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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1×2 pieces on a hexagonal grid board of 2n+1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the puzzle can always be solved. We also study the complexity of determining whether a given set of colored pieces can be placed on a colored hexagonal grid board with matching colors. We show this problem is NP-complete for arbitrarily many colors, but solvable in randomized polynomial time if the number of colors is a fixed constant.

Joep Hamersma, Marc van Kreveld, Yushi Uno, and Tom C. van der Zanden. Gourds: A Sliding-Block Puzzle with Turning. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hamersma_et_al:LIPIcs.ISAAC.2020.33, author = {Hamersma, Joep and van Kreveld, Marc and Uno, Yushi and van der Zanden, Tom C.}, title = {{Gourds: A Sliding-Block Puzzle with Turning}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {33:1--33:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.33}, URN = {urn:nbn:de:0030-drops-133773}, doi = {10.4230/LIPIcs.ISAAC.2020.33}, annote = {Keywords: computational complexity, divide-and-conquer, Hamiltonian cycle, puzzle game, (combinatorial) reconfiguration, sliding-block puzzle} }

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

Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisingly admits a subexponential algorithm. In this paper we give tight running time bounds for packing similarly-sized non-fat objects in higher dimensions.
We propose an alternative and very weak measure of fatness called the stabbing number, and show that the packing problem in Euclidean space of constant dimension d >=slant 3 for a family of similarly sized objects with stabbing number alpha can be solved in 2^O(n^(1-1/d) alpha) time. We prove that even in the case of axis-parallel boxes of fixed shape, there is no 2^o(n^(1-1/d) alpha) algorithm under ETH. This result smoothly bridges the whole range of having constant-fat objects on one extreme (alpha=1) and a subexponential algorithm of the usual running time, and having very "skinny" objects on the other extreme (alpha=n^(1/d)), where we cannot hope to improve upon the brute force running time of 2^O(n), and thereby characterizes the impact of fatness on the complexity of packing in case of similarly sized objects. We also study the same problem when parameterized by the solution size k, and give a n^O(k^(1-1/d) alpha) algorithm, with an almost matching lower bound: there is no algorithm with running time of the form f(k) n^o(k^(1-1/d) alpha/log k) under ETH. One of our main tools in these reductions is a new wiring theorem that may be of independent interest.

Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. How Does Object Fatness Impact the Complexity of Packing in d Dimensions?. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kisfaludibak_et_al:LIPIcs.ISAAC.2019.36, author = {Kisfaludi-Bak, S\'{a}ndor and Marx, D\'{a}niel and van der Zanden, Tom C.}, title = {{How Does Object Fatness Impact the Complexity of Packing in d Dimensions?}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {36:1--36:18}, 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.2019.36}, URN = {urn:nbn:de:0030-drops-115327}, doi = {10.4230/LIPIcs.ISAAC.2019.36}, annote = {Keywords: Geometric intersection graph, Independent Set, Object fatness} }

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

We show that the problem of deciding whether a collection of polyominoes, each fitting in a 2 x O(log n) rectangle, can be packed into a 3 x n box does not admit a 2^{o(n/log{n})}-time algorithm, unless the Exponential Time Hypothesis fails. We also give an algorithm that attains this lower bound, solving any instance of polyomino packing with total area n in 2^{O(n/log{n})} time. This establishes a tight bound on the complexity of Polyomino Packing, even in a very restricted case. In contrast, for a 2 x n box, we show that the problem can be solved in strongly subexponential time.

Hans L. Bodlaender and Tom C. van der Zanden. On the Exact Complexity of Polyomino Packing. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bodlaender_et_al:LIPIcs.FUN.2018.9, author = {Bodlaender, Hans L. and van der Zanden, Tom C.}, title = {{On the Exact Complexity of Polyomino Packing}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {9:1--9:10}, 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.2018.9}, URN = {urn:nbn:de:0030-drops-88001}, doi = {10.4230/LIPIcs.FUN.2018.9}, annote = {Keywords: polyomino packing, exact complexity, exponential time hypothesis} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

We present a parallel algorithm for computing the treewidth of a graph on a GPU. We implement this algorithm in OpenCL, and experimentally evaluate its performance. Our algorithm is based on an O*(2^n)-time algorithm that explores the elimination orderings of the graph using a Held-Karp like dynamic programming approach. We use Bloom filters to detect duplicate solutions.
GPU programming presents unique challenges and constraints, such as constraints on the use of memory and the need to limit branch divergence. We experiment with various optimizations to see if it is possible to work around these issues. We achieve a very large speed up (up to 77x) compared to running the same algorithm on the CPU.

Tom C. van der Zanden and Hans L. Bodlaender. Computing Treewidth on the GPU. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{vanderzanden_et_al:LIPIcs.IPEC.2017.29, author = {van der Zanden, Tom C. and Bodlaender, Hans L.}, title = {{Computing Treewidth on the GPU}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {29:1--29:13}, 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.2017.29}, URN = {urn:nbn:de:0030-drops-85671}, doi = {10.4230/LIPIcs.IPEC.2017.29}, annote = {Keywords: treewidth, GPU, GPGPU, exact algorithms, graph algorithms, algorithm engineering} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We establish the complexity of several graph embedding problems: Subgraph Isomorphism, Graph Minor, Induced Subgraph and Induced Minor, when restricted to H-minor free graphs. In each of these problems, we are given a pattern graph P and a host graph G, and want to determine whether P is a subgraph (minor, induced subgraph or induced minor) of G. We show that, for any fixed graph H and epsilon > 0, if P is H-Minor Free and G has treewidth tw, (induced) subgraph can be solved 2^{O(k^{epsilon}*tw+k/log(k))}*n^{O(1)} time and (induced) minor can be solved in 2^{O(k^{epsilon}*tw+tw*log(tw)+k/log(k))}*n^{O(1)} time, where k = |V(P)|.
We also show that this is optimal, in the sense that the existence of an algorithm for one of these problems running in 2^{o(n/log(n))} time would contradict the Exponential Time Hypothesis. This solves an open problem on the complexity of Subgraph Isomorphism for planar graphs.
The key algorithmic insight is that dynamic programming approaches can be sped up by identifying isomorphic connected components in the pattern graph. This technique seems widely applicable, and it appears that there is a relatively unexplored class of problems that share a similar upper and lower bound.

Hans L. Bodlaender, Jesper Nederlof, and Tom C. van der Zanden. Subexponential Time Algorithms for Embedding H-Minor Free Graphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bodlaender_et_al:LIPIcs.ICALP.2016.9, author = {Bodlaender, Hans L. and Nederlof, Jesper and van der Zanden, Tom C.}, title = {{Subexponential Time Algorithms for Embedding H-Minor Free Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.9}, URN = {urn:nbn:de:0030-drops-62756}, doi = {10.4230/LIPIcs.ICALP.2016.9}, annote = {Keywords: subgraph isomorphism, graph minors, subexponential time} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to k*n boards, is PSPACE-complete even when k is at most a constant.

Tom C. van der Zanden. Parameterized Complexity of Graph Constraint Logic. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 282-293, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{vanderzanden:LIPIcs.IPEC.2015.282, author = {van der Zanden, Tom C.}, title = {{Parameterized Complexity of Graph Constraint Logic}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {282--293}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.282}, URN = {urn:nbn:de:0030-drops-55906}, doi = {10.4230/LIPIcs.IPEC.2015.282}, annote = {Keywords: Nondeterministic Constraint Logic, Reconfiguration Problems, Parameterized Complexity, Treewidth, Bandwidth} }

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