8 Search Results for "Madathil, Jayakrishnan"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.13
Solving Partial Dominating Set and Related Problems Using Twin-Width

Authors: Jakub Balabán, Daniel Mock, and Peter Rossmanith

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are W[1]-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form ϕ≡∃ x₁⋯ ∃ x_k ∑_{α ∈ I} #y ψ_α(x₁,…,x_k,y) ≥ t, where ψ_α is a quantifier-free formula for each α ∈ I, t is an arbitrary number, and #y is a counting quantifier, can be evaluated in time f(d,k)n, where n is the number of vertices and d is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.
Cite as

Jakub Balabán, Daniel Mock, and Peter Rossmanith. Solving Partial Dominating Set and Related Problems Using Twin-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{balaban_et_al:LIPIcs.MFCS.2025.13,
  author =	{Balab\'{a}n, Jakub and Mock, Daniel and Rossmanith, Peter},
  title =	{{Solving Partial Dominating Set and Related Problems Using Twin-Width}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.13},
  URN =		{urn:nbn:de:0030-drops-241203},
  doi =		{10.4230/LIPIcs.MFCS.2025.13},
  annote =	{Keywords: Partial Dominating Set, Partial Vertex Cover, meta-algorithm, counting logic, twin-width}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.12
All-Subsets Important Separators with Applications to Sample Sets, Balanced Separators and Vertex Sparsifiers in Directed Graphs

Authors: Aditya Anand, Euiwoong Lee, Jason Li, and Thatchaphol Saranurak

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Given a directed graph G with n vertices and m edges, a parameter k and two disjoint subsets S,T ⊆ V(G), we show that the number of all-subsets important separators, which is the number of A-B important vertex separators of size at most k over all A ⊆ S and B ⊆ T, is at most β(|S|, |T|, k) = 4^k binom(|S|, ≤ k) binom(|T|, ≤ 2k), where binom(x, ≤ c) = ∑_{i = 1}^c binom(x,i), and that they can be enumerated in time 𝒪(β(|S|,|T|,k)k²(m+n)). This is a generalization of the folklore result stating that the number of A-B important separators for two fixed sets A and B is at most 4^k (first implicitly shown by Chen, Liu and Lu Algorithmica '09). From this result, we obtain the following applications: 1) We give a construction for detection sets and sample sets in directed graphs, generalizing the results of Kleinberg (Internet Mathematics' 03) and Feige and Mahdian (STOC' 06) to directed graphs. 2) Via our new sample sets, we give the first FPT algorithm for finding balanced separators in directed graphs parameterized by k, the size of the separator. Our algorithm runs in time 2^{𝒪(k)} ⋅ (m + n). 3) Additionally, we show a 𝒪(√{log k}) approximation algorithm for finding balanced separators in directed graphs in polynomial time. This improves the best known approximation guarantee of 𝒪(√{log n}) and matches the known guarantee in undirected graphs by Feige, Hajiaghayi and Lee (SICOMP' 08). 4) Finally, using our algorithm for listing all-subsets important separators, we give a deterministic construction of vertex cut sparsifiers in directed graphs when we are interested in preserving min-cuts of size upto c between bipartitions of the terminal set. Our algorithm constructs a sparsifier of size 𝒪(binom(t, ≤ 3c)2^{𝒪(c)}) and runs in time 𝒪(binom(t, ≤ 3c) 2^{𝒪(c)}(m + n)), where t is the number of terminals, and the sparsifier additionally preserves the set of important separators of size at most c between bipartitions of the terminals.
Cite as

Aditya Anand, Euiwoong Lee, Jason Li, and Thatchaphol Saranurak. All-Subsets Important Separators with Applications to Sample Sets, Balanced Separators and Vertex Sparsifiers in Directed Graphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{anand_et_al:LIPIcs.ICALP.2025.12,
  author =	{Anand, Aditya and Lee, Euiwoong and Li, Jason and Saranurak, Thatchaphol},
  title =	{{All-Subsets Important Separators with Applications to Sample Sets, Balanced Separators and Vertex Sparsifiers in Directed Graphs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.12},
  URN =		{urn:nbn:de:0030-drops-233892},
  doi =		{10.4230/LIPIcs.ICALP.2025.12},
  annote =	{Keywords: directed graphs, important separators, sample sets, balanced separators}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.44
Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances

Authors: Tim A. Hartmann and Dániel Marx

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

Abstract
The distance-d variants of Independent Set and Dominating Set problems have been extensively studied from different algorithmic viewpoints. In particular, the complexity of these problems are well understood on bounded-treewidth graphs [Katsikarelis, Lampis, and Paschos, Discret. Appl. Math 2022][Borradaile and Le, IPEC 2016]: given a tree decomposition of width t, the two problems can be solved in time d^t⋅ n^O(1) and (2d+1)^t⋅ n^O(1), respectively. Furthermore, assuming the Strong Exponential-Time Hypothesis (SETH), the base constants are best possible in these running times: they cannot be improved to d-ε and 2d+1-ε, respectively, for any ε > 0. We investigate continuous versions of these problems in a setting introduced by Megiddo and Tamir [SICOMP 1983], where every edge is modeled by a unit-length interval of points. In the δ-Dispersion problem, the task is to find a maximum number of points (possibly inside edges) that are pairwise at distance at least δ from each other. Similarly, in the δ-Covering problem, the task is to find a minimum number of points (possibly inside edges) such that every point of the graph (including those inside edges) is at distance at most δ from the selected point set. We provide a comprehensive understanding of these two problems on bounded-treewidth graphs. 1) Let δ = a/b with a and b being coprime. If a ≤ 2, then δ-Dispersion is polynomial-time solvable. For a ≥ 3, given a tree decomposition of width t, the problem can be solved in time (2a)^t⋅ n^O(1), and, assuming SETH, there is no (2a-ε)^t⋅n^{O(1)} time algorithm for any ε > 0. 2) Let δ = a/b with a and b being coprime. If a = 1, then δ-Covering is polynomial-time solvable. For a ≥ 2, given a tree decomposition of width t, the problem can be solved in time ((2+2(bod 2)) a)^t⋅ n^O(1), and, assuming SETH, there is no ((2+2(bod 2))a -ε)^t⋅n^O(1) time algorithm for any ε > 0. 3) For every fixed irrational number δ > 0 satisfying some mild computability condition, both δ-Dispersion and δ-Covering can be solved in time n^O(t) on graphs of treewidth t. We show a very explicitly defined irrational number δ = (4∑_{j=1}^∞ 2^{-2^j})^{-1} ≈ 0.790085 such that δ-Dispersion and δ/2-Covering are W[1]-hard parameterized by the treewidth t of the input graph, and, assuming ETH, cannot be solved in time f(t)⋅n^o(t). As a key step in obtaining these results, we extend earlier results on distance-d versions of Independent Set and Dominating Set: We determine the exact complexity of these problems in the special case when the input graph arises from some graph G' by subdividing every edge exactly b times.
Cite as

Tim A. Hartmann and Dániel Marx. Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hartmann_et_al:LIPIcs.STACS.2025.44,
  author =	{Hartmann, Tim A. and Marx, D\'{a}niel},
  title =	{{Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{44:1--44:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.44},
  URN =		{urn:nbn:de:0030-drops-228700},
  doi =		{10.4230/LIPIcs.STACS.2025.44},
  annote =	{Keywords: Independence, Domination, Irrationals, Treewidth, SETH}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.39
Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems

Authors: Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, and Saket Saurabh

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
For a positive integer c, a graph G is said to be c-closed if every pair of non-adjacent vertices in G have at most c-1 neighbours in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by Fox et al. [ICALP `18 and SICOMP `20]. Koana et al. [ESA `20] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set and (Threshold) Dominating Set, and showed that each of these problems admits a polynomial kernel, either w.r.t. the parameter k+c or w.r.t. the parameter k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable w.r.t. the parameter k+cl(G), whereas Partial Dominating Set, parameterized by k is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs, unless NP ⊆ co-NP/poly.
Cite as

Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, and Saket Saurabh. Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kanesh_et_al:LIPIcs.STACS.2022.39,
  author =	{Kanesh, Lawqueen and Madathil, Jayakrishnan and Roy, Sanjukta and Sahu, Abhishek and Saurabh, Saket},
  title =	{{Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra 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.2022.39},
  URN =		{urn:nbn:de:0030-drops-158494},
  doi =		{10.4230/LIPIcs.STACS.2022.39},
  annote =	{Keywords: c-closed graphs, domination problems, perfect code, connected dominating set, fixed-parameter tractable, polynomial kernel}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.23
A Polynomial Kernel for Bipartite Permutation Vertex Deletion

Authors: Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, and Shaily Verma

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

Abstract
In a permutation graph, vertices represent the elements of a permutation, and edges represent pairs of elements that are reversed by the permutation. In the Permutation Vertex Deletion problem, given an undirected graph G and an integer k, the objective is to test whether there exists a vertex subset S ⊆ V(G) such that |S| ≤ k and G-S is a permutation graph. The parameterized complexity of Permutation Vertex Deletion is a well-known open problem. Bożyk et al. [IPEC 2020] initiated a study towards this problem by requiring that G-S be a bipartite permutation graph (a permutation graph that is bipartite). They called this the Bipartite Permutation Vertex Deletion (BPVD) problem. They showed that the problem admits a factor 9-approximation algorithm as well as a fixed parameter tractable (FPT) algorithm running in time 𝒪(9^k |V(G)|⁹). And they posed the question {whether BPVD admits a polynomial kernel.} We resolve this question in the affirmative by designing a polynomial kernel for BPVD. In particular, we obtain the following: Given an instance (G,k) of BPVD, in polynomial time we obtain an equivalent instance (G',k') of BPVD such that k' ≤ k, and |V(G')|+|E(G')| ≤ k^{𝒪(1)}.
Cite as

Lawqueen Kanesh, Jayakrishnan Madathil, Abhishek Sahu, Saket Saurabh, and Shaily Verma. A Polynomial Kernel for Bipartite Permutation Vertex Deletion. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kanesh_et_al:LIPIcs.IPEC.2021.23,
  author =	{Kanesh, Lawqueen and Madathil, Jayakrishnan and Sahu, Abhishek and Saurabh, Saket and Verma, Shaily},
  title =	{{A Polynomial Kernel for Bipartite Permutation Vertex Deletion}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{23:1--23:18},
  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.2021.23},
  URN =		{urn:nbn:de:0030-drops-154065},
  doi =		{10.4230/LIPIcs.IPEC.2021.23},
  annote =	{Keywords: kernelization, bipartite permutation graph, bicliques}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.41
Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices

Authors: Akanksha Agrawal, Sudeshna Kolay, Jayakrishnan Madathil, and Saket Saurabh

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

Abstract
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}.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.28
A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Authors: Jayakrishnan Madathil, Roohani Sharma, and Meirav Zehavi

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

Abstract
Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k=0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixed-parameter tractable algorithms. That is, we show that the problem admits a 2^{O(sqrt{k} log k)}n^{O(1)} time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n,k,k^2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
Cite as

Jayakrishnan Madathil, Roohani Sharma, and Meirav Zehavi. A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{madathil_et_al:LIPIcs.MFCS.2019.28,
  author =	{Madathil, Jayakrishnan and Sharma, Roohani and Zehavi, Meirav},
  title =	{{A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{28:1--28:14},
  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.2019.28},
  URN =		{urn:nbn:de:0030-drops-109721},
  doi =		{10.4230/LIPIcs.MFCS.2019.28},
  annote =	{Keywords: bisection, semicomplete digraph, tournament, fpt algorithm, chromatic coding, polynomial kernel, splitters}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.7
Connecting the Dots (with Minimum Crossings)

Authors: Akanksha Agrawal, Grzegorz Guśpiel, Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi

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

Abstract
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.
Cite as

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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