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Documents authored by Włodarczyk, Michał

Document
DOI: 10.4230/LIPIcs.IPEC.2023.31
Sidestepping Barriers for Dominating Set in Parameterized Complexity

Authors: Ioannis Koutis, Michał Włodarczyk, and Meirav Zehavi

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

Abstract
We study the classic Dominating Set problem with respect to several prominent parameters. Specifically, we present algorithmic results that sidestep time complexity barriers by the incorporation of either approximation or larger parameterization. Our results span several parameterization regimes, including: (i,ii,iii) time/ratio-tradeoff for the parameters treewidth, vertex modulator to constant treewidth and solution size; (iv,v) FPT-algorithms for the parameters vertex cover number and feedback edge set number; and (vi) compression for the parameter feedback edge set number.
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Ioannis Koutis, Michał Włodarczyk, and Meirav Zehavi. Sidestepping Barriers for Dominating Set in Parameterized Complexity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{koutis_et_al:LIPIcs.IPEC.2023.31,
  author =	{Koutis, Ioannis and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav},
  title =	{{Sidestepping Barriers for Dominating Set in Parameterized Complexity}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{31:1--31:17},
  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.2023.31},
  URN =		{urn:nbn:de:0030-drops-194506},
  doi =		{10.4230/LIPIcs.IPEC.2023.31},
  annote =	{Keywords: Dominating Set, Parameterized Complexity, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.42
Single-Exponential FPT Algorithms for Enumerating Secluded ℱ-Free Subgraphs and Deleting to Scattered Graph Classes

Authors: Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk

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

Abstract
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.
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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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.65
Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large

Authors: Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi

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

Abstract
We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP.
Cite as

Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi. Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jacob_et_al:LIPIcs.ESA.2023.65,
  author =	{Jacob, Ashwin and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav},
  title =	{{Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{65:1--65:17},
  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.2023.65},
  URN =		{urn:nbn:de:0030-drops-187184},
  doi =		{10.4230/LIPIcs.ESA.2023.65},
  annote =	{Keywords: Hamiltonian cycle, longest path, directed feedback vertex set, directed graphs, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.66
5-Approximation for ℋ-Treewidth Essentially as Fast as ℋ-Deletion Parameterized by Solution Size

Authors: Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk

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

Abstract
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.
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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}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.106
Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth

Authors: Michał Włodarczyk

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
In Chordal/Interval Vertex Deletion we ask how many vertices one needs to remove from a graph to make it chordal (respectively: interval). We study these problems under the parameterization by treewidth tw of the input graph G. On the one hand, we present an algorithm for Chordal Vertex Deletion with running time 2^𝒪(tw)⋅|V(G)|, improving upon the running time 2^𝒪(tw²)⋅|V(G)|^𝒪(1) by Jansen, de Kroon, and Włodarczyk (STOC'21). When a tree decomposition of width tw is given, then the base of the exponent equals 2^{ω-1}⋅3 + 1. Our algorithm is based on a novel link between chordal graphs and graphic matroids, which allows us to employ the framework of representative families. On the other hand, we prove that the known 2^𝒪(tw log tw)⋅|V(G)|-time algorithm for Interval Vertex Deletion cannot be improved assuming Exponential Time Hypothesis.
Cite as

Michał Włodarczyk. Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 106:1-106:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{wlodarczyk:LIPIcs.ICALP.2023.106,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{106:1--106:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.106},
  URN =		{urn:nbn:de:0030-drops-181585},
  doi =		{10.4230/LIPIcs.ICALP.2023.106},
  annote =	{Keywords: fixed-parameter tractability, treewidth, chordal graphs, interval graphs, matroids, representative families}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.14
Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

Authors: Huib Donkers, Bart M. P. Jansen, and Michał Włodarczyk

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

Abstract
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.
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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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.64
On the Hardness of Compressing Weights

Authors: Bart M. P. Jansen, Shivesh K. Roy, and Michał Włodarczyk

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.63
Optimal Polynomial-Time Compression for Boolean Max CSP

Authors: Bart M. P. Jansen and Michał Włodarczyk

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

Abstract
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.
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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}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.104
Parameterized Inapproximability for Steiner Orientation by Gap Amplification

Authors: Michał Włodarczyk

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

Abstract
In the k-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of k terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than 𝒪(k) is known. We show that k-Steiner Orientation is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form (log k)^o(1) for FPT algorithms (assuming FPT ≠ W[1]) and (log n)^o(1) for purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). This constitutes a novel inapproximability result for polynomial-time algorithms obtained via tools from the FPT theory. Moreover, we prove k-Steiner Orientation to belong to W[1], which entails W[1]-completeness of (log k)^o(1)-approximation for k-Steiner Orientation. This provides an example of a natural approximation task that is complete in a parameterized complexity class. Finally, we apply our technique to the maximization version of directed multicut - Max (k,p)-Directed Multicut - where we are given a directed graph, k terminals pairs, and a budget p. The goal is to maximize the number of separated terminal pairs by removing p edges. We present a simple proof that the problem admits no FPT approximation with factor 𝒪(k^(1/2 - ε)) (assuming FPT ≠ W[1]) and no polynomial-time approximation with ratio 𝒪(|E(G)|^(1/2 - ε)) (assuming NP ⊈ co-RP).
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Michał Włodarczyk. Parameterized Inapproximability for Steiner Orientation by Gap Amplification. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 104:1-104:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{wlodarczyk:LIPIcs.ICALP.2020.104,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Parameterized Inapproximability for Steiner Orientation by Gap Amplification}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{104:1--104:19},
  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.2020.104},
  URN =		{urn:nbn:de:0030-drops-125110},
  doi =		{10.4230/LIPIcs.ICALP.2020.104},
  annote =	{Keywords: approximation algorithms, fixed-parameter tractability, hardness of approximation, gap amplification}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.1
Constant-Factor FPT Approximation for Capacitated k-Median

Authors: Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski, Syed M. Meesum, and Michał Włodarczyk

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

Abstract
Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon.
Cite as

Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski, Syed M. Meesum, and Michał Włodarczyk. Constant-Factor FPT Approximation for Capacitated k-Median. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{adamczyk_et_al:LIPIcs.ESA.2019.1,
  author =	{Adamczyk, Marek and Byrka, Jaros{\l}aw and Marcinkowski, Jan and Meesum, Syed M. and W{\l}odarczyk, Micha{\l}},
  title =	{{Constant-Factor FPT Approximation for Capacitated k-Median}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{1:1--1:14},
  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.2019.1},
  URN =		{urn:nbn:de:0030-drops-111225},
  doi =		{10.4230/LIPIcs.ESA.2019.1},
  annote =	{Keywords: K-median, Clustering, Approximation Algorithms, Fixed Parameter Tractability}
}
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