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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

A framework consists of an undirected graph G and a matroid M whose elements correspond to the vertices of G. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank k in frameworks. More precisely, for vertices s and t of G, and an integer k, they gave FPT algorithms parameterized by k deciding whether there is an (s,t)-path in G whose vertex set contains a subset of elements of M of rank k. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open.
We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph G is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid M is represented over rationals.
Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Giannos Stamoulis. Computing Paths of Large Rank in Planar Frameworks Deterministically. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 32:1-32:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ISAAC.2023.32, author = {Fomin, Fedor V. and Golovach, Petr A. and Korhonen, Tuukka and Stamoulis, Giannos}, title = {{Computing Paths of Large Rank in Planar Frameworks Deterministically}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {32:1--32:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.32}, URN = {urn:nbn:de:0030-drops-193341}, doi = {10.4230/LIPIcs.ISAAC.2023.32}, annote = {Keywords: Planar graph, longest path, linear matroid, irrelevant vertex} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We prove the following result about approximating the maximum independent set in a graph. Informally, we show that any approximation algorithm with a "non-trivial" approximation ratio (as a function of the number of vertices of the input graph G) can be turned into an approximation algorithm achieving almost the same ratio, albeit as a function of the treewidth of G. More formally, we prove that for any function f, the existence of a polynomial time (n/f(n))-approximation algorithm yields the existence of a polynomial time O(tw⋅log{f(tw)}/f(tw))-approximation algorithm, where n and tw denote the number of vertices and the width of a given tree decomposition of the input graph. By pipelining our result with the state-of-the-art O(n ⋅ (log log n)²/log³n)-approximation algorithm by Feige (2004), this implies an O(tw⋅(log log tw)³/log³tw)-approximation algorithm.

Parinya Chalermsook, Fedor Fomin, Thekla Hamm, Tuukka Korhonen, Jesper Nederlof, and Ly Orgo. Polynomial-Time Approximation of Independent Set Parameterized by Treewidth. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 33:1-33:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chalermsook_et_al:LIPIcs.ESA.2023.33, author = {Chalermsook, Parinya and Fomin, Fedor and Hamm, Thekla and Korhonen, Tuukka and Nederlof, Jesper and Orgo, Ly}, title = {{Polynomial-Time Approximation of Independent Set Parameterized by Treewidth}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {33:1--33:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.33}, URN = {urn:nbn:de:0030-drops-186865}, doi = {10.4230/LIPIcs.ESA.2023.33}, annote = {Keywords: Maximum Independent Set, Treewidth, Approximation Algorithms, Parameterized Approximation} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

We show that there is no 2^o(k²) n^O(1) time algorithm for Independent Set on n-vertex graphs with rank-width k, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the 2^O(k²) n^O(1) time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known 2^O(k²) n^O(1) time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width k are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for n-vertex graphs.

Benjamin Bergougnoux, Tuukka Korhonen, and Jesper Nederlof. Tight Lower Bounds for Problems Parameterized by Rank-Width. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 11:1-11:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.STACS.2023.11, author = {Bergougnoux, Benjamin and Korhonen, Tuukka and Nederlof, Jesper}, title = {{Tight Lower Bounds for Problems Parameterized by Rank-Width}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.11}, URN = {urn:nbn:de:0030-drops-176636}, doi = {10.4230/LIPIcs.STACS.2023.11}, annote = {Keywords: rank-width, exponential time hypothesis, Boolean-width, parameterized algorithms, independent set, dominating set, maximum induced matching, feedback vertex set} }

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Short Paper

**Published in:** LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)

Propositional model counting (#SAT), the problem of determining the number of satisfying assignments of a propositional formula, is the archetypical #P-complete problem with a wide range of applications in AI. In this paper, we show that integrating tree decompositions of low width into the decision heuristics of a reference exact model counter (SharpSAT) significantly improves its runtime performance. In particular, our modifications to SharpSAT (and its derivant GANAK) lift the runtime efficiency of SharpSAT to the extent that it outperforms state-of-the-art exact model counters, including earlier-developed model counters that exploit tree decompositions.

Tuukka Korhonen and Matti Järvisalo. Integrating Tree Decompositions into Decision Heuristics of Propositional Model Counters (Short Paper). In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 8:1-8:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{korhonen_et_al:LIPIcs.CP.2021.8, author = {Korhonen, Tuukka and J\"{a}rvisalo, Matti}, title = {{Integrating Tree Decompositions into Decision Heuristics of Propositional Model Counters}}, booktitle = {27th International Conference on Principles and Practice of Constraint Programming (CP 2021)}, pages = {8:1--8:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-211-2}, ISSN = {1868-8969}, year = {2021}, volume = {210}, editor = {Michel, Laurent D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.8}, URN = {urn:nbn:de:0030-drops-152992}, doi = {10.4230/LIPIcs.CP.2021.8}, annote = {Keywords: propositional model counting, decision heuristics, tree decompositions, empirical evaluation} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2^{Ω(w/d)} gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2^{Ω(w)} gates. An MWIS-formula is an MWIS-circuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2^{Ω(t/d)} gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.

Tuukka Korhonen. Lower Bounds on Dynamic Programming for Maximum Weight Independent Set. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 87:1-87:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{korhonen:LIPIcs.ICALP.2021.87, author = {Korhonen, Tuukka}, title = {{Lower Bounds on Dynamic Programming for Maximum Weight Independent Set}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {87:1--87:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.87}, URN = {urn:nbn:de:0030-drops-141562}, doi = {10.4230/LIPIcs.ICALP.2021.87}, annote = {Keywords: Maximum weight independent set, Treewidth, Tropical circuits, Dynamic programming, Treedepth, Monotone circuit complexity} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Many graph problems can be formulated as a task of finding an optimal triangulation of a given graph with respect to some notion of optimality. In this paper we give algorithms to such problems parameterized by the size of a minimum edge clique cover (cc) of the graph. The parameter cc is both natural and well-motivated in many problems on this setting. For example, in the perfect phylogeny problem cc is at most the number of taxa, in fractional hypertreewidth cc is at most the number of hyperedges, and in treewidth of Bayesian networks cc is at most the number of non-root nodes of the Bayesian network.
Our results are based on the framework of potential maximal cliques. We show that the number of minimal separators of graphs is at most 2^cc and the number of potential maximal cliques is at most 3^cc. Furthermore, these objects can be listed in times O^*(2^cc) and O^*(3^cc), respectively, even when no edge clique cover is given as input; the O^*(⋅) notation omits factors polynomial in the input size. Using these enumeration algorithms we obtain O^*(3^cc) time algorithms for problems in the potential maximal clique framework, including for example treewidth, minimum fill-in, and feedback vertex set. We also obtain an O^*(3^m) time algorithm for fractional hypertreewidth, where m is the number of hyperedges. In the case when an edge clique cover of size cc' is given as an input we further improve the time complexity to O^*(2^cc') for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^cc') and O^*(9^(cc + O(log^2 cc))) for problems in this framework.

Tuukka Korhonen. Finding Optimal Triangulations Parameterized by Edge Clique Cover. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{korhonen:LIPIcs.IPEC.2020.22, author = {Korhonen, Tuukka}, title = {{Finding Optimal Triangulations Parameterized by Edge Clique Cover}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.22}, URN = {urn:nbn:de:0030-drops-133253}, doi = {10.4230/LIPIcs.IPEC.2020.22}, annote = {Keywords: Treewidth, Minimum fill-in, Perfect phylogeny, Fractional hypertreewidth, Potential maximal cliques, Edge clique cover} }

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PACE Solver Description

**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

We describe SMS, our submission to the exact treedepth track of PACE 2020. SMS computes the treedepth of a graph by branching on the Small Minimal Separators of the graph.

Tuukka Korhonen. PACE Solver Description: SMS. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 30:1-30:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{korhonen:LIPIcs.IPEC.2020.30, author = {Korhonen, Tuukka}, title = {{PACE Solver Description: SMS}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {30:1--30:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.30}, URN = {urn:nbn:de:0030-drops-133338}, doi = {10.4230/LIPIcs.IPEC.2020.30}, annote = {Keywords: Treedepth, PACE 2020, SMS, Minimal separators} }

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