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PACE Solver Description

**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Twin-width (tww) is a parameter measuring the similarity of an undirected graph to a co-graph [Édouard Bonnet et al., 2022]. It is useful to analyze the parameterized complexity of various graph problems. This paper presents two algorithms to compute the twin-width and to provide a contraction sequence as witness. The two algorithms are motivated by the PACE 2023 challenge, one for the exact track and one for the heuristic track. Each algorithm produces a contraction sequence witnessing (i) the minimal twin-width admissible by the graph in the exact track (ii) an upper bound on the twin-width as tight as possible in the heuristic track.
Our heuristic algorithm relies on several greedy approaches with different performance characteristics to find and improve solutions. For large graphs we use locality sensitive hashing to approximately identify suitable contraction candidates. The exact solver follows a branch-and-bound design. It relies on the heuristic algorithm to provide initial upper bounds, and uses lower bounds via contraction sequences to show the optimality of a heuristic solution found in some branch.

Alexander Leonhardt, Holger Dell, Anselm Haak, Frank Kammer, Johannes Meintrup, Ulrich Meyer, and Manuel Penschuck. PACE Solver Description: Exact (GUTHMI) and Heuristic (GUTHM). In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 37:1-37:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{leonhardt_et_al:LIPIcs.IPEC.2023.37, author = {Leonhardt, Alexander and Dell, Holger and Haak, Anselm and Kammer, Frank and Meintrup, Johannes and Meyer, Ulrich and Penschuck, Manuel}, title = {{PACE Solver Description: Exact (GUTHMI) and Heuristic (GUTHM)}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {37:1--37:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.37}, URN = {urn:nbn:de:0030-drops-194563}, doi = {10.4230/LIPIcs.IPEC.2023.37}, annote = {Keywords: PACE 2023 Challenge, Heuristic, Exact, Twin-Width} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 11 (2023)

This report documents the program and the outcomes of Dagstuhl Seminar 22482 "Counting and Sampling: Algorithms and Complexity". We document the talks presented, covering many advances in the area made over the last five years. As well, we document the progress made by working groups on future projects.

Holger Dell, Mark R. Jerrum, Haiko Müller, Konrad Anand, and Marcus Pappik. Counting and Sampling: Algorithms and Complexity (Dagstuhl Seminar 22482). In Dagstuhl Reports, Volume 12, Issue 11, pp. 124-145, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@Article{dell_et_al:DagRep.12.11.124, author = {Dell, Holger and Jerrum, Mark R. and M\"{u}ller, Haiko and Anand, Konrad and Pappik, Marcus}, title = {{Counting and Sampling: Algorithms and Complexity (Dagstuhl Seminar 22482)}}, pages = {124--145}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {12}, number = {11}, editor = {Dell, Holger and Jerrum, Mark R. and M\"{u}ller, Haiko and Anand, Konrad and Pappik, Marcus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.11.124}, URN = {urn:nbn:de:0030-drops-178394}, doi = {10.4230/DagRep.12.11.124}, annote = {Keywords: Sampling, Counting, Algorithms, Complexity, Statistical Physics, Phase Transitions, Markov Chains, Graphs, Point Processes} }

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Complete Volume

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

LIPIcs, Volume 249, IPEC 2022, Complete Volume

Holger Dell and Jesper Nederlof. LIPIcs, Volume 249, IPEC 2022, Complete Volume. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 1-520, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{dell_et_al:LIPIcs.IPEC.2022, title = {{LIPIcs, Volume 249, IPEC 2022, Complete Volume}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {1--520}, 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}, URN = {urn:nbn:de:0030-drops-173553}, doi = {10.4230/LIPIcs.IPEC.2022}, annote = {Keywords: LIPIcs, Volume 249, IPEC 2022, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

Holger Dell and Jesper Nederlof. Front Matter, Table of Contents, Preface, Conference Organization. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 0:i-0:xviii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2022.0, author = {Dell, Holger and Nederlof, Jesper}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {0:i--0:xviii}, 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.2022.0}, URN = {urn:nbn:de:0030-drops-173562}, doi = {10.4230/LIPIcs.IPEC.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of k-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an n^{f(t,s)}-time algorithm to compute modulo 2^t the number of subgraph occurrences of patterns that are s vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo 2^t. Complementing our algorithm, we also give a simple and self-contained proof that counting k-matchings modulo odd integers q is {Mod}_q W[1]-complete and prove that counting k-paths modulo 2 is ⊕W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).

Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 34:1-34:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{curticapean_et_al:LIPIcs.ESA.2021.34, author = {Curticapean, Radu and Dell, Holger and Husfeldt, Thore}, title = {{Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {34:1--34:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.34}, URN = {urn:nbn:de:0030-drops-146154}, doi = {10.4230/LIPIcs.ESA.2021.34}, annote = {Keywords: Counting complexity, matchings, paths, subgraphs, parameterized complexity} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem’s parameterized and data complexity, where the query is considered to be small or even fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis (SETH) as well as a #W[2]-hardness result in parameterized complexity. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; in the language of parameterized complexity, this essentially establishes a #A[2]-completeness result.
Using ideas stemming from Lovász (1967), we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that refines and generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number (and thus the treewidth) is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.

Holger Dell, Marc Roth, and Philip Wellnitz. Counting Answers to Existential Questions (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 113:1-113:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dell_et_al:LIPIcs.ICALP.2019.113, author = {Dell, Holger and Roth, Marc and Wellnitz, Philip}, title = {{Counting Answers to Existential Questions}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {113:1--113:15}, 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.2019.113}, URN = {urn:nbn:de:0030-drops-106894}, doi = {10.4230/LIPIcs.ICALP.2019.113}, annote = {Keywords: Conjunctive queries, graph homomorphisms, counting complexity, parameterized complexity, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In this paper, we relate a beautiful theory by Lovász with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs G and H are indistinguishable by the color refinement algorithm if and only if, for all trees T, the number Hom(T,G) of homomorphisms from T to G equals the corresponding number Hom(T,H) for H.
There is a natural system of linear equations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The nonnegative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the nonnegativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all t, the two graphs have the same number of length-t walks.
We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of bounded path width and solutions to our system where we drop the nonnegativity constraints.
A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs G and H, there is a tree T with Hom(T,G)!=Hom(T,H).

Holger Dell, Martin Grohe, and Gaurav Rattan. Lovász Meets Weisfeiler and Leman. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 40:1-40:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dell_et_al:LIPIcs.ICALP.2018.40, author = {Dell, Holger and Grohe, Martin and Rattan, Gaurav}, title = {{Lov\'{a}sz Meets Weisfeiler and Leman}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.40}, URN = {urn:nbn:de:0030-drops-90444}, doi = {10.4230/LIPIcs.ICALP.2018.40}, annote = {Keywords: graph isomorphism, graph homomorphism numbers, tree width} }

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

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHPi_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHPi_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas. A Fixed-Parameter Perspective on #BIS. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 13:1-13:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{curticapean_et_al:LIPIcs.IPEC.2017.13, author = {Curticapean, Radu and Dell, Holger and Fomin, Fedor V. and Goldberg, Leslie Ann and Lapinskas, John}, title = {{A Fixed-Parameter Perspective on #BIS}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {13:1--13:13}, 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.2017.13}, URN = {urn:nbn:de:0030-drops-85613}, doi = {10.4230/LIPIcs.IPEC.2017.13}, annote = {Keywords: Approximate counting, parameterised complexity, independent sets} }

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

In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition.

Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 30:1-30:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2017.30, author = {Dell, Holger and Komusiewicz, Christian and Talmon, Nimrod and Weller, Mathias}, title = {{The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {30:1--30:12}, 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.2017.30}, URN = {urn:nbn:de:0030-drops-85582}, doi = {10.4230/LIPIcs.IPEC.2017.30}, annote = {Keywords: treewidth, minimum fill-in, contest, implementation challenge, FPT} }

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

We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.

Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding Detours is Fixed-Parameter Tractable. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2017.54, author = {Bez\'{a}kov\'{a}, Ivona and Curticapean, Radu and Dell, Holger and Fomin, Fedor V.}, title = {{Finding Detours is Fixed-Parameter Tractable}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {54:1--54:14}, 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.2017.54}, URN = {urn:nbn:de:0030-drops-74790}, doi = {10.4230/LIPIcs.ICALP.2017.54}, annote = {Keywords: longest path, fixed-parameter tractable algorithms, above-guarantee parameterization, graph minors} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)poly(n) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from H, is #W[1]-hard if no such bound exists.
Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

Radu Curticapean, Holger Dell, and Marc Roth. Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 25:1-25:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{curticapean_et_al:LIPIcs.STACS.2017.25, author = {Curticapean, Radu and Dell, Holger and Roth, Marc}, title = {{Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.25}, URN = {urn:nbn:de:0030-drops-70080}, doi = {10.4230/LIPIcs.STACS.2017.25}, annote = {Keywords: matchings, homomorphisms, line graphs, counting complexity, parameterized complexity} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Jaeger, Vertigan, and Welsh (1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (2010) and Husfeldt and Taslaman (2010), in combination with the results of Curticapean (2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp(o(n)) unless #ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann (1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases cannot be solved in time exp(o(n)) unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp(o(n)) unless #ETH fails.
In order to prove our results, we use the block interpolation idea by Curticapean (2015) and transfer it to systems of linear equations that might not directly correspond to interpolation.

Cornelius Brand, Holger Dell, and Marc Roth. Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brand_et_al:LIPIcs.IPEC.2016.9, author = {Brand, Cornelius and Dell, Holger and Roth, Marc}, title = {{Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {9:1--9:14}, 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.2016.9}, URN = {urn:nbn:de:0030-drops-69426}, doi = {10.4230/LIPIcs.IPEC.2016.9}, annote = {Keywords: computational complexity, counting problems, Tutte polynomial, exponential time hypothesis, forests, independent sets} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

In this article, the steering committee of the Parameterized Algorithms and Computational Experiments challenge (PACE) reports on the first iteration of the challenge. Where did PACE come from, how did it go, who won, and what's next?

Holger Dell, Thore Husfeldt, Bart M. P. Jansen, Petteri Kaski, Christian Komusiewicz, and Frances A. Rosamond. The First Parameterized Algorithms and Computational Experiments Challenge. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 30:1-30:9, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2016.30, author = {Dell, Holger and Husfeldt, Thore and Jansen, Bart M. P. and Kaski, Petteri and Komusiewicz, Christian and Rosamond, Frances A.}, title = {{The First Parameterized Algorithms and Computational Experiments Challenge}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {30:1--30:9}, 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.2016.30}, URN = {urn:nbn:de:0030-drops-69310}, doi = {10.4230/LIPIcs.IPEC.2016.30}, annote = {Keywords: treewidth, feedback vertex set, contest, implementation challenge, FPT} }

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

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance.
We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases:
1. The exact optimum can be computed in FPT-time.
2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n).
3. There is no FPT-AS unless FPT=W[1].
For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.

Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Tobias Mömke. Complexity and Approximability of Parameterized MAX-CSPs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 294-306, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2015.294, author = {Dell, Holger and Kim, Eun Jung and Lampis, Michael and Mitsou, Valia and M\"{o}mke, Tobias}, title = {{Complexity and Approximability of Parameterized MAX-CSPs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {294--306}, 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.2015.294}, URN = {urn:nbn:de:0030-drops-55910}, doi = {10.4230/LIPIcs.IPEC.2015.294}, annote = {Keywords: Approximation, Structural Parameters, Constraint Satisfaction} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

Consider the following two-player communication process to decide a
language $L$: The first player holds the entire input $x$ but is
polynomially bounded; the second player is computationally unbounded
but does not know any part of $x$; their goal is to cooperatively
decide whether $x$ belongs to $L$ at small cost, where the cost
measure is the number of bits of communication from the first player
to the second player.
For any integer $d geq 3$ and positive real $epsilon$ we show that
if satisfiability for $n$-variable $d$-CNF formulas has a protocol of
cost $O(n^{d-epsilon})$ then coNP is in NP/poly, which implies that
the polynomial-time hierarchy collapses to its third level. The result
even holds when the first player is conondeterministic, and is tight as
there exists a trivial protocol for $epsilon = 0$. Under the
hypothesis that coNP is not in NP/poly, our result implies tight lower
bounds for parameters of interest in several areas, namely
sparsification, kernelization in parameterized complexity, lossy
compression, and probabilistically checkable proofs.
By reduction, similar results hold for other NP-complete problems.
For the vertex cover problem on $n$-vertex $d$-uniform hypergraphs,
the above statement holds for any integer $d geq 2$. The case $d=2$
implies that no NP-hard vertex deletion problem based on a graph
property that is inherited by subgraphs can have kernels consisting of
$O(k^{2-epsilon})$ edges unless coNP is in NP/poly, where $k$ denotes
the size of the deletion set. Kernels consisting of $O(k^2)$ edges are
known for several problems in the class, including vertex cover,
feedback vertex set, and bounded-degree deletion.

Holger Dell and Dieter van Melkebeek. Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, pp. 1-29, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dell_et_al:DagSemProc.09511.7, author = {Dell, Holger and van Melkebeek, Dieter}, title = {{Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses}}, booktitle = {Parameterized complexity and approximation algorithms}, pages = {1--29}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9511}, editor = {Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.7}, URN = {urn:nbn:de:0030-drops-25043}, doi = {10.4230/DagSemProc.09511.7}, annote = {Keywords: Sparsification, Kernelization, Parameterized Complexity, Probabilistically Checkable Proofs, Satisfiability, Vertex Cover} }

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