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

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.

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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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

In k-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most k sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense.
In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set (DFVS). It is already known that, for all k ≥ 2, k-Digraph Coloring is NP-hard on digraphs of DFVS at most k+4. We strengthen this result to show that, for all k ≥ 2, k-Digraph Coloring is already NP-hard for DFVS exactly k. This immediately provides a dichotomy, as k-Digraph Coloring is trivial if DFVS is at most k-1. Refining our reduction we obtain two further consequences: (i) for all k ≥ 2, k-Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most k²; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by k if FAS is at most k²-1; (ii) k-Digraph Coloring is NP-hard for graphs of DFVS k, even if the maximum degree Δ is at most 4k-1; we show that this is also almost tight, as the problem becomes FPT for DFVS k and Δ ≤ 4k-3.
Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that k-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)k^{tw}. Since this is considerably worse than the k^{tw} dependence of (undirected) k-Coloring, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for k = 2. Specifically, we show that an FPT algorithm solving 2-Digraph Coloring with dependence td^o(td) would contradict the ETH.

Ararat Harutyunyan, Michael Lampis, and Nikolaos Melissinos. Digraph Coloring and Distance to Acyclicity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{harutyunyan_et_al:LIPIcs.STACS.2021.41, author = {Harutyunyan, Ararat and Lampis, Michael and Melissinos, Nikolaos}, title = {{Digraph Coloring and Distance to Acyclicity}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {41:1--41:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.41}, URN = {urn:nbn:de:0030-drops-136865}, doi = {10.4230/LIPIcs.STACS.2021.41}, annote = {Keywords: Digraph Coloring, Dichromatic number, NP-completeness, Parameterized complexity, Feedback vertex and arc sets} }