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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Treewidth serves as an important parameter that, when bounded, yields tractability for a wide class of problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and Quantified SAT or, more generally, Quantified CSP, are fixed-parameter tractable parameterized by the treewidth {of the input’s (primal) graph} plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], respectively. The algorithms generated by these (meta-)results have running times whose dependence on treewidth is a tower of exponents. A conditional lower bound by Fichte, Hecher, and Pfandler [LICS 2020] shows that, for Quantified SAT, the height of this tower is equal to the number of quantifier alternations. These types of lower bounds, which show that at least double-exponential factors in the running time are necessary, exhibit the extraordinary level of computational hardness for such problems, and are rare in the current literature: there are only a handful of such lower bounds (for treewidth and vertex cover parameterizations) and all of them are for problems that are #NP-complete, Σ₂^p-complete, Π₂^p-complete, or complete for even higher levels of the polynomial hierarchy.
Our results demonstrate, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to achieve double-exponential lower bounds: we derive double-exponential lower bounds in the treewidth (tw) and the vertex cover number (vc), for natural, important, and well-studied NP-complete graph problems. Specifically, we design a technique to obtain such lower bounds and show its versatility by applying it to three different problems: Metric Dimension, Strong Metric Dimension, and Geodetic Set. We prove that these problems do not admit 2^{2^o(tw)}⋅n^𝒪(1)-time algorithms, even on bounded diameter graphs, unless the ETH fails (here, n is the number of vertices in the graph). In fact, for Strong Metric Dimension, the double-exponential lower bound holds even for the vertex cover number. We further complement all our lower bounds with matching (and sometimes non-trivial) upper bounds.
For the conditional lower bounds, we design and use a novel, yet simple technique based on Sperner families of sets. We believe that the amenability of our technique will lead to obtaining such lower bounds for many other problems in NP.

Florent Foucaud, Esther Galby, Liana Khazaliya, Shaohua Li, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Problems in NP Can Admit Double-Exponential Lower Bounds When Parameterized by Treewidth or Vertex Cover. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 66:1-66:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{foucaud_et_al:LIPIcs.ICALP.2024.66, author = {Foucaud, Florent and Galby, Esther and Khazaliya, Liana and Li, Shaohua and Mc Inerney, Fionn and Sharma, Roohani and Tale, Prafullkumar}, title = {{Problems in NP Can Admit Double-Exponential Lower Bounds When Parameterized by Treewidth or Vertex Cover}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {66:1--66:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.66}, URN = {urn:nbn:de:0030-drops-202091}, doi = {10.4230/LIPIcs.ICALP.2024.66}, annote = {Keywords: Parameterized Complexity, ETH-based Lower Bounds, Double-Exponential Lower Bounds, Kernelization, Vertex Cover, Treewidth, Diameter, Metric Dimension, Strong Metric Dimension, Geodetic Sets} }

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

A bipartite graph is called a biclique if it is a complete bipartite graph and a biclique is called a balanced biclique if it has equal number of vertices in both parts of its bipartition. In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges in G to obtain a biclique and a balanced biclique, respectively. We first prove that these problems are NP-complete even when the input graph is bipartite. Next, we study the parameterized complexity of these problems and show that they admit single exponential-time FPT algorithms when parameterized by the number k of edge contractions. Then, we show that Balanced Biclique Contraction admits a quadratic vertex kernel while Biclique Contraction does not admit any polynomial compression (or kernel) unless NP ⊆ coNP/poly.

R. Krithika, V. K. Kutty Malu, Roohani Sharma, and Prafullkumar Tale. Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction. 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. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{krithika_et_al:LIPIcs.FSTTCS.2023.8, author = {Krithika, R. and Malu, V. K. Kutty and Sharma, Roohani and Tale, Prafullkumar}, title = {{Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {8:1--8:18}, 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.2023.8}, URN = {urn:nbn:de:0030-drops-193811}, doi = {10.4230/LIPIcs.FSTTCS.2023.8}, annote = {Keywords: contraction, bicliques, balanced bicliques, parameterized complexity} }

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

The leafage of a chordal graph G is the minimum integer 𝓁 such that G can be realized as an intersection graph of subtrees of a tree with 𝓁 leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1). We present a conceptually much simpler algorithm that runs in time 2^𝒪(𝓁) ⋅ n^𝒪(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n^𝒪(𝓁)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n^O(1)-time.

Esther Galby, Dániel Marx, Philipp Schepper, Roohani Sharma, and Prafullkumar Tale. Domination and Cut Problems on Chordal Graphs with Bounded Leafage. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{galby_et_al:LIPIcs.IPEC.2022.14, author = {Galby, Esther and Marx, D\'{a}niel and Schepper, Philipp and Sharma, Roohani and Tale, Prafullkumar}, title = {{Domination and Cut Problems on Chordal Graphs with Bounded Leafage}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {14:1--14:24}, 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.2022.14}, URN = {urn:nbn:de:0030-drops-173704}, doi = {10.4230/LIPIcs.IPEC.2022.14}, annote = {Keywords: Chordal Graphs, Leafage, FPT Algorithms, Dominating Set, MultiCut with Undeletable Terminals, Multiway Cut with Undeletable Terminals} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of k ≥ 1 agents. While the initial positions of Facilitator’s agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator’s goal to bring both her agents at same vertex and Divider’s goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with k agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents.
This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.

Neeldhara Misra, Manas Mulpuri, Prafullkumar Tale, and Gaurav Viramgami. Romeo and Juliet Meeting in Forest like Regions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{misra_et_al:LIPIcs.FSTTCS.2022.27, author = {Misra, Neeldhara and Mulpuri, Manas and Tale, Prafullkumar and Viramgami, Gaurav}, title = {{Romeo and Juliet Meeting in Forest like Regions}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {27:1--27:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.27}, URN = {urn:nbn:de:0030-drops-174194}, doi = {10.4230/LIPIcs.FSTTCS.2022.27}, annote = {Keywords: Games on Graphs, Dynamic Separators, W\lbrack1\rbrack-hardness, Structural Parametersization, Treewidth} }

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

For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson’s book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{galby_et_al:LIPIcs.MFCS.2022.51, author = {Galby, Esther and Khazaliya, Liana and Mc Inerney, Fionn and Sharma, Roohani and Tale, Prafullkumar}, title = {{Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.51}, URN = {urn:nbn:de:0030-drops-168496}, doi = {10.4230/LIPIcs.MFCS.2022.51}, annote = {Keywords: Metric Dimension, Parameterized Complexity, Feedback Vertex Set} }

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

The Contraction(vc) problem takes as input a graph G on n vertices and two integers k and d, and asks whether one can contract at most k edges to reduce the size of a minimum vertex cover of G by at least d. Recently, Lima et al. [MFCS 2020, JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, Contraction(vc) admits an XP algorithm running in time f(d) ⋅ n^O(d). They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results:
- Contraction(vc) is W[1]-hard parameterized by k + d. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f(k + d) ⋅ n^o(k + d) for any function f. In particular, this answers the open question stated in Lima et al. [MFCS 2020] in the negative.
- It is NP-hard to decide whether an instance (G, k, d) of {Contraction(vc)} is a Yes-instance even when k = d, hence enhancing our understanding of the classical complexity of the problem.
- Contraction(vc) can be solved in time 2^O(d) ⋅ n^{k - d + O(1)}. This XP algorithm improves the one of Lima et al. [MFCS 2020], which uses Courcelle’s theorem as a subroutine and hence, the f(d)-factor in the running time is non-explicit and probably very large. On the other hand, this shows that when k = d, the problem is FPT parameterized by d (or by k).

Paloma T. Lima, Vinicius F. dos Santos, Ignasi Sau, Uéverton S. Souza, and Prafullkumar Tale. Reducing the Vertex Cover Number via Edge Contractions. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lima_et_al:LIPIcs.MFCS.2022.69, author = {Lima, Paloma T. and dos Santos, Vinicius F. and Sau, Ignasi and Souza, U\'{e}verton S. and Tale, Prafullkumar}, title = {{Reducing the Vertex Cover Number via Edge Contractions}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {69:1--69:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.69}, URN = {urn:nbn:de:0030-drops-168671}, doi = {10.4230/LIPIcs.MFCS.2022.69}, annote = {Keywords: Blocker problems, edge contraction, vertex cover, parameterized complexity} }

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

In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n^𝒪(k). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH).
Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time (d + k)^(2k) ⋅ n^𝒪(1) = 2^𝒪(k log (k+d)) ⋅ n^𝒪(1), and remains NP-hard for every constant d ≥ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^𝒪(dk) ⋅ n^𝒪(1). In particular, our algorithm runs in time 2^𝒪(k) ⋅ n^𝒪(1), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly.

Saket Saurabh and Prafullkumar Tale. On the Parameterized Complexity of Maximum Degree Contraction Problem. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{saurabh_et_al:LIPIcs.IPEC.2020.26, author = {Saurabh, Saket and Tale, Prafullkumar}, title = {{On the Parameterized Complexity of Maximum Degree Contraction Problem}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {26:1--26:16}, 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.2020.26}, URN = {urn:nbn:de:0030-drops-133297}, doi = {10.4230/LIPIcs.IPEC.2020.26}, annote = {Keywords: Graph Contraction Problems, FPT Algorithm, Lower Bound, ETH, No Polynomial Kernel} }

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APPROX

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

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the F-Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k, F-Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. We obtain the following results for the F-Contraction problem.
- Clique Contraction is known to be FPT. However, unless NP ⊆ coNP/poly, it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (PSAKS). That is, it admits a (1 + ε)-approximate kernel with {O}(k^{f(ε)}) vertices for every ε > 0.
- Split Contraction is known to be W[1]-Hard. We deconstruct this intractability result in two ways. Firstly, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)-FPT-approximation algorithm for Split Contraction). Furthermore, we show that, assuming Gap-ETH, there is no (5/4-δ)-FPT-approximation algorithm for Split Contraction. Here, ε, δ > 0 are fixed constants.
- Chordal Contraction is known to be W[2]-Hard. We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1], there is no F(k)-FPT-approximation algorithm for Chordal Contraction. Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k)-FPT-approximation algorithm for the F-Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and |X| ≤ k, it outputs an edge set Y of size at most h(k) ⋅ k for which G/Y is in F. We find it extremely interesting that three closely related problems have different behavior with respect to FPT-approximation.

Spoorthy Gunda, Pallavi Jain, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. On the Parameterized Approximability of Contraction to Classes of Chordal Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gunda_et_al:LIPIcs.APPROX/RANDOM.2020.51, author = {Gunda, Spoorthy and Jain, Pallavi and Lokshtanov, Daniel and Saurabh, Saket and Tale, Prafullkumar}, title = {{On the Parameterized Approximability of Contraction to Classes of Chordal Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {51:1--51:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.51}, URN = {urn:nbn:de:0030-drops-126545}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.51}, annote = {Keywords: Graph Contraction, FPT-Approximation, Inapproximability, Lossy Kernels} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

For a family of graphs 𝒢, the 𝒢-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists F ⊆ E(G) of size at most k such that G/F belongs to 𝒢. Here, G/F is the graph obtained from G by contracting all the edges in F. In this article, we initiate the study of Grid Contraction from the parameterized complexity point of view. We present a fixed parameter tractable algorithm, running in time c^k ⋅ |V(G)|^{{O}(1)}, for this problem. We complement this result by proving that unless ETH fails, there is no algorithm for Grid Contraction with running time c^{o(k)} ⋅ |V(G)|^{{O}(1)}. We also present a polynomial kernel for this problem.

Saket Saurabh, Uéverton dos Santos Souza, and Prafullkumar Tale. On the Parameterized Complexity of Grid Contraction. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{saurabh_et_al:LIPIcs.SWAT.2020.34, author = {Saurabh, Saket and Souza, U\'{e}verton dos Santos and Tale, Prafullkumar}, title = {{On the Parameterized Complexity of Grid Contraction}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {34:1--34:17}, 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.2020.34}, URN = {urn:nbn:de:0030-drops-122810}, doi = {10.4230/LIPIcs.SWAT.2020.34}, annote = {Keywords: Grid Contraction, FPT, Kernelization, Lower Bound} }

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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 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 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740^k*n^{O(1)} (polynomial space) and 1.1277^k*n^{O(1)} (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667^k*n^{O(1)} running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2^k*n^{O(1)} running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

R. Krithika, Abhishek Sahu, and Prafullkumar Tale. Dynamic Parameterized Problems. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{krithika_et_al:LIPIcs.IPEC.2016.19, author = {Krithika, R. and Sahu, Abhishek and Tale, Prafullkumar}, title = {{Dynamic Parameterized Problems}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {19:1--19:14}, 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.2016.19}, URN = {urn:nbn:de:0030-drops-69366}, doi = {10.4230/LIPIcs.IPEC.2016.19}, annote = {Keywords: dynamic problems, fixed-parameter tractability, kernelization} }

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

We study some well-known graph contraction problems in the recently introduced framework of lossy kernelization. In classical kernelization, given an instance (I,k) of a parameterized problem, we are interested in obtaining (in polynomial time) an equivalent instance (I',k') of the same problem whose size is bounded by a function in k. This notion however has a major limitation. Given an approximate solution to the instance (I',k'), we can say nothing about the original instance (I,k). To handle this issue, among others, the framework of lossy kernelization was introduced. In this framework, for a constant alpha, given an instance (I,k) we obtain an instance (I',k') of the same problem such that, for every c>1, any c-approximate solution to (I',k') can be turned into a (c*alpha)-approximate solution to the original instance (I, k) in polynomial time. Naturally, we are interested in a polynomial time algorithm for this task, and further require that |I'| + k' = k^{O(1)}. Akin to the notion of polynomial time approximation schemes in approximation algorithms, a parameterized problem is said to admit a polynomial size approximate kernelization scheme (PSAKS) if it admits a polynomial size alpha-approximate kernel for every approximation parameter alpha > 1. In this work, we design PSAKSs for Tree Contraction, Star Contraction, Out-Tree Contraction and Cactus Contraction problems. These problems do not admit polynomial kernels, and we show that each of them admit a PSAKS with running time k^{f(alpha)}|I|^{O(1)} that returns an instance of size k^{g(alpha)} where f(alpha) and g(alpha) are constants depending on alpha.

R. Krithika, Pranabendu Misra, Ashutosh Rai, and Prafullkumar Tale. Lossy Kernels for Graph Contraction Problems. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{krithika_et_al:LIPIcs.FSTTCS.2016.23, author = {Krithika, R. and Misra, Pranabendu and Rai, Ashutosh and Tale, Prafullkumar}, title = {{Lossy Kernels for Graph Contraction Problems}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.23}, URN = {urn:nbn:de:0030-drops-68587}, doi = {10.4230/LIPIcs.FSTTCS.2016.23}, annote = {Keywords: parameterized complexity, lossy kernelization, graph theory, edge contraction problems} }

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