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

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015).
In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{βtail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{βtail} L-hard and can be written as the difference of two functions in #para_{βtail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{βtail} L under parameterised logspace parsimonious reductions coincides with #para_β L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism.
Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions.
Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes.
Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

Anselm Haak, Arne Meier, Om Prakash, and Raghavendra Rao B. V.. Parameterised Counting in Logspace. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haak_et_al:LIPIcs.STACS.2021.40, author = {Haak, Anselm and Meier, Arne and Prakash, Om and Rao B. V., Raghavendra}, title = {{Parameterised Counting in Logspace}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {40:1--40:17}, 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.2021.40}, URN = {urn:nbn:de:0030-drops-136859}, doi = {10.4230/LIPIcs.STACS.2021.40}, annote = {Keywords: Parameterized Complexity, Counting Complexity, Logspace} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We study descriptive complexity of counting complexity classes in the range from #P to #*NP. A corollary of Fagin’s characterization of NP by existential second-order logic is that #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski’s semantics. Our results show that the class #*NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #*NP and #P, respectively. We also study the function class generated by inclusion logic and relate it to the complexity class TotP, which is a subclass of #P. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Sigma_1-formulae is #*NP-complete with respect to Turing reductions as well as complete for the function class generated by dependence logic with respect to first-order reductions.

Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, and Fan Yang. Counting of Teams in First-Order Team Logics. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haak_et_al:LIPIcs.MFCS.2019.19, author = {Haak, Anselm and Kontinen, Juha and M\"{u}ller, Fabian and Vollmer, Heribert and Yang, Fan}, title = {{Counting of Teams in First-Order Team Logics}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.19}, URN = {urn:nbn:de:0030-drops-109634}, doi = {10.4230/LIPIcs.MFCS.2019.19}, annote = {Keywords: team-based logics, counting classes, finite model theory, descriptive complexity} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.

Arnaud Durand, Anselm Haak, Juha Kontinen, and Heribert Vollmer. Descriptive Complexity of #AC^0 Functions. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{durand_et_al:LIPIcs.CSL.2016.20, author = {Durand, Arnaud and Haak, Anselm and Kontinen, Juha and Vollmer, Heribert}, title = {{Descriptive Complexity of #AC^0 Functions}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {20:1--20:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.20}, URN = {urn:nbn:de:0030-drops-65601}, doi = {10.4230/LIPIcs.CSL.2016.20}, annote = {Keywords: finite model theory, Fagin's theorem, arithmetic circuits, counting classes, Skolem function} }

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