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DOI: 10.4230/LIPIcs.ISAAC.2021.36

Filling Crosswords Is Very Hard

Authors: Laurent Gourvès, Ararat Harutyunyan, Michael Lampis, and Nikolaos Melissinos

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract

We revisit a classical crossword filling puzzle which already appeared in Garey&Jonhson’s book. We are given a grid with n vertical and horizontal slots and a dictionary with m words and are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by carefully taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we discover that the problem remains NP-hard under very severe structural restrictions, namely, if the grid graph is a union of stars and the alphabet has size 2, or the grid graph is a matching (so the crossword is a collection of disjoint crosses) and the alphabet has size 3. The problem does become slightly more tractable if word reuse is allowed, as we obtain an m^{tw} algorithm in this case, where tw is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved to obtain fixed-parameter tractability. More strongly, we show that under the ETH the problem cannot be solved in time m^o(k), where k is the number of horizontal slots of the instance (which trivially bounds tw). Motivated by these mostly negative results, we also consider the much more restricted case where the problem is parameterized by the number of slots n. Here, we show that the problem does become FPT (if the alphabet has constant size), but the parameter dependence is exponential in n². We show that this dependence is also justified: the existence of an algorithm with running time 2^o(n²), even for binary alphabet, would contradict the randomized ETH. Finally, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a 1/2-approximation, even on weighted instances, simply by considering only horizontal or only vertical slots. We show that this trivial algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture. The latter two results apply whether word reuse is allowed or not.

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Laurent Gourvès, Ararat Harutyunyan, Michael Lampis, and Nikolaos Melissinos. Filling Crosswords Is Very Hard. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gourves_et_al:LIPIcs.ISAAC.2021.36,
  author =	{Gourv\`{e}s, Laurent and Harutyunyan, Ararat and Lampis, Michael and Melissinos, Nikolaos},
  title =	{{Filling Crosswords Is Very Hard}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.36},
  URN =		{urn:nbn:de:0030-drops-154690},
  doi =		{10.4230/LIPIcs.ISAAC.2021.36},
  annote =	{Keywords: Crossword Puzzle, Treewidth, ETH}
}

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DOI: 10.4230/LIPIcs.WABI.2021.5

The Maximum Duo-Preservation String Mapping Problem with Bounded Alphabet

Authors: Nicolas Boria, Laurent Gourvès, Vangelis Th. Paschos, and Jérôme Monnot

Published in: LIPIcs, Volume 201, 21st International Workshop on Algorithms in Bioinformatics (WABI 2021)

Abstract

Given two strings A and B such that B is a permutation of A, the max duo-preservation string mapping (MPSM) problem asks to find a mapping π between them so as to preserve a maximum number of duos. A duo is any pair of consecutive characters in a string and it is preserved by π if its two consecutive characters in A are mapped to same two consecutive characters in B. This problem has received a growing attention in recent years, partly as an alternative way to produce approximation algorithms for its minimization counterpart, min common string partition, a widely studied problem due its applications in comparative genomics. Considering this favored field of application with short alphabet, it is surprising that MPSM^𝓁, the variant of MPSM with bounded alphabet, has received so little attention, with a single yet impressive work that provides a 2.67-approximation achieved in O(n) [Brubach, 2018], where n = |A| = |B|. Our work focuses on MPSM^𝓁, and our main contribution is the demonstration that this problem admits a Polynomial Time Approximation Scheme (PTAS) when 𝓁 = O(1). We also provide an alternate, somewhat simpler, proof of NP-hardness for this problem compared with the NP-hardness proof presented in [Haitao Jiang et al., 2012].

Cite as

Nicolas Boria, Laurent Gourvès, Vangelis Th. Paschos, and Jérôme Monnot. The Maximum Duo-Preservation String Mapping Problem with Bounded Alphabet. In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 201, pp. 5:1-5:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{boria_et_al:LIPIcs.WABI.2021.5,
  author =	{Boria, Nicolas and Gourv\`{e}s, Laurent and Paschos, Vangelis Th. and Monnot, J\'{e}r\^{o}me},
  title =	{{The Maximum Duo-Preservation String Mapping Problem with Bounded Alphabet}},
  booktitle =	{21st International Workshop on Algorithms in Bioinformatics (WABI 2021)},
  pages =	{5:1--5:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-200-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{201},
  editor =	{Carbone, Alessandra and El-Kebir, Mohammed},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2021.5},
  URN =		{urn:nbn:de:0030-drops-143586},
  doi =		{10.4230/LIPIcs.WABI.2021.5},
  annote =	{Keywords: Maximum-Duo Preservation String Mapping, Bounded alphabet, Polynomial Time Approximation Scheme}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2018.73

Covering Clients with Types and Budgets

Authors: Dimitris Fotakis, Laurent Gourvès, Claire Mathieu, and Abhinav Srivastav

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract

In this paper, we consider a variant of the facility location problem. Imagine the scenario where facilities are categorized into multiple types such as schools, hospitals, post offices, etc. and the cost of connecting a client to a facility is realized by the distance between them. Each client has a total budget on the distance she/he is willing to travel. The goal is to open the minimum number of facilities such that the aggregate distance of each client to multiple types is within her/his budget. This problem closely resembles to the set cover and r-domination problems. Here, we study this problem in different settings. Specifically, we present some positive and negative results in the general setting, where no assumption is made on the distance values. Then we show that better results can be achieved when clients and facilities lie in a metric space.

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Dimitris Fotakis, Laurent Gourvès, Claire Mathieu, and Abhinav Srivastav. Covering Clients with Types and Budgets. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 73:1-73:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fotakis_et_al:LIPIcs.ISAAC.2018.73,
  author =	{Fotakis, Dimitris and Gourv\`{e}s, Laurent and Mathieu, Claire and Srivastav, Abhinav},
  title =	{{Covering Clients with Types and Budgets}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{73:1--73:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.73},
  URN =		{urn:nbn:de:0030-drops-100213},
  doi =		{10.4230/LIPIcs.ISAAC.2018.73},
  annote =	{Keywords: Facility Location, Geometric Set Cover, Local Search}
}

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Documents authored by Gourvès, Laurent

Filling Crosswords Is Very Hard

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The Maximum Duo-Preservation String Mapping Problem with Bounded Alphabet

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Covering Clients with Types and Budgets

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