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Documents authored by Savani, Rahul

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.32
Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents

Authors: Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the α-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our α-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.
Cite as

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{borzechowski_et_al:LIPIcs.ICALP.2024.32,
  author =	{Borzechowski, Michaela and Fearnley, John and Gordon, Spencer and Savani, Rahul and Schnider, Patrick and Weber, Simon},
  title =	{{Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.32},
  URN =		{urn:nbn:de:0030-drops-201751},
  doi =		{10.4230/LIPIcs.ICALP.2024.32},
  annote =	{Keywords: P-LCP, Unique Sink Orientation, \alpha-Ham Sandwich, search complexity, TFNP, UEOPL}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSTTCS.2021.5
The Complexity of Gradient Descent (Invited Talk)

Authors: Rahul Savani

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
PPAD and PLS are successful classes that capture the complexity of important game-theoretic problems. For example, finding a mixed Nash equilibrium in a bimatrix game is PPAD-complete, and finding a pure Nash equilibrium in a congestion game is PLS-complete. Many important problems, such as solving a Simple Stochastic Game or finding a mixed Nash equilibrium of a congestion game, lie in both classes. It was strongly believed that their intersection, PPAD ∩ PLS, does not have natural complete problems. We show that it does: any problem that lies in both classes can be reduced in polynomial time to the problem of finding a stationary point of a continuously differentiable function on the domain [0,1]². Thus, as PPAD captures problems that can be solved by Lemke-Howson type complementary pivoting algorithms, and PLS captures problems that can be solved by local search, we show that PPAD ∩ PLS exactly captures problems that can be solved by Gradient Descent. This is joint work with John Fearnley, Paul Goldberg, and Alexandros Hollender. It appeared at STOC'21, where it was given a Best Paper Award [Fearnley et al., 2021].
Cite as

Rahul Savani. The Complexity of Gradient Descent (Invited Talk). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 5:1-5:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{savani:LIPIcs.FSTTCS.2021.5,
  author =	{Savani, Rahul},
  title =	{{The Complexity of Gradient Descent}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{5:1--5:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.5},
  URN =		{urn:nbn:de:0030-drops-155167},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.5},
  annote =	{Keywords: Computational Complexity, Continuous Optimization, TFNP, PPAD, PLS, CLS, UEOPL}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.29
A Faster Algorithm for Finding Tarski Fixed Points

Authors: John Fearnley and Rahul Savani

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log^k n) queries [Chuangyin Dang et al., 2020]. Multiple authors have conjectured that this algorithm is optimal [Chuangyin Dang et al., 2020; Kousha Etessami et al., 2020], and indeed this has been proven for two-dimensional instances [Kousha Etessami et al., 2020]. We show that these conjectures are false in dimension three or higher by giving an O(log² n) query algorithm for the three-dimensional Tarski problem, which generalises to give an O(log^{k-1} n) query algorithm for the k-dimensional problem when k ≥ 3.
Cite as

John Fearnley and Rahul Savani. A Faster Algorithm for Finding Tarski Fixed Points. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fearnley_et_al:LIPIcs.STACS.2021.29,
  author =	{Fearnley, John and Savani, Rahul},
  title =	{{A Faster Algorithm for Finding Tarski Fixed Points}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus 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.2021.29},
  URN =		{urn:nbn:de:0030-drops-136741},
  doi =		{10.4230/LIPIcs.STACS.2021.29},
  annote =	{Keywords: query complexity, Tarski fixed points, total function problem}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.38
Tree Polymatrix Games Are PPAD-Hard

Authors: Argyrios Deligkas, John Fearnley, and Rahul Savani

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

Abstract
We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an ε-fixed point of a 2D-LinearFIXP instance, when ε is any constant less than (√2 - 1)/2 ≈ 0.2071. This lifts the hardness regime from polynomially small approximations in k-dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of 0.5.
Cite as

Argyrios Deligkas, John Fearnley, and Rahul Savani. Tree Polymatrix Games Are PPAD-Hard. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{deligkas_et_al:LIPIcs.ICALP.2020.38,
  author =	{Deligkas, Argyrios and Fearnley, John and Savani, Rahul},
  title =	{{Tree Polymatrix Games Are PPAD-Hard}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{38:1--38:14},
  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.2020.38},
  URN =		{urn:nbn:de:0030-drops-124458},
  doi =		{10.4230/LIPIcs.ICALP.2020.38},
  annote =	{Keywords: Nash Equilibria, Polymatrix Games, PPAD, Brouwer Fixed Points}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.56
Unique End of Potential Line

Authors: John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The complexity class CLS was proposed by Daskalakis and Papadimitriou in 2011 to understand the complexity of important NP search problems that admit both path following and potential optimizing algorithms. Here we identify a subclass of CLS - called UniqueEOPL - that applies a more specific combinatorial principle that guarantees unique solutions. We show that UniqueEOPL contains several important problems such as the P-matrix Linear Complementarity Problem, finding Fixed Point of Contraction Maps, and solving Unique Sink Orientations (USOs). UniqueEOPL seems to a proper subclass of CLS and looks more likely to be the right class for the problems of interest. We identify a problem - closely related to solving contraction maps and USOs - that is complete for UniqueEOPL. Our results also give the fastest randomised algorithm for P-matrix LCP.
Cite as

John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique End of Potential Line. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fearnley_et_al:LIPIcs.ICALP.2019.56,
  author =	{Fearnley, John and Gordon, Spencer and Mehta, Ruta and Savani, Rahul},
  title =	{{Unique End of Potential Line}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{56:1--56:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.56},
  URN =		{urn:nbn:de:0030-drops-106327},
  doi =		{10.4230/LIPIcs.ICALP.2019.56},
  annote =	{Keywords: P-matrix linear complementarity problem, unique sink orientation, contraction map, TFNP, total search problems, continuous local search}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.124
Reachability Switching Games

Authors: John Fearnley, Martin Gairing, Matthias Mnich, and Rahul Savani

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

Abstract
In this paper, we study the problem of deciding the winner of reachability switching games. We study zero-, one-, and two-player variants of these games. We show that the zero-player case is NL-hard, the one-player case is NP-complete, and that the two-player case is PSPACE-hard and in EXPTIME. For the zero-player case, we also show P-hardness for a succinctly-represented model that maintains the upper bound of NP n coNP. For the one- and two-player cases, our results hold in both the natural, explicit model and succinctly-represented model. We also study the structure of winning strategies in these games, and in particular we show that exponential memory is required in both the one- and two-player settings.
Cite as

John Fearnley, Martin Gairing, Matthias Mnich, and Rahul Savani. Reachability Switching Games. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 124:1-124:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fearnley_et_al:LIPIcs.ICALP.2018.124,
  author =	{Fearnley, John and Gairing, Martin and Mnich, Matthias and Savani, Rahul},
  title =	{{Reachability Switching Games}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{124:1--124:14},
  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.2018.124},
  URN =		{urn:nbn:de:0030-drops-91282},
  doi =		{10.4230/LIPIcs.ICALP.2018.124},
  annote =	{Keywords: Deterministic Random Walks, Model Checking, Reachability, Simple Stochastic Game, Switching Systems}
}
Document
DOI: 10.4230/DagRep.4.8.73
Equilibrium Computation (Dagstuhl Seminar 14342)

Authors: Nimrod Megiddo, Kurt Mehlhorn, Rahul Savani, and Vijay V. Vazirani

Published in: Dagstuhl Reports, Volume 4, Issue 8 (2015)

Abstract
This report documents the program and outcomes of Dagstuhl Seminar 14342 "Equilibrium Computation". The seminar was at the leading edge of current topics related to equilibrium computation for games and markets. We summarize these topics, give the talk abstracts, and give brief summaries of the problems that were discussed in the open problem sessions.
Cite as

Nimrod Megiddo, Kurt Mehlhorn, Rahul Savani, and Vijay V. Vazirani. Equilibrium Computation (Dagstuhl Seminar 14342). In Dagstuhl Reports, Volume 4, Issue 8, pp. 73-88, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{megiddo_et_al:DagRep.4.8.73,
  author =	{Megiddo, Nimrod and Mehlhorn, Kurt and Savani, Rahul and Vazirani, Vijay V.},
  title =	{{Equilibrium Computation (Dagstuhl Seminar 14342)}},
  pages =	{73--88},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2014},
  volume =	{4},
  number =	{8},
  editor =	{Megiddo, Nimrod and Mehlhorn, Kurt and Savani, Rahul and Vazirani, Vijay V.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.8.73},
  URN =		{urn:nbn:de:0030-drops-47990},
  doi =		{10.4230/DagRep.4.8.73},
  annote =	{Keywords: Algorithms, Computational Complexity, Equilibrium Computation, Game Theory, Market Equilibrium, Nash Equilibrium}
}
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