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Documents authored by Opler, Michal

Document
DOI: 10.4230/LIPIcs.ISAAC.2025.23
Precoloring Extension with Demands on Paths

Authors: Arun Kumar Das, Michal Opler, and Tomáš Valla

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
Let G be a graph with a set of precolored vertices, and let us be given an integer distance parameter d and a set of integer demands d₁,… ,d_c. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex c-coloring of G such that the following three conditions hold: (i) the resulting coloring respects the colors of the precolored vertices, (ii) the distance of two vertices of the same color is at least d, and (iii) the number of vertices colored by color i is exactly d_i. This problem is motivated by a program scheduling in commercial broadcast channels with constraints on content repetition and placement, which leads precisely to the DPED problem for paths. In this paper, we study DPED on paths and present a polynomial time exact algorithm when precolored vertices are restricted to the two ends of the path and devise an approximation algorithm for DPED with an additive approximation factor polynomially bounded by d and the number of precolored vertices. Then, we prove that the Distance Precoloring Extension problem on paths, a less restrictive version of DPED without the demand constraints, and then DPED itself, is NP-complete. Motivated by this result, we further study the parameterized complexity of DPED on paths. We establish that the DPED problem on paths is W[1]-hard when parameterized by the number of colors and the distance. On the positive side, we devise a fixed parameter tractable (FPT) algorithm for DPED on paths when the number of colors, the distance, and the number of precolored vertices are considered as the parameters. Moreover, we prove that Distance Precoloring Extension is FPT parameterized by the distance. As a byproduct, we also obtain several results for the Distance List Coloring problem on paths.
Cite as

Arun Kumar Das, Michal Opler, and Tomáš Valla. Precoloring Extension with Demands on Paths. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{das_et_al:LIPIcs.ISAAC.2025.23,
  author =	{Das, Arun Kumar and Opler, Michal and Valla, Tom\'{a}\v{s}},
  title =	{{Precoloring Extension with Demands on Paths}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.23},
  URN =		{urn:nbn:de:0030-drops-249319},
  doi =		{10.4230/LIPIcs.ISAAC.2025.23},
  annote =	{Keywords: precoloring extension, distance coloring, FPT, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.28
Pathfinding in Self-Deleting Graphs

Authors: Michal Dvořák, Dušan Knop, Michal Opler, Jan Pokorný, Ondřej Suchý, and Krisztina Szilágyi

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
In this paper, we study the problem of pathfinding on traversal-dependent graphs, i.e., graphs whose edges change depending on the previously visited vertices. In particular, we study self-deleting graphs, introduced by Carmesin et al. [Sarah Carmesin et al., 2023], which consist of a graph G = (V, E) and a function f: V → 2^E, where f(v) is the set of edges that will be deleted after visiting the vertex v. In the (Shortest) Self-Deleting s-t-path problem we are given a self-deleting graph and its vertices s and t, and we are asked to find a (shortest) path from s to t, such that it does not traverse an edge in f(v) after visiting v for any vertex v. We prove that Self-Deleting s-t-path is NP-hard even if the given graph is outerplanar, bipartite, has maximum degree 3, bandwidth 2 and |f(v)| ≤ 1 for each vertex v. We show that Shortest Self-Deleting s-t-path is W[1]-complete parameterized by the length of the sought path and that Self-Deleting s-t-path is W[1]-complete parameterized by the vertex cover number, feedback vertex set number and treedepth. We also show that the problem becomes FPT when we parameterize by the maximum size of f(v) and several structural parameters. Lastly, we show that the problem does not admit a polynomial kernel even for parameterization by the vertex cover number and the maximum size of f(v) combined already on 2-outerplanar graphs.
Cite as

Michal Dvořák, Dušan Knop, Michal Opler, Jan Pokorný, Ondřej Suchý, and Krisztina Szilágyi. Pathfinding in Self-Deleting Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2025.28,
  author =	{Dvo\v{r}\'{a}k, Michal and Knop, Du\v{s}an and Opler, Michal and Pokorn\'{y}, Jan and Such\'{y}, Ond\v{r}ej and Szil\'{a}gyi, Krisztina},
  title =	{{Pathfinding in Self-Deleting Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{28:1--28:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.28},
  URN =		{urn:nbn:de:0030-drops-249365},
  doi =		{10.4230/LIPIcs.ISAAC.2025.28},
  annote =	{Keywords: Parameterized complexity, self-deleting graphs, pathfinding}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.22
Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder

Authors: Vít Jelínek, Michal Opler, and Jakub Pekárek

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

Abstract
We study the counting problem known as #PPM, whose input is a pair of permutations π and τ (called pattern and text, respectively), and the task is to find the number of subsequences of τ that have the same relative order as π. A simple brute-force approach solves #PPM for a pattern of length k and a text of length n in time O(n^{k+1}), while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time f(k) n^{o(k/log k)} for any function f. In this paper, we consider the restriction of #PPM, known as 𝒞-Pattern #PPM, where the pattern π must belong to a hereditary permutation class 𝒞. Our goal is to identify the structural properties of 𝒞 that determine the complexity of 𝒞-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1) If 𝒞 has the LPP, then 𝒞-Pattern #PPM cannot be solved in time f(k)n^{o(√k)} for any function f, and 2) if 𝒞 has the DTP, then 𝒞-Pattern #PPM cannot be solved in time f(k)n^{o(k/log² k)} for any function f. Furthermore, when 𝒞 is one of the so-called monotone grid classes, we show that if 𝒞 has the LPP but not the DTP, then 𝒞-Pattern #PPM can be solved in time f(k)n^{O(√ k)}. In particular, the lower bounds above are tight up to the polylog terms in the exponents.
Cite as

Vít Jelínek, Michal Opler, and Jakub Pekárek. Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jelinek_et_al:LIPIcs.IPEC.2021.22,
  author =	{Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub},
  title =	{{Long Paths Make Pattern-Counting Hard, and Deep Trees Make It Harder}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{22:1--22:17},
  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.2021.22},
  URN =		{urn:nbn:de:0030-drops-154050},
  doi =		{10.4230/LIPIcs.IPEC.2021.22},
  annote =	{Keywords: Permutation pattern matching, subexponential algorithm, conditional lower bounds, tree-width}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.65
Griddings of Permutations and Hardness of Pattern Matching

Authors: Vít Jelínek, Michal Opler, and Jakub Pekárek

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

Abstract
We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations τ (the "text") and π (the "pattern"), and the goal is to decide whether τ contains π as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern σ; this restriction is known as Av(σ)-PPM. It has been previously shown that Av(σ)-PPM is polynomial for any σ of size at most 3, while it is NP-hard for any σ containing a monotone subsequence of length four. In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av(σ)-PPM is hard for every σ of size at least 6, for every σ of size 5 except the symmetry class of 41352, as well as for every σ symmetric to one of the three permutations 4321, 4312 and 4231. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av(σ)-PPM can be solved in time 2^o(n/log n). Previously, such conditional lower bound was not known even for the unconstrained PPM problem. On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class 𝒞, PPM is polynomial when the text is restricted to a permutation from 𝒞.
Cite as

Vít Jelínek, Michal Opler, and Jakub Pekárek. Griddings of Permutations and Hardness of Pattern Matching. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 65:1-65:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jelinek_et_al:LIPIcs.MFCS.2021.65,
  author =	{Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub},
  title =	{{Griddings of Permutations and Hardness of Pattern Matching}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{65:1--65:22},
  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.2021.65},
  URN =		{urn:nbn:de:0030-drops-145050},
  doi =		{10.4230/LIPIcs.MFCS.2021.65},
  annote =	{Keywords: Permutation, pattern matching, NP-hardness}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.52
A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes

Authors: Vít Jelínek, Michal Opler, and Jakub Pekárek

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations π and τ whether the pattern π is contained in the text τ. Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern π to a fixed permutation class 𝒞; this is known as the 𝒞-Pattern PPM problem. There have been several results in this direction, namely the work of Jelínek and Kynčl who completely resolved the hardness of 𝒞-Pattern PPM when 𝒞 is taken to be the class of σ-avoiding permutations for some σ. Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in which each building block is a monotone sequence. Recently, it has been discovered that grid classes, especially the monotone ones, play a fundamental role in the understanding of the structure of general permutation classes. This motivates us to study the hardness of 𝒞-Pattern PPM for a (monotone) grid class 𝒞. We provide a complexity dichotomy for 𝒞-Pattern PPM when 𝒞 is taken to be a monotone grid class. Specifically, we show that the problem is polynomial-time solvable if a certain graph associated with 𝒞, called the cell graph, is a forest, and it is NP-complete otherwise. We further generalize our results to grid classes whose blocks belong to classes of bounded grid-width. We show that the 𝒞-Pattern PPM for such a grid class 𝒞 is polynomial-time solvable if the cell graph of 𝒞 avoids a cycle or a certain special type of path, and it is NP-complete otherwise.
Cite as

Vít Jelínek, Michal Opler, and Jakub Pekárek. A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jelinek_et_al:LIPIcs.MFCS.2020.52,
  author =	{Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Pek\'{a}rek, Jakub},
  title =	{{A Complexity Dichotomy for Permutation Pattern Matching on Grid Classes}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{52:1--52:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.52},
  URN =		{urn:nbn:de:0030-drops-127186},
  doi =		{10.4230/LIPIcs.MFCS.2020.52},
  annote =	{Keywords: permutations, pattern matching, grid classes}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.50
Generalized Coloring of Permutations

Authors: Vít Jelínek, Michal Opler, and Pavel Valtr

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

Abstract
A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D. We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.
Cite as

Vít Jelínek, Michal Opler, and Pavel Valtr. Generalized Coloring of Permutations. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jelinek_et_al:LIPIcs.ESA.2018.50,
  author =	{Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Valtr, Pavel},
  title =	{{Generalized Coloring of Permutations}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{50:1--50:14},
  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.2018.50},
  URN =		{urn:nbn:de:0030-drops-95137},
  doi =		{10.4230/LIPIcs.ESA.2018.50},
  annote =	{Keywords: Permutations, merge, generalized coloring}
}
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