishek}, title = {{Parameterized Complexity of Perfectly Matched Sets}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {2:1--2:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.2}, URN = {urn:nbn:de:0030-drops-173580}, doi = {10.4230/LIPIcs.IPEC.2022.2}, annote = {Keywords: Perfectly Matched Sets, Parameterized Complexity, Apex-minor-free graphs, d-degenerate graphs, Planar graphs, Interval Graphs} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

For a family of graphs F, given a graph G and an integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in the family F. The F-Deletion problems for all non-trivial families F that satisfy the hereditary property on induced subgraphs are known to be NP-hard by a result of Yannakakis (STOC'78). Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. Equivalently, they form the set of graphs that do not contain any chordless cycles or a gem as an induced subgraph. (A gem is the graph on 5 vertices, where four vertices form an induced path, and the fifth vertex is adjacent to all the vertices of this induced path.) The Ptolemaic Deletion problem is the F-Deletion problem, where F is the family of Ptolemaic graphs. In this paper we study Ptolemaic Deletion from the viewpoint of Kernelization Complexity, and obtain a kernel with 𝒪(k⁶) vertices for the problem.

Akanksha Agrawal, Aditya Anand, and Saket Saurabh. A Polynomial Kernel for Deletion to Ptolemaic Graphs. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2021.1, author = {Agrawal, Akanksha and Anand, Aditya and Saurabh, Saket}, title = {{A Polynomial Kernel for Deletion to Ptolemaic Graphs}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.1}, URN = {urn:nbn:de:0030-drops-153840}, doi = {10.4230/LIPIcs.IPEC.2021.1}, annote = {Keywords: Ptolemaic Deletion, Kernelization, Parameterized Complexity, Gem-free chordal graphs} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

In this article we study a well-known problem, called Bipartite Token Jumping and not-so-well known problem(s), which we call, Half (Induced-) Subgraph, and show that under Gap-ETH, these problems do not admit FPT algorithms. The problem Bipartite Token Jumping takes as input a bipartite graph G and two independent sets S,T in G, where |S| = |T| = k, and the objective is to test if there is a sequence of exactly k-sized independent sets ⟨ I₀, I₁,⋯, I_𝓁 ⟩ in G, such that: i) I₀ = S and I_𝓁 = T, and ii) for every j ∈ [𝓁], I_{j} is obtained from I_{j-1} by replacing a vertex in I_{j-1} by a vertex in V(G) ⧵ I_{j-1}. We show that, assuming Gap-ETH, Bipartite Token Jumping does not admit an FPT algorithm. We note that this result resolves one of the (two) open problems posed by Bartier et al. (ISAAC 2020), under Gap-ETH. Most of the known reductions related to Token Jumping exploit the property given by triangles (i.e., C₃s), to obtain the correctness, and our results refutes FPT algorithm for Bipartite Token Jumping, where the input graph cannot have any triangles.
For an integer k ∈ ℕ, the half graph S_{k,k} is the graph with vertex set V(S_{k,k}) = A_k ∪ B_k, where A_k = {a₁,a₂,⋯, a_k} and B_k = {b₁,b₂,⋯, b_k}, and for i,j ∈ [k], {a_i,b_j} ∈ E(T_{k,k}) if and only if j ≥ i. We also study the Half (Induced-)Subgraph problem where we are given a graph G and an integer k, and the goal is to check if G contains S_{k,k} as an (induced-)subgraph. Again under Gap-ETH, we show that Half (Induced-)Subgraph does not admit an FPT algorithm, even when the input is a bipartite graph. We believe that the above problem (and its negative) result maybe of independent interest and could be useful obtaining new fixed parameter intractability results.
There are very few reductions known in the literature which refute FPT algorithms for a parameterized problem based on assumptions like Gap-ETH. Thus our technique (and simple reductions) exhibits the potential of such conjectures in obtaining new (and possibly easier) proofs for refuting FPT algorithms for parameterized problems.

Akanksha Agrawal, Ravi Kiran Allumalla, and Varun Teja Dhanekula. Refuting FPT Algorithms for Some Parameterized Problems Under Gap-ETH. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2021.2, author = {Agrawal, Akanksha and Allumalla, Ravi Kiran and Dhanekula, Varun Teja}, title = {{Refuting FPT Algorithms for Some Parameterized Problems Under Gap-ETH}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.2}, URN = {urn:nbn:de:0030-drops-153851}, doi = {10.4230/LIPIcs.IPEC.2021.2}, annote = {Keywords: Token Jumping, Bipartite Graphs, Fixed Parameter Intractability, Half Graphs, Gap-Exponential Time Hypothesis} }

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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 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery, it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm.

Akanksha Agrawal, Sudeshna Kolay, and Meirav Zehavi. Parameter Analysis for Guarding Terrains. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2020.4, author = {Agrawal, Akanksha and Kolay, Sudeshna and Zehavi, Meirav}, title = {{Parameter Analysis for Guarding Terrains}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {4:1--4:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.4}, URN = {urn:nbn:de:0030-drops-122514}, doi = {10.4230/LIPIcs.SWAT.2020.4}, annote = {Keywords: Terrain Guarding, Reflex Vertices, Terrain Minima, FPT Algorithm, XP Algorithm, Kernelization} }

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

The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ℝ-complete [Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP.
Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time r^O(r²)n^O(1). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. The Parameterized Complexity of Guarding Almost Convex Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agrawal_et_al:LIPIcs.SoCG.2020.3, author = {Agrawal, Akanksha and Knudsen, Kristine V. K. and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav}, title = {{The Parameterized Complexity of Guarding Almost Convex Polygons}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {3:1--3:16}, 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.2020.3}, URN = {urn:nbn:de:0030-drops-121614}, doi = {10.4230/LIPIcs.SoCG.2020.3}, annote = {Keywords: Art Gallery, Reflex vertices, Monotone 2-CSP, Parameterized Complexity, Fixed Parameter Tractability} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k.
We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses.
To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned.
In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p).

Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh. Parameterized Streaming Algorithms for Min-Ones d-SAT. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2019.8, author = {Agrawal, Akanksha and Biswas, Arindam and Bonnet, \'{E}douard and Brettell, Nick and Curticapean, Radu and Marx, D\'{a}niel and Miltzow, Tillmann and Raman, Venkatesh and Saurabh, Saket}, title = {{Parameterized Streaming Algorithms for Min-Ones d-SAT}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.8}, URN = {urn:nbn:de:0030-drops-115708}, doi = {10.4230/LIPIcs.FSTTCS.2019.8}, annote = {Keywords: min, ones, sat, d-sat, parameterized, kernelization, streaming, space, efficient, algorithm, parameter} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Given a symmetric l x l matrix M=(m_{i,j}) with entries in {0,1,*}, a graph G and a function L : V(G) - > 2^{[l]} (where [l] = {1,2,...,l}), a list M-partition of G with respect to L is a partition of V(G) into l parts, say, V_1, V_2, ..., V_l such that for each i,j in {1,2,...,l}, (i) if m_{i,j}=0 then for any u in V_i and v in V_j, uv not in E(G), (ii) if m_{i,j}=1 then for any (distinct) u in V_i and v in V_j, uv in E(G), (iii) for each v in V(G), if v in V_i then i in L(v). We consider the Deletion to List M-Partition problem that takes as input a graph G, a list function L:V(G) - > 2^[l] and a positive integer k. The aim is to determine whether there is a k-sized set S subseteq V(G) such that G-S has a list M-partition. Many important problems like Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, Multiway Cut and Deletion to List Homomorphism are special cases of the Deletion to List M-Partition problem. In this paper, we provide a classification of the parameterized complexity of Deletion to List M-Partition, parameterized by k, (a) when M is of order at most 3, and (b) when M is of order 4 with all diagonal entries belonging to {0,1}.

Akanksha Agrawal, Sudeshna Kolay, Jayakrishnan Madathil, and Saket Saurabh. Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.ISAAC.2019.41, author = {Agrawal, Akanksha and Kolay, Sudeshna and Madathil, Jayakrishnan and Saurabh, Saket}, title = {{Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.41}, URN = {urn:nbn:de:0030-drops-115372}, doi = {10.4230/LIPIcs.ISAAC.2019.41}, annote = {Keywords: list matrix partitions, parameterized classification, Almost 2-SAT, important separators, iterative compression} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid.

Akanksha Agrawal, Pallavi Jain, Lawqueen Kanesh, and Saket Saurabh. Parameterized Complexity of Conflict-Free Matchings and Paths. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.MFCS.2019.35, author = {Agrawal, Akanksha and Jain, Pallavi and Kanesh, Lawqueen and Saurabh, Saket}, title = {{Parameterized Complexity of Conflict-Free Matchings and Paths}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.35}, URN = {urn:nbn:de:0030-drops-109798}, doi = {10.4230/LIPIcs.MFCS.2019.35}, annote = {Keywords: Conflict-free, Matching, Shortest Path, FPT algorithm, W\lbrack1\rbrack-hard, Matroid} }

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Track A: Algorithms, Complexity and Games

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

A graph G is contractible to a graph H if there is a set X subseteq E(G), such that G/X is isomorphic to H. Here, G/X is the graph obtained from G by contracting all the edges in X. For a family of graphs F, the F-Contraction problem takes as input a graph G on n vertices, and the objective is to output the largest integer t, such that G is contractible to a graph H in F, where |V(H)|=t. When F is the family of paths, then the corresponding F-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time 2^n * n^{O(1)}. In spite of the deceptive simplicity of the problem, beating the 2^n * n^{O(1)} bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time 1.99987^n * n^{O(1)}. We also define a problem called 3-Disjoint Connected Subgraphs, and design an algorithm for it that runs in time 1.88^n * n^{O(1)}. The above algorithm is used as a sub-routine in our algorithm for Path Contraction.

Akanksha Agrawal, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. Path Contraction Faster Than 2^n. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.ICALP.2019.11, author = {Agrawal, Akanksha and Fomin, Fedor V. and Lokshtanov, Daniel and Saurabh, Saket and Tale, Prafullkumar}, title = {{Path Contraction Faster Than 2^n}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {11:1--11:13}, 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.2019.11}, URN = {urn:nbn:de:0030-drops-105874}, doi = {10.4230/LIPIcs.ICALP.2019.11}, annote = {Keywords: path contraction, exact exponential time algorithms, graph algorithms, enumerating connected sets, 3-disjoint connected subgraphs} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L'subseteq L such that the graph G=(P,L') is isomorphic to a graph in F and L' has at most k crossings. By G=(P,L'), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L'. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces.
As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP.

Akanksha Agrawal, Grzegorz Guśpiel, Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi. Connecting the Dots (with Minimum Crossings). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.SoCG.2019.7, author = {Agrawal, Akanksha and Gu\'{s}piel, Grzegorz and Madathil, Jayakrishnan and Saurabh, Saket and Zehavi, Meirav}, title = {{Connecting the Dots (with Minimum Crossings)}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.7}, URN = {urn:nbn:de:0030-drops-104117}, doi = {10.4230/LIPIcs.SoCG.2019.7}, annote = {Keywords: crossing minimization, parameterized complexity, FPT algorithm, polynomial kernel, W\lbrack1\rbrack-hardness} }

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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

A generalization of classical cycle hitting problems, called conflict version of the problem, is defined as follows. An input is undirected graphs G and H on the same vertex set, and a positive integer k, and the objective is to decide whether there exists a vertex subset X subseteq V(G) such that it intersects all desired "cycles" (all cycles or all odd cycles or all even cycles) and X is an independent set in H. In this paper we study the conflict version of classical Feedback Vertex Set, and Odd Cycle Transversal problems, from the view point of kernelization complexity. In particular, we obtain the following results, when the conflict graph H belongs to the family of d-degenerate graphs.
1) CF-FVS admits a O(k^{O(d)}) kernel.
2) CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP subseteq coNP/poly.
For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for the classical Feedback Vertex Set problem, as well as, devise new reduction rules that exploit degeneracy crucially. Our main conceptual contribution here is the notion of "k-independence preserver". Informally, it is a set of "important" vertices for a given subset X subseteq V(H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size k^{O(d)} exists, and can be used in designing polynomial kernel.

Akanksha Agrawal, Pallavi Jain, Lawqueen Kanesh, Pranabendu Misra, and Saket Saurabh. Exploring the Kernelization Borders for Hitting Cycles. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2018.14, author = {Agrawal, Akanksha and Jain, Pallavi and Kanesh, Lawqueen and Misra, Pranabendu and Saurabh, Saket}, title = {{Exploring the Kernelization Borders for Hitting Cycles}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.14}, URN = {urn:nbn:de:0030-drops-102158}, doi = {10.4230/LIPIcs.IPEC.2018.14}, annote = {Keywords: Parameterized Complexity, Kernelization, Conflict-free problems, Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

In this paper we study recently introduced conflict version of the classical Feedback Vertex Set (FVS) problem. For a family of graphs F, we consider the problem F-CF-Feedback Vertex Set (F-CF-FVS, for short). The F-CF-FVS problem takes as an input a graph G, a graph H in F (where V(G)=V(H)), and an integer k, and the objective is to decide if there is a set S subseteq V(G) of size at most k such that G-S is a forest and S is an independent set in H. Observe that if we instantiate F to be the family of edgeless graphs then we get the classical FVS problem. Jain, Kanesh, and Misra [CSR 2018] showed that in contrast to FVS, F-CF-FVS is W[1]-hard on general graphs and admits an FPT algorithm if F is the family of d-degenerate graphs. In this paper, we relate F-CF-FVS to the Independent Set problem on special classes of graphs, and obtain a complete dichotomy result on the Parameterized Complexity of the problem F-CF-FVS, when F is a hereditary graph family. In particular, we show that F-CF-FVS is FPT parameterized by the solution size if and only if F+Cluster IS is FPT parameterized by the solution size. Here, F+Cluster IS is the Independent Set problem in the (edge) union of a graph G in F and a cluster graph H (G and H are explicitly given). Next, we exploit this characterization to obtain new FPT results as well as intractability results for F-CF-FVS. In particular, we give an FPT algorithm for F+Cluster IS when F is the family of K_{i,j}-free graphs. We show that for the family of bipartite graph B, B-CF-FVS is W[1]-hard, when parameterized by the solution size. Finally, we consider, for each 0< epsilon<1, the family of graphs F_epsilon, which comprise of graphs G such that |E(G)| <= |V(G)|^(2-epsilon), and show that F_epsilon-CF-FVS is W[1]-hard, when parameterized by the solution size, for every 0<epsilon<1.

Akanksha Agrawal, Pallavi Jain, Lawqueen Kanesh, Daniel Lokshtanov, and Saket Saurabh. Conflict Free Feedback Vertex Set: A Parameterized Dichotomy. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agrawal_et_al:LIPIcs.MFCS.2018.53, author = {Agrawal, Akanksha and Jain, Pallavi and Kanesh, Lawqueen and Lokshtanov, Daniel and Saurabh, Saket}, title = {{Conflict Free Feedback Vertex Set: A Parameterized Dichotomy}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {53:1--53:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.53}, URN = {urn:nbn:de:0030-drops-96355}, doi = {10.4230/LIPIcs.MFCS.2018.53}, annote = {Keywords: Conflict-free, Feedback Vertex Set, FPT algorithm, W\lbrack1\rbrack-hardness} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G) - >R^+, find a minimum weight subset S subseteq V(G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^{O(1)} n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^{O(1)} n)-approximation algorithms for the following vertex deletion problems.
- Let {F} be a finite set of graphs containing a planar graph, and F=G(F) be the family of graphs such that every graph H in G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^{1.5} n) and O(log^2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012].
- We give an O(log^2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs.
- We give an O(log^3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one.
We believe that our recursive scheme can be applied to obtain O(log^{O(1)} n)-approximation algorithms for many other problems as well.

Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agrawal_et_al:LIPIcs.APPROX-RANDOM.2018.1, author = {Agrawal, Akanksha and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav}, title = {{Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.1}, URN = {urn:nbn:de:0030-drops-94058}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.1}, annote = {Keywords: Approximation Algorithms, Planar- F-Deletion, Separator} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S \subseteq E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al.[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied \cal F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T_\ell be the family of graphs such that each graph in T_\ell can be made into a tree by deleting at most \ell edges. Thus, the problem we study is T_\ell-Contraction. We design an FPT algorithm for T_\ell-Contraction running in time O((\ncol)^{O(k + \ell)} * n^{O(1)}). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T_\ell-Contraction of size O([k(k + 2\ell)] ^{(\lceil {\frac{\alpha}{\alpha-1}\rceil + 1)}}).

Akanksha Agrawal, Saket Saurabh, and Prafullkumar Tale. On the Parameterized Complexity of Contraction to Generalization of Trees. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 1:1-1:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2017.1, author = {Agrawal, Akanksha and Saurabh, Saket and Tale, Prafullkumar}, title = {{On the Parameterized Complexity of Contraction to Generalization of Trees}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {1:1--1:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.1}, URN = {urn:nbn:de:0030-drops-85446}, doi = {10.4230/LIPIcs.IPEC.2017.1}, annote = {Keywords: Graph Contraction, Fixed Parameter Tractability, Graph Algorithms, Generalization of Trees} }

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

The duality between packing and covering problems lies at the heart of fundamental combinatorial proofs, as well as well-known algorithmic methods such as the primal-dual method for approximation and win/win-approach for parameterized analysis. The very essence of this duality is encompassed by a well-known property called the Erdös-Pósa property, which has been extensively studied for over five decades. Informally, we say that a class of graphs F admits the Erdös-Pósa property if there exists f such that for any graph G, either G has vertex-disjoint "copies" of the graphs in F, or there is a set S \subseteq V(G) of f(k) vertices that intersects all copies of the graphs in F. In the context of any graph class G, the most natural question that arises in this regard is as follows - do obstructions to G have the Erdös-Pósa property? Having this view in mind, we focus on the class of interval graphs. Structural properties of interval graphs are intensively studied, also as they lead to the design of polynomial-time algorithms for classic problems that are NP-hard on general graphs. Nevertheless, about one of the most basic properties of such graphs, namely, the Erdös-Pósa property, nothing is known. In this paper, we settle this anomaly: we prove that the family of obstructions to interval graphs - namely, the family of chordless cycles and ATs---admits the Erdös-Pósa property. Our main theorem immediately results in an algorithm to decide whether an input graph G has vertex-disjoint ATs and chordless cycles, or there exists a set of O(k^2 log k) vertices in G that hits all ATs and chordless cycles.

Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Erdös-Pósa Property of Obstructions to Interval Graphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agrawal_et_al:LIPIcs.STACS.2018.7, author = {Agrawal, Akanksha and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav}, title = {{Erd\"{o}s-P\'{o}sa Property of Obstructions to Interval Graphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {7:1--7:15}, 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.2018.7}, URN = {urn:nbn:de:0030-drops-84815}, doi = {10.4230/LIPIcs.STACS.2018.7}, annote = {Keywords: Interval Graphs, Obstructions, Erd\"{o}s-P\'{o}sa Property} }

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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)

Consider a graph G and an edge-coloring c_R:E(G) \rightarrow [k]. A rainbow path between u,v \in V(G) is a path P from u to v such that for all e,e' \in E(P), where e \neq e' we have c_R(e) \neq c_R(e'). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G) \times V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all (u,v) \in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results.
- For every k \geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails.
- For every k \geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails.
- Subset Rainbow k-Coloring admits an algorithm running in time 2^{\OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.

Akanksha Agrawal. Fine-Grained Complexity of Rainbow Coloring and its Variants. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agrawal:LIPIcs.MFCS.2017.60, author = {Agrawal, Akanksha}, title = {{Fine-Grained Complexity of Rainbow Coloring and its Variants}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {60:1--60:14}, 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.2017.60}, URN = {urn:nbn:de:0030-drops-80990}, doi = {10.4230/LIPIcs.MFCS.2017.60}, annote = {Keywords: Rainbow Coloring, Lower bound, ETH, Fine-grained Complexity} }

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

The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this paper, we examine an important family of graphs, namely the family of split graphs, which in the context of edge contractions, is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, the Split Contraction problem asks whether there exists a subset X of edges of G such that G/X is a split graph and X has at most k elements. Here, G/X is the graph obtained from G by contracting edges in X. It was previously claimed that the Split Contraction problem is fixed-parameter tractable. However, we show that, despite its deceptive simplicity, it is W[1]-hard. Our main result establishes the following conditional lower bound: under the Exponential Time Hypothesis, the Split Contraction problem cannot be solved in time 2^(o(l^2)) * poly(n) where l is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2^(o(l^2)) * poly(n) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.

Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split Contraction: The Untold Story. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agrawal_et_al:LIPIcs.STACS.2017.5, author = {Agrawal, Akanksha and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav}, title = {{Split Contraction: The Untold Story}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {5:1--5:14}, 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.2017.5}, URN = {urn:nbn:de:0030-drops-70297}, doi = {10.4230/LIPIcs.STACS.2017.5}, annote = {Keywords: Split Graph, Parameterized Complexity, Edge Contraction} }

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

In this paper we study the "independent" version of the classic Feedback Vertex Set problem in the realm of parameterized algorithms and moderately exponential time algorithms. More precisely, we study the Independent Feedback Vertex Set problem, where we are given an undirected graph G on n vertices and a positive integer k, and the objective is to check if there is an independent feedback vertex set of size at most k. A set S subseteq V(G) is called an independent feedback vertex set (ifvs) if S is an independent set and G\S is a forest. In this paper we design two deterministic exact algorithms for Independent Feedback Vertex Set with running times O*(4.1481^k) and O*(1.5981^n). In fact, the algorithm with O*(1.5981^n) running time finds the smallest sized ifvs, if an ifvs exists. Both the algorithms are based on interesting measures and improve the best known algorithms for the problem in their respective domains. In particular, the algorithm with running time O*(4.1481^k) is an improvement over the previous algorithm that ran in time O*(5^k). On the other hand, the algorithm with running time O*(1.5981^n) is the first moderately exponential time algorithm that improves over the naive algorithm that enumerates all the subsets of V(G). Additionally, we show that the number of minimal ifvses in any graph on n vertices is upper bounded by 1.7485^n.

Akanksha Agrawal, Sushmita Gupta, Saket Saurabh, and Roohani Sharma. Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2016.2, author = {Agrawal, Akanksha and Gupta, Sushmita and Saurabh, Saket and Sharma, Roohani}, title = {{Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {2:1--2:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.2}, URN = {urn:nbn:de:0030-drops-69400}, doi = {10.4230/LIPIcs.IPEC.2016.2}, annote = {Keywords: independent feedback vertex set, fixed parameter tractable, exact algorithm, enumeration} }

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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 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions.

Akanksha Agrawal, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Kernels for Deletion to Classes of Acyclic Digraphs. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{agrawal_et_al:LIPIcs.ISAAC.2016.6, author = {Agrawal, Akanksha and Saurabh, Saket and Sharma, Roohani and Zehavi, Meirav}, title = {{Kernels for Deletion to Classes of Acyclic Digraphs}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {6:1--6:12}, 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.2016.6}, URN = {urn:nbn:de:0030-drops-67777}, doi = {10.4230/LIPIcs.ISAAC.2016.6}, annote = {Keywords: out-forest, pumpkin, parameterized complexity, kernelization} }

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

A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1.

Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, and Saket Saurabh. Kernelization of Cycle Packing with Relaxed Disjointness Constraints. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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

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

For a family of graphs F, a 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 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 alpha-SIMULTANEOUS F-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 <= i <= alpha, is in F. In this work, we study alpha-SIMULTANEOUS F-DELETION for F being the family of forests. In other words, we focus on the alpha-SIMULTANEOUS FEEDBACK VERTEX SET (alpha-SIMFVS) problem. Algorithmically, we show that, like its classical counterpart, alpha-SIMFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant alpha. In particular, we give an algorithm running in 2^{O(alpha * k)} * n^{O(1)} time and a kernel with O(alpha * k^{3(alpha + 1)}) vertices. The running time of our algorithm implies that alpha-SIMFVS is FPT even when alpha in o(log(n)). We complement this positive result by showing that for alpha in O(log(n)), where n is the number of vertices in the input graph, alpha-SIMFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

Akanksha Agrawal, Daniel Lokshtanov, Amer E. Mouawad, and Saket Saurabh. Simultaneous Feedback Vertex Set: A Parameterized Perspective. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{agrawal_et_al:LIPIcs.STACS.2016.7, author = {Agrawal, Akanksha and Lokshtanov, Daniel and Mouawad, Amer E. and Saurabh, Saket}, title = {{Simultaneous Feedback Vertex Set: A Parameterized Perspective}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {7:1--7:15}, 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.2016.7}, URN = {urn:nbn:de:0030-drops-57084}, doi = {10.4230/LIPIcs.STACS.2016.7}, annote = {Keywords: parameterized complexity ,feedback vertex set, kernel, edge-colored graphs} }

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