6 Search Results for "Paták, Pavel"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.85
ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes

Authors: Geevarghese Philip and Erlend Raa Vågset

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth k, OMM has long been known to be solvable on triangulations of 3-manifolds in 2^O(k²) n^O(1) time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on k has remained an open question. We resolve this by giving a new 2^O(k log k) n-time algorithm for any finite regular CW complex, and show that no 2^o(k log k) n^O(1)-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.
Cite as

Geevarghese Philip and Erlend Raa Vågset. ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{philip_et_al:LIPIcs.SoCG.2026.85,
  author =	{Philip, Geevarghese and V\r{a}gset, Erlend Raa},
  title =	{{ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{85:1--85:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.85},
  URN =		{urn:nbn:de:0030-drops-258926},
  doi =		{10.4230/LIPIcs.SoCG.2026.85},
  annote =	{Keywords: Discrete Morse Theory, Simplicial Complexes, Optimal Morse Matching, Treewidth, Parameterized Algorithms, Computational Topology, Dynamic Programming, Exponential Time Hypothesis, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.9
Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions

Authors: Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A theorem of Matoušek asserts that for any k ≥ 2, any set system whose shatter function is o(n^k) enjoys a fractional Helly theorem of order k: in the k-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and whose ground set has a forbidden homological minor (which includes ℝ^d by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture to sufficiently slowly growing homological shatter functions.
Cite as

Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc. Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{avvakumov_et_al:LIPIcs.SoCG.2026.9,
  author =	{Avvakumov, Sergey and Bin, Marguerite and Goaoc, Xavier},
  title =	{{Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.9},
  URN =		{urn:nbn:de:0030-drops-258152},
  doi =		{10.4230/LIPIcs.SoCG.2026.9},
  annote =	{Keywords: Fractional Helly theorem, homological minor, combinatorial convexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.61
Bounding Radon Number via Betti Numbers

Authors: Zuzana Patáková

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.
Cite as

Zuzana Patáková. Bounding Radon Number via Betti Numbers. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{patakova:LIPIcs.SoCG.2020.61,
  author =	{Pat\'{a}kov\'{a}, Zuzana},
  title =	{{Bounding Radon Number via Betti Numbers}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{61:1--61:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.61},
  URN =		{urn:nbn:de:0030-drops-122198},
  doi =		{10.4230/LIPIcs.SoCG.2020.61},
  annote =	{Keywords: Radon number, topological complexity, constrained chain maps, fractional Helly theorem, convexity spaces}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.41
Shellability is NP-Complete

Authors: Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We prove that for every d >= 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d >= 2 and k >= 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d >= 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.
Cite as

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. Shellability is NP-Complete. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2018.41,
  author =	{Goaoc, Xavier and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Shellability is NP-Complete}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{41:1--41:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.41},
  URN =		{urn:nbn:de:0030-drops-87542},
  doi =		{10.4230/LIPIcs.SoCG.2018.41},
  annote =	{Keywords: Shellability, simplicial complexes, NP-completeness, collapsibility}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.476
On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result

Authors: Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
Cite as

Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 476-490, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goaoc_et_al:LIPIcs.SOCG.2015.476,
  author =	{Goaoc, Xavier and Mabillard, Isaac and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{476--490},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.476},
  URN =		{urn:nbn:de:0030-drops-51256},
  doi =		{10.4230/LIPIcs.SOCG.2015.476},
  annote =	{Keywords: Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.507
Bounding Helly Numbers via Betti Numbers

Authors: Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.
Cite as

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. Bounding Helly Numbers via Betti Numbers. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 507-521, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goaoc_et_al:LIPIcs.SOCG.2015.507,
  author =	{Goaoc, Xavier and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Bounding Helly Numbers via Betti Numbers}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{507--521},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.507},
  URN =		{urn:nbn:de:0030-drops-51297},
  doi =		{10.4230/LIPIcs.SOCG.2015.507},
  annote =	{Keywords: Helly-type theorem, Ramsey’s theorem, Embedding of simplicial complexes, Homological almost-embedding, Betti numbers}
}
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