47 Search Results for "G�teau, Benjamin"

Document
DOI: 10.4230/LIPIcs.ISAAC.2023.11
Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width

Authors: Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski

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

Abstract
Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.
Cite as

Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ISAAC.2023.11,
  author =	{Bergougnoux, Benjamin and Gajarsk\'{y}, Jakub and Gu\'{s}piel, Grzegorz and Hlin\v{e}n\'{y}, Petr and Pokr\'{y}vka, Filip and Soko{\l}owski, Marek},
  title =	{{Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{11:1--11:13},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.11},
  URN =		{urn:nbn:de:0030-drops-193130},
  doi =		{10.4230/LIPIcs.ISAAC.2023.11},
  annote =	{Keywords: twin-width, tree-width, excluded grid, sparsity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.18
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

Authors: Benjamin Bergougnoux, Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Matthias Mnich, Sang-il Oum, Michał Pilipczuk, and Erik Jan van Leeuwen

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

Abstract
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using 𝒪(dk²log n) space; - Max Cut can be solved in time n^𝒪(dk) using 𝒪(dk log n) space; and - Dominating Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using n^𝒪(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.
Cite as

Benjamin Bergougnoux, Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Matthias Mnich, Sang-il Oum, Michał Pilipczuk, and Erik Jan van Leeuwen. Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ESA.2023.18,
  author =	{Bergougnoux, Benjamin and Chekan, Vera and Ganian, Robert and Kant\'{e}, Mamadou Moustapha and Mnich, Matthias and Oum, Sang-il and Pilipczuk, Micha{\l} and van Leeuwen, Erik Jan},
  title =	{{Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{18:1--18:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.18},
  URN =		{urn:nbn:de:0030-drops-186710},
  doi =		{10.4230/LIPIcs.ESA.2023.18},
  annote =	{Keywords: Parameterized complexity, shrubdepth, space complexity, algebraic methods}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.61
Solving Edge Clique Cover Exactly via Synergistic Data Reduction

Authors: Anthony Hevia, Benjamin Kallus, Summer McClintic, Samantha Reisner, Darren Strash, and Johnathan Wilson

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

Abstract
The edge clique cover (ECC) problem - where the goal is to find a minimum cardinality set of cliques that cover all the edges of a graph - is a classic NP-hard problem that has received much attention from both the theoretical and experimental algorithms communities. While small sparse graphs can be solved exactly via the branch-and-reduce algorithm of Gramm et al. [JEA 2009], larger instances can currently only be solved inexactly using heuristics with unknown overall solution quality. We revisit computing minimum ECCs exactly in practice by combining data reduction for both the ECC and vertex clique cover (VCC) problems. We do so by modifying the polynomial-time reduction of Kou et al. [Commun. ACM 1978] to transform a reduced ECC instance to a VCC instance; alternatively, we show it is possible to "lift" some VCC reductions to the ECC problem. Our experiments show that combining data reduction for both problems (which we call synergistic data reduction) enables finding exact minimum ECCs orders of magnitude faster than the technique of Gramm et al., and allows solving large sparse graphs on up to millions of vertices and edges that have never before been solved. With these new exact solutions, we evaluate the quality of recent heuristic algorithms on large instances for the first time. The most recent of these, EO-ECC by Abdullah et al. [ICCS 2022], solves 8 of the 27 instances for which we have exact solutions. It is our hope that our strategy rallies researchers to seek improved algorithms for the ECC problem.
Cite as

Anthony Hevia, Benjamin Kallus, Summer McClintic, Samantha Reisner, Darren Strash, and Johnathan Wilson. Solving Edge Clique Cover Exactly via Synergistic Data Reduction. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hevia_et_al:LIPIcs.ESA.2023.61,
  author =	{Hevia, Anthony and Kallus, Benjamin and McClintic, Summer and Reisner, Samantha and Strash, Darren and Wilson, Johnathan},
  title =	{{Solving Edge Clique Cover Exactly via Synergistic Data Reduction}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{61:1--61:19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.61},
  URN =		{urn:nbn:de:0030-drops-187148},
  doi =		{10.4230/LIPIcs.ESA.2023.61},
  annote =	{Keywords: Edge clique cover, Vertex clique cover, Data reduction, Degeneracy}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.79
Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

Authors: Benjamin Scheidt and Nicole Schweikardt

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We introduce the 2-sorted counting logic GC^k and its restriction RGC^k that express properties of hypergraphs. These logics have available k variables to address hyperedges, an unbounded number of variables to address vertices of a hypergraph, and atomic formulas E(e,v) to express that a vertex v is contained in a hyperedge e. We show that two hypergraphs H,H' satisfy the same sentences of the logic RGC^k if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of generalised hypertree width at most k. Here, H,H' are called homomorphism indistinguishable over a class 𝒞 if for every hypergraph G ∈ 𝒞 the number of homomorphisms from G to H equals the number of homomorphisms from G to H'. This result can be viewed as a lifting (from graphs to hypergraphs) of a result by Dvořák (2010) stating that any two (undirected, simple, finite) graphs H,H' are indistinguishable by the k+1-variable counting logic C^{k+1} if, and only if, they are homomorphism indistinguishable over the class of graphs of tree-width at most k.
Cite as

Benjamin Scheidt and Nicole Schweikardt. Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{scheidt_et_al:LIPIcs.MFCS.2023.79,
  author =	{Scheidt, Benjamin and Schweikardt, Nicole},
  title =	{{Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{79:1--79:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.79},
  URN =		{urn:nbn:de:0030-drops-186131},
  doi =		{10.4230/LIPIcs.MFCS.2023.79},
  annote =	{Keywords: counting logics, guarded logics, homomorphism counting, hypertree decompositions, hypergraphs, incidence graphs, quantum graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.43
Reconfiguration of Digraph Homomorphisms

Authors: Benjamin Lévêque, Moritz Mühlenthaler, and Thomas Suzan

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

Abstract
For a fixed graph H, the H-Recoloring problem asks whether, given two homomorphisms from a graph G to H, one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to H throughout. The most general algorithmic result for H-Recoloring so far has been proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph H. We show that the topological approach can be used to recover essentially all previous algorithmic results for H-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that H-Recoloring admits a polynomial-time algorithm i) if H is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and ii) if H is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0.
Cite as

Benjamin Lévêque, Moritz Mühlenthaler, and Thomas Suzan. Reconfiguration of Digraph Homomorphisms. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 43:1-43:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{leveque_et_al:LIPIcs.STACS.2023.43,
  author =	{L\'{e}v\^{e}que, Benjamin and M\"{u}hlenthaler, Moritz and Suzan, Thomas},
  title =	{{Reconfiguration of Digraph Homomorphisms}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{43:1--43:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.43},
  URN =		{urn:nbn:de:0030-drops-176958},
  doi =		{10.4230/LIPIcs.STACS.2023.43},
  annote =	{Keywords: Digraph Homomorphisms, Combinatorial Reconfiguration}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.68
Symmetric Formulas for Products of Permutations

Authors: William He and Benjamin Rossman

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We study the formula complexity of the word problem Word_{S_n,k} : {0,1}^{kn²} → {0,1}: given n-by-n permutation matrices M₁,… ,M_k, compute the (1,1)-entry of the matrix product M₁⋯ M_k. An important feature of this function is that it is invariant under action of S_n^{k-1} given by (π₁,… ,π_{k-1})(M₁,… ,M_k) = (M₁π₁^{-1},π₁M₂π₂^{-1},… ,π_{k-2}M_{k-1}π_{k-1}^{-1},π_{k-1}M_k). This symmetry is also exhibited in the smallest known unbounded fan-in {and,or,not}-formulas for Word_{S_n,k}, which have size n^O(log k). In this paper we prove a matching n^{Ω(log k)} lower bound for S_n^{k-1}-invariant formulas computing Word_{S_n,k}. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes NC¹ and Logspace. Our more general main theorem gives a nearly tight n^d(k^{1/d}-1) lower bound on the G^{k-1}-invariant depth-d {maj,and,or,not}-formula size of Word_{G,k} for any finite simple group G whose minimum permutation representation has degree n. We also give nearly tight lower bounds on the G^{k-1}-invariant depth-d {and,or,not}-formula size in the case where G is an abelian group.
Cite as

William He and Benjamin Rossman. Symmetric Formulas for Products of Permutations. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 68:1-68:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{he_et_al:LIPIcs.ITCS.2023.68,
  author =	{He, William and Rossman, Benjamin},
  title =	{{Symmetric Formulas for Products of Permutations}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{68:1--68:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.68},
  URN =		{urn:nbn:de:0030-drops-175717},
  doi =		{10.4230/LIPIcs.ITCS.2023.68},
  annote =	{Keywords: circuit complexity, group-invariant formulas}
}
Document
DOI: 10.4230/LIPIcs.CSL.2023.30
Gödel’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability

Authors: Dominik Kirst and Benjamin Peters

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Abstract
Gödel published his groundbreaking first incompleteness theorem in 1931, stating that a large class of formal logics admits independent sentences which are neither provable nor refutable. This result, in conjunction with his second incompleteness theorem, established the impossibility of concluding Hilbert’s program, which pursued a possible path towards a single formal system unifying all of mathematics. Using a technical trick to refine Gödel’s original proof, the incompleteness result was strengthened further by Rosser in 1936 regarding the conditions imposed on the formal systems. Computability theory, which also originated in the 1930s, was quickly applied to formal logics by Turing, Kleene, and others to yield incompleteness results similar in strength to Gödel’s original theorem, but weaker than Rosser’s refinement. Only much later, Kleene found an improved but far less well-known proof based on computational notions, yielding a result as strong as Rosser’s. In this expository paper, we work in constructive type theory to reformulate Kleene’s incompleteness results abstractly in the setting of synthetic computability theory and assuming a form of Church’s thesis, an axiom internalising the fact that all functions definable in such a setting are computable. Our novel, greatly condensed reformulation showcases the simplicity of the computational argument while staying formally entirely precise, a combination hard to achieve in typical textbook presentations. As an application, we instantiate the abstract result to first-order logic in order to derive essential incompleteness and, along the way, essential undecidability of Robinson arithmetic. This paper is accompanied by a Coq mechanisation covering all our results and based on existing libraries of undecidability proofs and first-order logic, complementing the extensive work on mechanised incompleteness using the Gödel-Rosser approach. In contrast to the related mechanisations, our development follows Kleene’s ideas and utilises Church’s thesis for additional simplicity.
Cite as

Dominik Kirst and Benjamin Peters. Gödel’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kirst_et_al:LIPIcs.CSL.2023.30,
  author =	{Kirst, Dominik and Peters, Benjamin},
  title =	{{G\"{o}del’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{30:1--30:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.30},
  URN =		{urn:nbn:de:0030-drops-174911},
  doi =		{10.4230/LIPIcs.CSL.2023.30},
  annote =	{Keywords: incompleteness, undecidability, synthetic computability theory}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.1
Enumerating Minimal Connected Dominating Sets

Authors: Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann

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

Abstract
The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.
Cite as

Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann. Enumerating Minimal Connected Dominating Sets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{abukhzam_et_al:LIPIcs.ESA.2022.1,
  author =	{Abu-Khzam, Faisal N. and Fernau, Henning and Gras, Benjamin and Liedloff, Mathieu and Mann, Kevin},
  title =	{{Enumerating Minimal Connected Dominating Sets}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{1:1--1:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.1},
  URN =		{urn:nbn:de:0030-drops-169390},
  doi =		{10.4230/LIPIcs.ESA.2022.1},
  annote =	{Keywords: enumeration problems, connected domination, degenerate graphs}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.14
The Diameter of Caterpillar Associahedra

Authors: Benjamin Aram Berendsohn

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
The caterpillar associahedron 𝒜(G) is a polytope arising from the rotation graph of search trees on a caterpillar tree G, generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of 𝒜(G) is Θ(n + m ⋅ (H+1)), where n is the number of vertices, m is the number of leaves, and H is the entropy of the leaf distribution of G. Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber’s first lower bound for dynamic BSTs, and the upper bound by reducing the problem to searching in static BSTs.
Cite as

Benjamin Aram Berendsohn. The Diameter of Caterpillar Associahedra. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berendsohn:LIPIcs.SWAT.2022.14,
  author =	{Berendsohn, Benjamin Aram},
  title =	{{The Diameter of Caterpillar Associahedra}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.14},
  URN =		{urn:nbn:de:0030-drops-161743},
  doi =		{10.4230/LIPIcs.SWAT.2022.14},
  annote =	{Keywords: Graph Associahedra, Binary Search Trees, Elimination Trees}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2022.3
Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring (Invited Talk)

Authors: Fabian Kuhn

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
The problem of obtaining fast deterministic algorithms for distributed symmetry breaking problems in graphs has long been considered one of the most challenging problems in the area of distributed graph algorithms. Consider for example the distributed coloring problem, where a (computer) network is modeled by an arbitrary graph G = (V,E) and the objective is to compute a vertex coloring of G by running a distributed algorithm on the graph G. It is maybe not surprising that randomization can be a helpful tool to efficiently compute such a coloring. In fact, as long as each node v ∈ V can choose among deg(v)+1 different colors, even an almost trivial algorithm in which all nodes keep trying a random available color allows to color all nodes in O(log n) parallel steps. How to obtain a similarly efficient deterministic distributed coloring algorithm is far less obvious. In fact, for a long time, there has been an exponential gap between the time complexities of the best randomized and the best deterministic distributed algorithms for various graph coloring variants and for many other basic graph problems. In the last few years, there however has been substantial progress on deterministic distributed graph algorithms that are nearly as fast as randomized algorithms for the same tasks. In particular, in a recent breakthrough, Rozhoň and Ghaffari managed to reduce the gap between the randomized and deterministic complexities of locally checkable graph problems to at most polylog n. In the talk, we give a brief overview of the history of the problem of finding fast deterministic algorithms for distributed symmetry breaking problems and of what we know about the relation between deterministic and randomized distributed algorithms for such problems. Together with some additional recent developments, the result of Rozhoň and Ghaffari provides a generic, somewhat brute-force way to efficiently derandomize randomized distributed algorithms. Apart from this, there has also been substantial progress on more direct, problem-specific algorithms. In the talk, we in particular discuss some novel deterministic distributed graph coloring algorithms. The algorithms are signficantly faster and we believe also simpler than previous algorithms for the same coloring problems.
Cite as

Fabian Kuhn. Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring (Invited Talk). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kuhn:LIPIcs.STACS.2022.3,
  author =	{Kuhn, Fabian},
  title =	{{Deterministic Distributed Symmetry Breaking at the Example of Distributed Graph Coloring}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.3},
  URN =		{urn:nbn:de:0030-drops-158131},
  doi =		{10.4230/LIPIcs.STACS.2022.3},
  annote =	{Keywords: distributed graph algorithms, derandomization, distributed coloring}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.12
Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers

Authors: Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem. We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.
Cite as

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12,
  author =	{Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko},
  title =	{{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.12},
  URN =		{urn:nbn:de:0030-drops-158221},
  doi =		{10.4230/LIPIcs.STACS.2022.12},
  annote =	{Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.14
On Explicit Constructions of Extremely Depth Robust Graphs

Authors: Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee, and Jin Young Son

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
A directed acyclic graph G = (V,E) is said to be (e,d)-depth robust if for every subset S ⊆ V of |S| ≤ e nodes the graph G-S still contains a directed path of length d. If the graph is (e,d)-depth-robust for any e,d such that e+d ≤ (1-ε)|V| then the graph is said to be ε-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents running in time polylog|V| which takes as input a node v ∈ V and returns the set of v’s parents. We give the first explicit construction of locally navigable ε-extreme depth-robust graphs with indegree O(log |V|). Previous constructions of ε-extreme depth-robust graphs either had indegree ω̃(log² |V|) or were not explicit.
Cite as

Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee, and Jin Young Son. On Explicit Constructions of Extremely Depth Robust Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blocki_et_al:LIPIcs.STACS.2022.14,
  author =	{Blocki, Jeremiah and Cinkoske, Mike and Lee, Seunghoon and Son, Jin Young},
  title =	{{On Explicit Constructions of Extremely Depth Robust Graphs}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{14:1--14:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.14},
  URN =		{urn:nbn:de:0030-drops-158241},
  doi =		{10.4230/LIPIcs.STACS.2022.14},
  annote =	{Keywords: Depth-Robust Graphs, Explicit Constructions, Data-Independent Memory Hard Functions, Proofs of Space and Replication}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.20
Symmetry and Quantum Query-To-Communication Simulation

Authors: Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways. - We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication). - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.
Cite as

Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf. Symmetry and Quantum Query-To-Communication Simulation. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2022.20,
  author =	{Chakraborty, Sourav and Chattopadhyay, Arkadev and H{\o}yer, Peter and Mande, Nikhil S. and Paraashar, Manaswi and de Wolf, Ronald},
  title =	{{Symmetry and Quantum Query-To-Communication Simulation}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{20:1--20:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.20},
  URN =		{urn:nbn:de:0030-drops-158309},
  doi =		{10.4230/LIPIcs.STACS.2022.20},
  annote =	{Keywords: Classical and quantum communication complexity, query-to-communication-simulation, quantum computing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.23
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths

Authors: Yung-Chung Chiu and Hsueh-I Lu

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
For vertices u and v of an n-vertex graph G, a uv-trail of G is an induced uv-path of G that is not a shortest uv-path of G. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in O(n^{18}) time, to either output a uv-trail of G or ensure that G admits no uv-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of n²× n² Boolean matrices, leading to a largely improved O(n^{4.75})-time algorithm.
Cite as

Yung-Chung Chiu and Hsueh-I Lu. Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chiu_et_al:LIPIcs.STACS.2022.23,
  author =	{Chiu, Yung-Chung and Lu, Hsueh-I},
  title =	{{Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.23},
  URN =		{urn:nbn:de:0030-drops-158333},
  doi =		{10.4230/LIPIcs.STACS.2022.23},
  annote =	{Keywords: Induced subgraph, induced path, non-shortest path, dynamic data structure}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.27
Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication

Authors: Michael Elkin and Ofer Neiman

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Consider an undirected weighted graph G = (V,E,w). We study the problem of computing (1+ε)-approximate shortest paths for S × V, for a subset S ⊆ V of |S| = n^r sources, for some 0 < r ≤ 1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time Õ(|E| ⋅ n^{o(1)} + n^{ω(r)}), where n^{ω(r)} is the time required to multiply an n^r × n matrix by an n × n one. Our PRAM algorithm has polylogarithmic time (log n)^{O(1/ρ)}, and its work complexity is Õ(|E| ⋅ n^ρ + n^{ω(r)}), for any arbitrarily small constant ρ > 0. In particular, for r ≤ 0.313…, our centralized algorithm computes S × V (1+ε)-approximate shortest paths in n^{2 + o(1)} time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E| ⋅ n^ρ + n^{2+o(1)}), for any arbitrarily small constant ρ > 0. Previously existing solutions either require centralized time/parallel work of O(|E| ⋅ |S|) or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S| = n^r sources, for r ≤ 0.655, while previous state-of-the-art algorithms did so only for r ≤ 1/2. Moreover, it improves previous bounds for all r > 1/2. For unweighted graphs, the running time is improved further to poly(log log n) for r ≤ 0.655. Previously this running time was known for r ≤ 1/2.
Cite as

Michael Elkin and Ofer Neiman. Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{elkin_et_al:LIPIcs.STACS.2022.27,
  author =	{Elkin, Michael and Neiman, Ofer},
  title =	{{Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{27:1--27:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.27},
  URN =		{urn:nbn:de:0030-drops-158379},
  doi =		{10.4230/LIPIcs.STACS.2022.27},
  annote =	{Keywords: Shortest paths, matrix multiplication, hopsets}
}
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