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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

For a fixed graph H, the H-Subgraph Hitting problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of H as a subgraph. This problem can be seen as a generalization of Vertex Cover, which corresponds to the case H = K₂. We initiate a study of H-Subgraph Hitting from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class 𝒞. Our main contribution is to identify graph parameters that, when H-Subgraph Hitting is parameterized by the vertex-deletion distance to a class 𝒞 where any of these parameters is bounded, and assuming standard complexity assumptions and that H is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if H is a clique. These new graph parameters are inspired by the notion of 𝒞-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit H as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Kernelization Dichotomies for Hitting Subgraphs Under Structural Parameterizations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bougeret_et_al:LIPIcs.ICALP.2024.33, author = {Bougeret, Marin and Jansen, Bart M. P. and Sau, Ignasi}, title = {{Kernelization Dichotomies for Hitting Subgraphs Under Structural Parameterizations}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.33}, URN = {urn:nbn:de:0030-drops-201766}, doi = {10.4230/LIPIcs.ICALP.2024.33}, annote = {Keywords: hitting subgraphs, hitting induced subgraphs, parameterized complexity, polynomial kernel, complexity dichotomy, elimination distance} }

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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

For a positive real γ ≥ 1, a γ-certified algorithm for a vertex-weighted graph optimization problem is an algorithm that, given a weighted graph (G,w), outputs a re-weighting of the graph obtained by scaling each weight individually with a factor between 1 and γ, along with a solution which is optimal for the perturbed weight function. Here we provide (1+ε)-certified algorithms for Dominating Set and H-Subgraph-Free-Deletion which, for any ε > 0, run in time f(1/ε)⋅n^𝒪(1) on minor-closed classes of graphs of bounded local tree-width with polynomially-bounded weights. We obtain our algorithms as corollaries of a more general result establishing FPT-time certified algorithms for problems admitting, at an intuitive level, certain "local solution-improvement properties". These results improve - in terms of generality, running time and parameter dependence - on Angelidakis, Awasthi, Blum, Chatziafratis and Dan’s XP-time (1+ε)-certified algorithm for Independent Set on planar graphs (ESA2019). Furthermore, our methods are also conceptually simpler: our algorithm is based on elementary local re-optimizations inspired by Baker’s technique, as opposed to the heavy machinery of the Sherali-Adams hierarchy required in previous work.

Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jaime Venne. Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bumpus_et_al:LIPIcs.SWAT.2024.19, author = {Bumpus, Benjamin Merlin and Jansen, Bart M. P. and Venne, Jaime}, title = {{Fixed-Parameter Tractable Certified Algorithms for Covering and Dominating in Planar Graphs and Beyond}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.19}, URN = {urn:nbn:de:0030-drops-200595}, doi = {10.4230/LIPIcs.SWAT.2024.19}, annote = {Keywords: fixed-parameter tractability, certified algorithms} }

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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most c ⋅ opt contain v. Recent work showed that polynomial-time algorithms to detect c-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected n-vertex graph G in time 2^𝒪(𝓁³)⋅n^𝒪(1), where 𝓁 is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that (2-ε)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.

Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jansen_et_al:LIPIcs.SWAT.2024.28, author = {Jansen, Bart M. P. and Verhaegh, Ruben F. A.}, title = {{Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.28}, URN = {urn:nbn:de:0030-drops-200683}, doi = {10.4230/LIPIcs.SWAT.2024.28}, annote = {Keywords: fixed-parameter tractability, essential vertices, integrality gap} }

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

A kernelization for a parameterized decision problem 𝒬 is a polynomial-time preprocessing algorithm that reduces any parameterized instance (x,k) into an instance (x',k') whose size is bounded by a function of k alone and which has the same yes/no answer for 𝒬. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to k. However, we show that for counting minimum feedback vertex sets of size at most k, and for counting minimum dominating sets of size at most k in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance (G',k') of size polynomial in k with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds 2^{poly(k)}, the size of the input is exponential in terms of k so that the running time of a parameterized counting algorithm can be bounded by poly(n). Otherwise, we can use gadgets that slightly increase k to represent choices among 2^{𝒪(k)} options by only poly(k) vertices.

Bart M. P. Jansen and Bart van der Steenhoven. Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2023.27, author = {Jansen, Bart M. P. and van der Steenhoven, Bart}, title = {{Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {27:1--27:15}, 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.2023.27}, URN = {urn:nbn:de:0030-drops-194466}, doi = {10.4230/LIPIcs.IPEC.2023.27}, annote = {Keywords: kernelization, counting problems, feedback vertex set, dominating set, protrusion decomposition} }

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

We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph G, integer k, and terminal set T ⊆ V(G), it asks whether there is a vertex set S ⊆ V(G) ⧵ T of size at most k such that in graph G-S, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in G-S by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of G-S. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time 2^{𝒪(k log k)} ⋅ n^{𝒪(1)}. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size k plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph G and terminal set T ⊆ V(G) along with a single vertex x ∈ V(G) that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing x.

Bart M. P. Jansen and Shivesh K. Roy. On the Parameterized Complexity of Multiway Near-Separator. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2023.28, author = {Jansen, Bart M. P. and Roy, Shivesh K.}, title = {{On the Parameterized Complexity of Multiway Near-Separator}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {28:1--28:18}, 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.2023.28}, URN = {urn:nbn:de:0030-drops-194470}, doi = {10.4230/LIPIcs.IPEC.2023.28}, annote = {Keywords: fixed-parameter tractability, multiway cut, near-separator} }

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

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w : V(G) → ℕ, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H₁, …, H_k of G such that H_i is isomorphic to H for each i ∈ [k] and the total weight of these k ⋅ |V(H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with 𝒪(k^{|V(H)|-1}) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted K_h-Packing on graphs of bounded expansion, along with a kernel with 𝒪(k^{1+ε}) vertices on nowhere-dense graphs for all ε > 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdős-Rado Sunflower lemma and the theory of sparsity.

Bart M. P. Jansen and Shivesh K. Roy. Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2023.29, author = {Jansen, Bart M. P. and Roy, Shivesh K.}, title = {{Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {29:1--29:13}, 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.2023.29}, URN = {urn:nbn:de:0030-drops-194488}, doi = {10.4230/LIPIcs.IPEC.2023.29}, annote = {Keywords: kernelization, weighted problems, graph packing, sunflower lemma, bounded expansion, nowhere dense} }

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

The celebrated notion of important separators bounds the number of small (S,T)-separators in a graph which are "farthest from S" in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k.
In this terminology, the bound on important separators says that there are at most 4^k maximal k-secluded connected vertex sets C containing S but disjoint from T. We generalize this statement significantly: even when we demand that G[C] avoids a finite set ℱ of forbidden induced subgraphs, the number of such maximal subgraphs is 2^𝒪(k) and they can be enumerated efficiently. This enumeration algorithm allows us to make significant improvements for two problems from the literature.
Our first application concerns the Connected k-Secluded ℱ-free subgraph problem, where ℱ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected k-secluded vertex set C ⊆ V(G) such that G[C] does not contain an induced subgraph isomorphic to any F ∈ ℱ. The parameterization by k is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential.
Our second application concerns the deletion problem to scattered graph classes. A scattered graph class is defined by demanding that every connected component is contained in at least one of the prescribed graph classes Π_1, …, Π_d. The deletion problem to a scattered graph class is to find a vertex set of size at most k whose removal yields a graph from the class. We obtain a single-exponential algorithm whenever each class Π_i is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jansen_et_al:LIPIcs.ISAAC.2023.42, author = {Jansen, Bart M. P. and de Kroon, Jari J. H. and W{\l}odarczyk, Micha{\l}}, title = {{Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {42:1--42:18}, 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.2023.42}, URN = {urn:nbn:de:0030-drops-193440}, doi = {10.4230/LIPIcs.ISAAC.2023.42}, annote = {Keywords: fixed-parameter tractability, important separators, secluded subgraphs} }

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

The notion of ℋ-treewidth, where ℋ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ℋ-treewidth at most k can be decomposed into (arbitrarily large) ℋ-subgraphs which interact only through vertex sets of size 𝒪(k) which can be organized in a tree-like fashion. ℋ-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ℋ-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ℋ. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ℋ-decompositions.
We present FPT-approximation algorithms to compute tree ℋ-decompositions for hereditary and union-closed graph classes ℋ. Given a graph of ℋ-treewidth k, we can compute a 5-approximate tree ℋ-decomposition in time f(𝒪(k)) ⋅ n^𝒪(1) whenever ℋ-deletion parameterized by solution size can be solved in time f(k) ⋅ n^𝒪(1) for some function f(k) ≥ 2^k. The current-best algorithms either achieve an approximation factor of k^𝒪(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^𝒪(k) ⋅ n^𝒪(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^𝒪(k log k) ⋅ n^𝒪(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.

Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. 5-Approximation for ℋ-Treewidth Essentially as Fast as ℋ-Deletion Parameterized by Solution Size. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 66:1-66:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2023.66, author = {Jansen, Bart M. P. and de Kroon, Jari J. H. and W{\l}odarczyk, Micha{\l}}, title = {{5-Approximation for ℋ-Treewidth Essentially as Fast as ℋ-Deletion Parameterized by Solution Size}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {66:1--66:16}, 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.2023.66}, URN = {urn:nbn:de:0030-drops-187195}, doi = {10.4230/LIPIcs.ESA.2023.66}, annote = {Keywords: fixed-parameter tractability, treewidth, graph decompositions} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.

Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-Space Reduction via Essential Vertices. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bumpus_et_al:LIPIcs.ESA.2022.30, author = {Bumpus, Benjamin Merlin and Jansen, Bart M. P. and de Kroon, Jari J. H.}, title = {{Search-Space Reduction via Essential Vertices}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.30}, URN = {urn:nbn:de:0030-drops-169687}, doi = {10.4230/LIPIcs.ESA.2022.30}, annote = {Keywords: fixed-parameter tractability, essential vertices, covering versus packing} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

In the ℱ-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G-S contains no graph from the finite family ℱ as a minor. It is known that whenever ℱ contains at least one planar graph, then ℱ-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size k^{𝒪(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size.
We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of ℱ-Minor-Free Deletion for the family ℱ = {K₄, K_{2,3}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with 𝒪(k⁴) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has 𝒪(k⁴) vertices and edges.

Huib Donkers, Bart M. P. Jansen, and Michał Włodarczyk. Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{donkers_et_al:LIPIcs.IPEC.2021.14, author = {Donkers, Huib and Jansen, Bart M. P. and W{\l}odarczyk, Micha{\l}}, title = {{Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {14:1--14:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.14}, URN = {urn:nbn:de:0030-drops-153979}, doi = {10.4230/LIPIcs.IPEC.2021.14}, annote = {Keywords: fixed-parameter tractability, kernelization, outerplanar graphs} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices n, is significantly harder than the unweighted problem by presenting an 𝒪(n^{3 - ε}) lower bound on the size of the kernel, under the assumption that NP ̸ ⊆ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by 2^𝒪(n), which yields a randomized kernel of 𝒪(n³) bits.
We generalize these results to the weighted d-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to {preserve the collection of} optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range [1, n], but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as 2^Ω(n) are necessary.

Bart M. P. Jansen, Shivesh K. Roy, and Michał Włodarczyk. On the Hardness of Compressing Weights. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 64:1-64:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jansen_et_al:LIPIcs.MFCS.2021.64, author = {Jansen, Bart M. P. and Roy, Shivesh K. and W{\l}odarczyk, Micha{\l}}, title = {{On the Hardness of Compressing Weights}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {64:1--64:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.64}, URN = {urn:nbn:de:0030-drops-145049}, doi = {10.4230/LIPIcs.MFCS.2021.64}, annote = {Keywords: kernelization, compression, edge-weighted clique, constraint satisfaction problems} }

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

We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize 𝒪(n^(2-ε)) for some ε > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ⊆ coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs.

Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, and Paweł Rzążewski. Sparsification Lower Bounds for List H-Coloring. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2020.58, author = {Chen, Hubie and Jansen, Bart M. P. and Okrasa, Karolina and Pieterse, Astrid and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Sparsification Lower Bounds for List H-Coloring}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {58:1--58:17}, 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.2020.58}, URN = {urn:nbn:de:0030-drops-134027}, doi = {10.4230/LIPIcs.ISAAC.2020.58}, annote = {Keywords: List H-Coloring, Sparsification, Constraint Satisfaction Problem} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

In the Boolean maximum constraint satisfaction problem - Max CSP(Γ) - one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP(Γ) with respect to the optimal compression size. Namely, we prove that Max CSP(Γ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that:
1) An instance of Max CSP(Γ) can be compressed into an equivalent instance with 𝒪(n^d log n) bits in polynomial time,
2) Max CSP(Γ) does not admit such a compression to 𝒪(n^{d-ε}) bits unless NP ⊆ co-NP / poly.
Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of constraint implementations. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP(Γ). More precisely, we show that obtaining a running time of the form 𝒪(2^{(1-ε)n}) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d.

Bart M. P. Jansen and Michał Włodarczyk. Optimal Polynomial-Time Compression for Boolean Max CSP. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 63:1-63:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2020.63, author = {Jansen, Bart M. P. and W{\l}odarczyk, Micha{\l}}, title = {{Optimal Polynomial-Time Compression for Boolean Max CSP}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {63:1--63:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.63}, URN = {urn:nbn:de:0030-drops-129297}, doi = {10.4230/LIPIcs.ESA.2020.63}, annote = {Keywords: constraint satisfaction problem, kernelization, exponential time algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ℱ, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ℱ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ℱ, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ℱ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ℱ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bougeret_et_al:LIPIcs.ICALP.2020.16, author = {Bougeret, Marin and Jansen, Bart M. P. and Sau, Ignasi}, title = {{Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {16:1--16:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.16}, URN = {urn:nbn:de:0030-drops-124238}, doi = {10.4230/LIPIcs.ICALP.2020.16}, annote = {Keywords: vertex cover, parameterized complexity, polynomial kernel, structural parameterization, bridge-depth} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We consider the Π-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties Π. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from Π as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property Π by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for Π-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. [JCSS 2014]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over 𝔽₂ implies graph-theoretic statements about the presence of forbidden induced subgraphs.

Bart M. P. Jansen and Jari J. H. de Kroon. Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jansen_et_al:LIPIcs.SWAT.2020.27, author = {Jansen, Bart M. P. and de Kroon, Jari J. H.}, title = {{Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.27}, URN = {urn:nbn:de:0030-drops-122748}, doi = {10.4230/LIPIcs.SWAT.2020.27}, annote = {Keywords: kernelization, vertex-deletion, graph modification, structural parameterization} }

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Complete Volume

**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

LIPIcs, Volume 148, IPEC'19, Complete Volume

14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{jansen_et_al:LIPIcs.IPEC.2019, title = {{LIPIcs, Volume 148, IPEC'19, Complete Volume}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019}, URN = {urn:nbn:de:0030-drops-116409}, doi = {10.4230/LIPIcs.IPEC.2019}, annote = {Keywords: Theory of computation, Parameterized complexity and exact algorithms} }

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Front Matter

**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Front Matter, Table of Contents, Preface, Conference Organization

14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2019.0, author = {Jansen, Bart M. P. and Telle, Jan Arne}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.0}, URN = {urn:nbn:de:0030-drops-114618}, doi = {10.4230/LIPIcs.IPEC.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C' by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs.

Édouard Bonnet, Yoichi Iwata, Bart M. P. Jansen, and Łukasz Kowalik. Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonnet_et_al:LIPIcs.ESA.2019.23, author = {Bonnet, \'{E}douard and Iwata, Yoichi and Jansen, Bart M. P. and Kowalik, {\L}ukasz}, title = {{Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.23}, URN = {urn:nbn:de:0030-drops-111444}, doi = {10.4230/LIPIcs.ESA.2019.23}, annote = {Keywords: traveling salesman problem, k-OPT, bounded degree} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Steiner trees after adding all edges between pairs of vertices that lie on a common face. We also show connections between Vertex Multiway Cut and the Vertex Planarization problem, where the existence of a polynomial kernel remains an important open problem.

Bart M. P. Jansen, Marcin Pilipczuk, and Erik Jan van Leeuwen. A Deterministic Polynomial Kernel for Odd Cycle Transversal and Vertex Multiway Cut in Planar Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jansen_et_al:LIPIcs.STACS.2019.39, author = {Jansen, Bart M. P. and Pilipczuk, Marcin and van Leeuwen, Erik Jan}, title = {{A Deterministic Polynomial Kernel for Odd Cycle Transversal and Vertex Multiway Cut in Planar Graphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {39:1--39:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.39}, URN = {urn:nbn:de:0030-drops-102783}, doi = {10.4230/LIPIcs.STACS.2019.39}, annote = {Keywords: planar graphs, kernelization, odd cycle transversal, multiway cut} }

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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

Many combinatorial problems can be solved in time O^*(c^{tw}) on graphs of treewidth tw, for a problem-specific constant c. In several cases, matching upper and lower bounds on c are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in O^*((2-epsilon)^{ctw}) time, and Dominating Set cannot be solved in O^*((3-epsilon)^{ctw}) time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving Independent Set or Dominating Set on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.

Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{vangeffen_et_al:LIPIcs.IPEC.2018.3, author = {van Geffen, Bas A. M. and Jansen, Bart M. P. and de Kroon, Arnoud A. W. M. and Morel, Rolf}, title = {{Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.3}, URN = {urn:nbn:de:0030-drops-102049}, doi = {10.4230/LIPIcs.IPEC.2018.3}, annote = {Keywords: planarization, dominating set, cutwidth, lower bounds, strong exponential time hypothesis} }

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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with O(n) constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language.

Hubie Chen, Bart M. P. Jansen, and Astrid Pieterse. Best-Case and Worst-Case Sparsifiability of Boolean CSPs. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.IPEC.2018.15, author = {Chen, Hubie and Jansen, Bart M. P. and Pieterse, Astrid}, title = {{Best-Case and Worst-Case Sparsifiability of Boolean CSPs}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.15}, URN = {urn:nbn:de:0030-drops-102169}, doi = {10.4230/LIPIcs.IPEC.2018.15}, annote = {Keywords: constraint satisfaction problems, kernelization, sparsification, lower bounds} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Computing the smallest number q such that the vertices of a given graph can be properly q-colored is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth {tw} in time O^*(q^{tw}), while a running time of O^*((q-epsilon)^{tw}) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators.
We present two algorithms for q-Coloring parameterized by cutwidth {ctw}: a deterministic one that runs in time O^*(2^{omega * {ctw}}), where omega is the matrix multiplication constant, and a randomized one with runtime O^*(2^{{ctw}}). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O^*((2-epsilon)^{{ctw}}) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O^*((floor[d/2]+1)^{{pw}}) time randomized algorithm for q-Coloring on graphs of pathwidth {pw} and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O^*((floor[d/2]+1-epsilon)^{{pw}})-time algorithm exists assuming SETH.

Bart M. P. Jansen and Jesper Nederlof. Computing the Chromatic Number Using Graph Decompositions via Matrix Rank. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2018.47, author = {Jansen, Bart M. P. and Nederlof, Jesper}, title = {{Computing the Chromatic Number Using Graph Decompositions via Matrix Rank}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.47}, URN = {urn:nbn:de:0030-drops-95104}, doi = {10.4230/LIPIcs.ESA.2018.47}, annote = {Keywords: Parameterized Complexity, Chromatic Number, Graph Decompositions} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size k^{O(1)}. In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth eta.
We prove that for each set F of connected graphs and constant eta, the F-Deletion problem parameterized by the size of a treedepth-eta modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G-X. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal F-Deletion solution from a single connected component of G-X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for F-Deletion parameterized by a vertex cover, which is a treedepth-one modulator.

Bart M. P. Jansen and Astrid Pieterse. Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jansen_et_al:LIPIcs.ESA.2018.48, author = {Jansen, Bart M. P. and Pieterse, Astrid}, title = {{Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {48:1--48:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.48}, URN = {urn:nbn:de:0030-drops-95119}, doi = {10.4230/LIPIcs.ESA.2018.48}, annote = {Keywords: Kernelization, F-minor free deletion, Treedepth modulator, Structural parameterization} }

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

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring.
First, we use a recent technique of finding redundant constraints by representing them as low-degree polynomials, to obtain a kernel of bitsize O(k^(q-1) log k) for q-Coloring parameterized by Vertex Cover for any q >= 3. This size bound is optimal up to k^o(1) factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^(2-e)) for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors.

Bart M. P. Jansen and Astrid Pieterse. Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 22:1-22:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2017.22, author = {Jansen, Bart M. P. and Pieterse, Astrid}, title = {{Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {22:1--22:12}, 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.2017.22}, URN = {urn:nbn:de:0030-drops-85500}, doi = {10.4230/LIPIcs.IPEC.2017.22}, annote = {Keywords: graph coloring, kernelization, sparsification} }

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

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-PATH admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{O(1)}?
We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-PATH even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path has a separation that can safely be reduced after communication with the oracle.

Bart M. P. Jansen, Marcin Pilipczuk, and Marcin Wrochna. Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2017.23, author = {Jansen, Bart M. P. and Pilipczuk, Marcin and Wrochna, Marcin}, title = {{Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {23:1--23:13}, 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.2017.23}, URN = {urn:nbn:de:0030-drops-85576}, doi = {10.4230/LIPIcs.IPEC.2017.23}, annote = {Keywords: Turing kernel, long path, k-path, excluded topological minor, modulator} }

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

Garnero et al. [SIAM J. Discrete Math. 2015, 29(4):1864-1894] recently introduced a framework based on dynamic programming to make applications of the protrusion replacement technique constructive and to obtain explicit upper bounds on the involved constants. They show that for several graph problems, for every boundary size t one can find an explicit set R_t of representatives. Any subgraph H with a boundary of size t can be replaced with a representative H' in R_t such that the effect of this replacement on the optimum can be deduced from H and H' alone. Their upper bounds on the size of the graphs in R_t grow triple-exponentially with t. In this paper we complement their results by lower bounds on the sizes of representatives, in terms of the boundary size t. For example, we show that each set of planar representatives R_t for the Independent Set problem contains a graph with Omega(2^t / sqrt{4t}) vertices. This lower bound even holds for sets that only represent the planar subgraphs of bounded pathwidth. To obtain our results we provide a lower bound on the number of equivalence classes of the canonical equivalence relation for Independent Set on t-boundaried graphs. We also find an elegant characterization of the number of equivalence classes in general graphs, in terms of the number of monotone functions of a certain kind. Our results show that the number of equivalence classes is at most 2^{2^t}, improving on earlier bounds of the form (t+1)^{2^t}.

Bart M. P. Jansen and Jules J. H. M. Wulms. Lower Bounds for Protrusion Replacement by Counting Equivalence Classes. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2016.17, author = {Jansen, Bart M. P. and Wulms, Jules J. H. M.}, title = {{Lower Bounds for Protrusion Replacement by Counting Equivalence Classes}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {17:1--17:12}, 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.2016.17}, URN = {urn:nbn:de:0030-drops-69275}, doi = {10.4230/LIPIcs.IPEC.2016.17}, annote = {Keywords: protrusions, boundaried graphs, independent set, equivalence classes, finite integer index} }

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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 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly k vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most k fewer vertices than the initial solution. We are interested in determining the minimum value of k for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large.

Mark de Berg, Bart M. P. Jansen, and Debankur Mukherjee. Independent-Set Reconfiguration Thresholds of Hereditary Graph Classes. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deberg_et_al:LIPIcs.FSTTCS.2016.34, author = {de Berg, Mark and Jansen, Bart M. P. and Mukherjee, Debankur}, title = {{Independent-Set Reconfiguration Thresholds of Hereditary Graph Classes}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.34}, URN = {urn:nbn:de:0030-drops-68694}, doi = {10.4230/LIPIcs.FSTTCS.2016.34}, annote = {Keywords: Reconfiguration, Independent set, Token Addition Removal, Token Sliding} }

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

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity.
Our first set of results is motivated by the Bitonic tsp problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamicprogramming exercise to solve this problem in O(n^2) time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Fréchet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic tsp in O(n*log^2(n)) time and its bottleneck version in O(n*log^3(n)) time. In the more general pyramidal tsp problem, the points to be visited are labeled 1, ..., n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) tsp problem also work for the pyramidal tsp problem in the plane.
Our second set of results concerns the popular k-opt heuristic for tsp in the graph setting. More precisely, we study the k-opt decision problem, which asks whether a given tour can be improved by a k-opt move that replaces k edges in the tour by k new edges. A simple algorithm solves k-opt in O(n^k) time for fixed k. For 2-opt, this is easily seen to be optimal. For k = 3 we prove that an algorithm with a runtime of the form ~O(n^{3-epsilon}) exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. For general k-opt, it is known that a runtime of f(k)*n^{o(k/log(k))} would contradict the Exponential Time Hypothesis. The results for k = 2, 3 may suggest that the actual time complexity of k-opt is Theta(n^k). We show that this is not the case, by presenting an algorithm that finds the best k-move in O(n^{lfoor 2k/3 rfloor +1}) time for fixed k >= 3. This implies that 4-opt can be solved in O(n^3) time, matching the best-known algorithm for 3-opt. Finally, we show how to beat the quadratic barrier for k = 2 in two important settings, namely for points in the plane and when we want to solve 2-opt repeatedly

Mark de Berg, Kevin Buchin, Bart M. P. Jansen, and Gerhard Woeginger. Fine-Grained Complexity Analysis of Two Classic TSP Variants. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deberg_et_al:LIPIcs.ICALP.2016.5, author = {de Berg, Mark and Buchin, Kevin and Jansen, Bart M. P. and Woeginger, Gerhard}, title = {{Fine-Grained Complexity Analysis of Two Classic TSP Variants}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {5:1--5:14}, 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.2016.5}, URN = {urn:nbn:de:0030-drops-62770}, doi = {10.4230/LIPIcs.ICALP.2016.5}, annote = {Keywords: Traveling salesman problem, fine-grained complexity, bitonic tours, k-opt} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with O(n^{d-epsilon}) bits for any epsilon > 0. For the Not-All-Equal SAT problem, a compression to size tilde-O(n^{d-1}) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n^{2-epsilon}) bits for any epsilon > 0, unless NP is contained in coNP/poly.

Bart M. P. Jansen and Astrid Pieterse. Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jansen_et_al:LIPIcs.MFCS.2016.71, author = {Jansen, Bart M. P. and Pieterse, Astrid}, title = {{Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.71}, URN = {urn:nbn:de:0030-drops-64821}, doi = {10.4230/LIPIcs.MFCS.2016.71}, annote = {Keywords: constraint satisfaction problem, sparsification, satisfiability, kernelization} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

The CONSTRAINED BIPARTITE VERTEX COVER problem asks, for a bipartite graph G with partite sets A and B, and integers k_A and k_B, whether there is a vertex cover for G containing at most k_A vertices from A and k_B vertices from B. The problem has an easy kernel with 2 * k_A * k_B edges and 4 k_A * k_B vertices, based on the fact that every vertex in A of degree more than k_B has to be included in the solution, together with every vertex in B of degree more than k_A. We show that the number of vertices and edges in this kernel are asymptotically essentially optimal in terms of the product k_A * k_B. We prove that if there is a polynomial-time algorithm that reduces any instance (G,A,B,k_A,k_B) of CONSTRAINED BIPARTITE VERTEX COVER to an equivalent instance (G',A',B',k'_A,k'_B) such that k'_A in (k_A)^{O(1)}, k'_B in (k_B)^{O(1)}, and |V(G')| in O((k_A * k_B)^{1 - epsilon}), for some epsilon > 0, then NP subseteq coNP/poly and the polynomial-time hierarchy collapses. Using a different construction, we prove that if there is a polynomial-time algorithm that reduces any n-vertex instance into an equivalent instance (of a possibly different problem) that can be encoded in O(n^{2- epsilon}) bits, then NP subseteq coNP/poly.

Bart M. P. Jansen. Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jansen:LIPIcs.STACS.2016.45, author = {Jansen, Bart M. P.}, title = {{Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.45}, URN = {urn:nbn:de:0030-drops-57463}, doi = {10.4230/LIPIcs.STACS.2016.45}, annote = {Keywords: kernel lower bounds, constrained bipartite vertex cover} }

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

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.

Bart M. P. Jansen and Astrid Pieterse. Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2015.163, author = {Jansen, Bart M. P. and Pieterse, Astrid}, title = {{Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {163--174}, 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.2015.163}, URN = {urn:nbn:de:0030-drops-55806}, doi = {10.4230/LIPIcs.IPEC.2015.163}, annote = {Keywords: sparsification, graph coloring, Hamiltonian cycle, satisfiability} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem $Q$ if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we show that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless the polynomial hierarchy collapses.
Our technique generalizes and strengthens the recent techniques of using OR-composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.

Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Cross-Composition: A New Technique for Kernelization Lower Bounds. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 165-176, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bodlaender_et_al:LIPIcs.STACS.2011.165, author = {Bodlaender, Hans L. and Jansen, Bart M. P. and Kratsch, Stefan}, title = {{Cross-Composition: A New Technique for Kernelization Lower Bounds}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {165--176}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.165}, URN = {urn:nbn:de:0030-drops-30082}, doi = {10.4230/LIPIcs.STACS.2011.165}, annote = {Keywords: kernelization, lower bounds, parameterized complexity} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. Intensive research into the Vertex Cover problem has shown that there is a preprocessing algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G',k') in polynomial time with the guarantee that G' has at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number VC(G) since fvs(G) <= VC(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G',X',k') such that k' <= k, |X'| <= |X| and most importantly |V(G')| <= 2k and |V(G')| in O(|X'|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have polynomial kernel when parameterized by fvs(G) unless the polynomial hierarchy collapses to the third level (PH=Sigma_3^p). Our work is one of the first examples of research in kernelization using a non-standard parameter, and shows that this approach can yield interesting computational insights. To obtain our results we make extensive use of the combinatorial structure of independent sets in forests.

Bart M. P. Jansen and Hans L. Bodlaender. Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 177-188, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{jansen_et_al:LIPIcs.STACS.2011.177, author = {Jansen, Bart M. P. and Bodlaender, Hans L.}, title = {{Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {177--188}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.177}, URN = {urn:nbn:de:0030-drops-30097}, doi = {10.4230/LIPIcs.STACS.2011.177}, annote = {Keywords: kernelization, lower bounds, parameterized complexity} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Computing the Dodgson Score of a candidate in an election is a hard computational problem, which has been analyzed using classical and parameterized analysis. In this paper we resolve two open problems regarding the parameterized complexity of DODGSON SCORE. We show that DODGSON SCORE parameterized by the target score value $k$ does not have a polynomial kernel unless the polynomial hierarchy collapses to the third level; this complements a result of Fellows, Rosamond and Slinko who obtain a non-trivial kernel of exponential size for a generalization of this problem. We also prove that DODGSON SCORE parameterized by the number $n$ of votes is hard for $W[1]$.

Michael Fellows, Bart M. P. Jansen, Daniel Lokshtanov, Frances A. Rosamond, and Saket Saurabh. Determining the Winner of a Dodgson Election is Hard. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 459-468, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{fellows_et_al:LIPIcs.FSTTCS.2010.459, author = {Fellows, Michael and Jansen, Bart M. P. and Lokshtanov, Daniel and Rosamond, Frances A. and Saurabh, Saket}, title = {{Determining the Winner of a Dodgson Election is Hard}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {459--468}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.459}, URN = {urn:nbn:de:0030-drops-28866}, doi = {10.4230/LIPIcs.FSTTCS.2010.459}, annote = {Keywords: Dodgson Score, Parameterized Complexity, Kernelization Lower Bounds} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7462, Assisted Living Systems - Models, Architectures and Engineering Approaches (2008)

This position paper argues in favor of three approaches for advancing the field of monitoring physical activity of elderly patients.

Bart Jansen. Position statement: Physical activity monitoring of elderly patients - 3 tricks to advance the field?. In Assisted Living Systems - Models, Architectures and Engineering Approaches. Dagstuhl Seminar Proceedings, Volume 7462, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{jansen:DagSemProc.07462.18, author = {Jansen, Bart}, title = {{Position statement: Physical activity monitoring of elderly patients - 3 tricks to advance the field?}}, booktitle = {Assisted Living Systems - Models, Architectures and Engineering Approaches}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7462}, editor = {Arthur I. Karshmer and J\"{u}rgen Nehmer and Hartmut Raffler and Gerhard Tr\"{o}ster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07462.18}, URN = {urn:nbn:de:0030-drops-14648}, doi = {10.4230/DagSemProc.07462.18}, annote = {Keywords: Elderly patients, physical activity, robot imitation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7462, Assisted Living Systems - Models, Architectures and Engineering Approaches (2008)

This position paper argues for using a 3D camera for the monitoring of physical activity of elderly patients.

Bart Jansen. Position statement: Telemonitoring - a too limited view on the wellbeing of the patient. In Assisted Living Systems - Models, Architectures and Engineering Approaches. Dagstuhl Seminar Proceedings, Volume 7462, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{jansen:DagSemProc.07462.19, author = {Jansen, Bart}, title = {{Position statement: Telemonitoring - a too limited view on the wellbeing of the patient}}, booktitle = {Assisted Living Systems - Models, Architectures and Engineering Approaches}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {7462}, editor = {Arthur I. Karshmer and J\"{u}rgen Nehmer and Hartmut Raffler and Gerhard Tr\"{o}ster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07462.19}, URN = {urn:nbn:de:0030-drops-14636}, doi = {10.4230/DagSemProc.07462.19}, annote = {Keywords: Telemonitoring, physical activity} }