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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

For any fixed positive integer r and a given budget k, the r-Eigenvalue Vertex Deletion (r-EVD) problem asks if a graph G admits a subset S of at most k vertices such that the adjacency matrix of G⧵S has at most r distinct eigenvalues. The edge deletion, edge addition, and edge editing variants are defined analogously. For r = 1, r-EVD is equivalent to the Vertex Cover problem. For r = 2, it turns out that r-EVD amounts to removing a subset S of at most k vertices so that G⧵ S is a cluster graph where all connected components have the same size.
We show that r-EVD is NP-complete even on bipartite graphs with maximum degree four for every fixed r > 2, and FPT when parameterized by the solution size and the maximum degree of the graph.
We also establish several results for the special case when r = 2. For the vertex deletion variant, we show that 2-EVD is NP-complete even on triangle-free and 3d-regular graphs for any d ≥ 2, and also NP-complete on d-regular graphs for any d ≥ 8. The edge deletion, addition, and editing variants are all NP-complete for r = 2. The edge deletion problem admits a polynomial time algorithm if the input is a cluster graph, while - in contrast - the edge addition variant is hard even when the input is a cluster graph. We show that the edge addition variant has a quadratic kernel. The edge deletion and vertex deletion variants admit a single-exponential FPT algorithm when parameterized by the solution size alone.
Our main contribution is to develop the complexity landscape for the problem of modifying a graph with the aim of reducing the number of distinct eigenvalues in the spectrum of its adjacency matrix. It turns out that this captures, apart from Vertex Cover, also a natural variation of the problem of modifying to a cluster graph as a special case, which we believe may be of independent interest.

Neeldhara Misra, Harshil Mittal, Saket Saurabh, and Dhara Thakkar. On the Complexity of the Eigenvalue Deletion Problem. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 53:1-53:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{misra_et_al:LIPIcs.ISAAC.2023.53, author = {Misra, Neeldhara and Mittal, Harshil and Saurabh, Saket and Thakkar, Dhara}, title = {{On the Complexity of the Eigenvalue Deletion Problem}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {53:1--53:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.53}, URN = {urn:nbn:de:0030-drops-193555}, doi = {10.4230/LIPIcs.ISAAC.2023.53}, annote = {Keywords: Graph Modification, Rank Reduction, Eigenvalues} }

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

We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph G and is denoted by evc(G). It is clear that evc(G) is at least mvc(G), the size of a minimum vertex cover of G. We say that G is Spartan if evc(G) = mvc(G). The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on 2n vertices where every edge belongs to a perfect matching, an easy strategy is to have n guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.

Neeldhara Misra and Saraswati Girish Nanoti. Spartan Bipartite Graphs Are Essentially Elementary. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 68:1-68:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{misra_et_al:LIPIcs.MFCS.2023.68, author = {Misra, Neeldhara and Nanoti, Saraswati Girish}, title = {{Spartan Bipartite Graphs Are Essentially Elementary}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {68:1--68:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.68}, URN = {urn:nbn:de:0030-drops-186025}, doi = {10.4230/LIPIcs.MFCS.2023.68}, annote = {Keywords: Bipartite Graphs, Eternal Vertex Cover, Perfect Matchings, Elementary, Spartan} }

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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 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 226, 11th International Conference on Fun with Algorithms (FUN 2022)

We introduce a generalization of "Solo Chess", a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 × 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e, make a sequence of captures after which only one piece is left.
We generalize this game to unbounded boards with n pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board.
Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.

N.R. Aravind, Neeldhara Misra, and Harshil Mittal. Chess Is Hard Even for a Single Player. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 5:1-5:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aravind_et_al:LIPIcs.FUN.2022.5, author = {Aravind, N.R. and Misra, Neeldhara and Mittal, Harshil}, title = {{Chess Is Hard Even for a Single Player}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.5}, URN = {urn:nbn:de:0030-drops-159753}, doi = {10.4230/LIPIcs.FUN.2022.5}, annote = {Keywords: chess, strategy, board games, NP-complete} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Neeldhara Misra. On the Complexity of Two Dots for Narrow Boards and Few Colors. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 7:1-7:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.7, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Misra, Neeldhara}, title = {{On the Complexity of Two Dots for Narrow Boards and Few Colors}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.7}, URN = {urn:nbn:de:0030-drops-87988}, doi = {10.4230/LIPIcs.FUN.2018.7}, annote = {Keywords: puzzle, NP-complete, perfect information, combinatorial game theory} }

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

We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule.
The Possible Winner problem is well-known to be NP-Complete in general, and it is in fact known to be NP-Complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the Possible Winner problem remains NP-Complete. In particular, we find the exact values of t for which the Possible Winner problem transitions to being NP-Complete from being in P, where t is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad subclass of scoring rules which includes all the commonly used scoring rules (such as plurality, veto, Borda, and k-approval), Copeland^\alpha for every \alpha in [0,1], maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the Possible Winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.

Palash Dey and Neeldhara Misra. On the Exact Amount of Missing Information that Makes Finding Possible Winners Hard. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 57:1-57:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dey_et_al:LIPIcs.MFCS.2017.57, author = {Dey, Palash and Misra, Neeldhara}, title = {{On the Exact Amount of Missing Information that Makes Finding Possible Winners Hard}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {57:1--57:14}, 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.2017.57}, URN = {urn:nbn:de:0030-drops-81354}, doi = {10.4230/LIPIcs.MFCS.2017.57}, annote = {Keywords: Computational Social Choice, Dichotomy, NP-completeness, Maxflow, Voting, Possible winner} }

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

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

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

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@InProceedings{choudhari_et_al:LIPIcs.ICALP.2017.135, author = {Choudhari, Jayesh and Dasgupta, Anirban and Misra, Neeldhara and Ramanujan, M. S.}, title = {{Saving Critical Nodes with Firefighters is FPT}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {135:1--135:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.135}, URN = {urn:nbn:de:0030-drops-74968}, doi = {10.4230/LIPIcs.ICALP.2017.135}, annote = {Keywords: firefighting, cuts, FPT, kernelization} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Two Dots is a popular single-player puzzle video game for iOS and Android. In its simplest form, the game consists of a board with dots of different colors, and a valid move consists of connecting a sequence of adjacent dots of the same color. We say that dots engaged in a move are "hit" by the player. After every move, the connected dots disappear, and the void is filled by new dots (the entire board shifts downwards and new dots appear on top). Typically the game provides a limited number of moves and varying goals (such as hitting a required number of dots of a particular color). We show that the perfect information version of the game (where the sequence of incoming dots is known) is NP-complete, even for fairly restricted goal types.

Neeldhara Misra. Two Dots is NP-complete. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 24:1-24:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{misra:LIPIcs.FUN.2016.24, author = {Misra, Neeldhara}, title = {{Two Dots is NP-complete}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {24:1--24:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.24}, URN = {urn:nbn:de:0030-drops-58831}, doi = {10.4230/LIPIcs.FUN.2016.24}, annote = {Keywords: combinatorial game theory, NP-complete, perfect information, puzzle} }

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

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

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

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

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We study a general class of problems called F-Deletion problems. In an F-Deletion problem, we are asked whether a subset of at most k
vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-Deletion problem when F contains a planar graph. We give
- a linear vertex kernel on graphs excluding t-claw K_(1,t), the star with t leaves, as an induced subgraph, where t is a fixed integer.
- an approximation algorithm achieving an approximation ratio of O(log^(3/2) OPT), where $OPT$ is the size of an optimal solution on general undirected graphs.
Finally, we obtain polynomial kernels for the case when F only contains graph theta_c as a minor for a fixed integer c. The graph theta_c consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and Kernelization. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 189-200, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2011.189, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Misra, Neeldhara and Philip, Geevarghese and Saurabh, Saket}, title = {{Hitting forbidden minors: Approximation and Kernelization}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {189--200}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.189}, URN = {urn:nbn:de:0030-drops-30103}, doi = {10.4230/LIPIcs.STACS.2011.189}, annote = {Keywords: kernelization} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set.
Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard'' on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$.
While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.

Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. The effect of girth on the kernelization complexity of Connected Dominating Set. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 96-107, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{misra_et_al:LIPIcs.FSTTCS.2010.96, author = {Misra, Neeldhara and Philip, Geevarghese and Raman, Venkatesh and Saurabh, Saket}, title = {{The effect of girth on the kernelization complexity of Connected Dominating Set}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {96--107}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.96}, URN = {urn:nbn:de:0030-drops-28567}, doi = {10.4230/LIPIcs.FSTTCS.2010.96}, annote = {Keywords: Connected Dominating Set, parameterized complexity, kernelization, girth} }

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