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DOI: 10.4230/LIPIcs.ICALP.2018.60

ARRIVAL: Next Stop in CLS

Authors: Bernd Gärtner, Thomas Dueholm Hansen, Pavel Hubácek, Karel Král, Hagar Mosaad, and Veronika Slívová

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract

We study the computational complexity of Arrival, a zero-player game on n-vertex switch graphs introduced by Dohrau, Gärtner, Kohler, Matousek, and Welzl. They showed that the problem of deciding termination of this game is contained in NP n coNP. Karthik C. S. recently introduced a search variant of Arrival and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of Arrival. First, we resolve a question suggested by Dohrau et al. and show that the decision variant of Arrival is in UP n coUP. Second, we prove that the search variant of Arrival is contained in CLS. Third, we give a randomized O(1.4143^n)-time algorithm to solve both variants. Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the Arrival game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime O(c^n) with c<2.

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Bernd Gärtner, Thomas Dueholm Hansen, Pavel Hubácek, Karel Král, Hagar Mosaad, and Veronika Slívová. ARRIVAL: Next Stop in CLS. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 60:1-60:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gartner_et_al:LIPIcs.ICALP.2018.60,
  author =	{G\"{a}rtner, Bernd and Hansen, Thomas Dueholm and Hub\'{a}cek, Pavel and Kr\'{a}l, Karel and Mosaad, Hagar and Sl{\'\i}vov\'{a}, Veronika},
  title =	{{ARRIVAL: Next Stop in CLS}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{60:1--60:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.60},
  URN =		{urn:nbn:de:0030-drops-90641},
  doi =		{10.4230/LIPIcs.ICALP.2018.60},
  annote =	{Keywords: CLS, switch graphs, zero-player game, UP n coUP}
}

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DOI: 10.4230/LIPIcs.ICALP.2016.51

Random-Edge Is Slower Than Random-Facet on Abstract Cubes

Authors: Thomas Dueholm Hansen and Uri Zwick

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

Abstract

Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2^{Omega(sqrt(n*log(n)))} steps. This improves on a 2^{Omega(sqrt^3(n))} lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2^{O(sqrt(n)} steps on any n-dimensional AUSO, this established our result. Improving our 2^{Omega(sqrt(n*log(n)))} lower bound seems to require radically new techniques.

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Thomas Dueholm Hansen and Uri Zwick. Random-Edge Is Slower Than Random-Facet on Abstract Cubes. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hansen_et_al:LIPIcs.ICALP.2016.51,
  author =	{Hansen, Thomas Dueholm and Zwick, Uri},
  title =	{{Random-Edge Is Slower Than Random-Facet on Abstract Cubes}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.51},
  URN =		{urn:nbn:de:0030-drops-63316},
  doi =		{10.4230/LIPIcs.ICALP.2016.51},
  annote =	{Keywords: Linear programming, the Simplex Algorithm, Pivoting rules, Acyclic Unique Sink Orientations}
}

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ARRIVAL: Next Stop in CLS

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Random-Edge Is Slower Than Random-Facet on Abstract Cubes

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