12 Search Results for "Goldberg, Paul"

Document
DOI: 10.4230/OASIcs.ATMOS.2024.2
Indexing Graphs for Shortest Beer Path Queries

Authors: David Coudert, Andrea D'Ascenzo, and Mattia D'Emidio

Published in: OASIcs, Volume 123, 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)

Abstract
A beer graph is an edge-weighted graph G = (V,E,ω) with beer vertices B ⊆ V. A beer path between two vertices s and t of a beer graph is a path that connects s and t and visits at least one vertex in B. The beer distance between two vertices is the weight of a shortest beer path, i.e. a beer path having minimum total weight. A graph indexing scheme is a two-phase method that constructs an index data structure by a one-time preprocessing of an input graph and then exploits it to compute (or accelerate the computation of) answers to queries on structures of the graph dataset. In the last decade, such indexing schemes have been designed to perform, effectively, many relevant types of queries, e.g. on reachability, and have gained significant popularity in essentially all data-intensive application domains where large number of queries have to be routinely answered (e.g. journey planners), since they have been shown, through many experimental studies, to offer extremely low query times at the price of limited preprocessing time and space overheads. In this paper, we showcase that an indexing scheme, to efficiently execute queries on beer distances or shortest beer paths for pairs of vertices of a beer graph, can be obtained by adapting the highway labeling, a recently introduced indexing method to accelerate the computation of classical shortest paths. We design a preprocessing algorithm to build a whl index, i.e. a weighted highway labeling of a beer graph, and show how it can be queried to compute beer distances and shortest beer paths. Through extensive experimentation on real networks, we empirically demonstrate its practical effectiveness and superiority, in terms of offered trade-off between preprocessing time, space overhead and query time, with respect to the state-of-the-art.
Cite as

David Coudert, Andrea D'Ascenzo, and Mattia D'Emidio. Indexing Graphs for Shortest Beer Path Queries. In 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024). Open Access Series in Informatics (OASIcs), Volume 123, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coudert_et_al:OASIcs.ATMOS.2024.2,
  author =	{Coudert, David and D'Ascenzo, Andrea and D'Emidio, Mattia},
  title =	{{Indexing Graphs for Shortest Beer Path Queries}},
  booktitle =	{24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)},
  pages =	{2:1--2:18},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-350-8},
  ISSN =	{2190-6807},
  year =	{2024},
  volume =	{123},
  editor =	{Bouman, Paul C. and Kontogiannis, Spyros C.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2024.2},
  URN =		{urn:nbn:de:0030-drops-211907},
  doi =		{10.4230/OASIcs.ATMOS.2024.2},
  annote =	{Keywords: Graph Algorithms, Indexing Schemes, Beer Distances, Algorithms Engineering}
}
Document
DOI: 10.4230/LIPIcs.CP.2024.20
Constraint Modelling with LLMs Using In-Context Learning

Authors: Kostis Michailidis, Dimos Tsouros, and Tias Guns

Published in: LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

Abstract
Constraint Programming (CP) allows for the modelling and solving of a wide range of combinatorial problems. However, modelling such problems using constraints over decision variables still requires significant expertise, both in conceptual thinking and syntactic use of modelling languages. In this work, we explore the potential of using pre-trained Large Language Models (LLMs) as coding assistants, to transform textual problem descriptions into concrete and executable CP specifications. We present different transformation pipelines with explicit intermediate representations, and we investigate the potential benefit of various retrieval-augmented example selection strategies for in-context learning. We evaluate our approach on 2 datasets from the literature, namely NL4Opt (optimisation) and Logic Grid Puzzles (satisfaction), and a heterogeneous set of exercises from a CP course. The results show that pre-trained LLMs have promising potential for initialising the modelling process, with retrieval-augmented in-context learning significantly enhancing their modelling capabilities.
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Kostis Michailidis, Dimos Tsouros, and Tias Guns. Constraint Modelling with LLMs Using In-Context Learning. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 20:1-20:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{michailidis_et_al:LIPIcs.CP.2024.20,
  author =	{Michailidis, Kostis and Tsouros, Dimos and Guns, Tias},
  title =	{{Constraint Modelling with LLMs Using In-Context Learning}},
  booktitle =	{30th International Conference on Principles and Practice of Constraint Programming (CP 2024)},
  pages =	{20:1--20:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-336-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{307},
  editor =	{Shaw, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.20},
  URN =		{urn:nbn:de:0030-drops-207053},
  doi =		{10.4230/LIPIcs.CP.2024.20},
  annote =	{Keywords: Constraint Modelling, Constraint Acquisition, Constraint Programming, Large Language Models, In-Context Learning, Natural Language Processing, Named Entity Recognition, Retrieval-Augmented Generation, Optimisation}
}
Document
DOI: 10.4230/LIPIcs.WABI.2024.5
Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests

Authors: Lukas Hübner and Alexandros Stamatakis

Published in: LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

Abstract
The field of population genetics attempts to advance our understanding of evolutionary processes. It has applications, for example, in medical research, wildlife conservation, and - in conjunction with recent advances in ancient DNA sequencing technology - studying human migration patterns over the past few thousand years. The basic toolbox of population genetics includes genealogical trees, which describe the shared evolutionary history among individuals of the same species. They are calculated on the basis of genetic variations. However, in recombining organisms, a single tree is insufficient to describe the evolutionary history of the whole genome. Instead, a collection of correlated trees can be used, where each describes the evolutionary history of a consecutive region of the genome. The current corresponding state of-the-art data structure, tree sequences, compresses these genealogical trees via edit operations when moving from one tree to the next along the genome instead of storing the full, often redundant, description for each tree. We propose a new data structure, genealogical forests, which compresses the set of genealogical trees into a DAG. In this DAG identical subtrees that are shared across the input trees are encoded only once, thereby allowing for straight-forward memoization of intermediate results. Additionally, we provide a C++ implementation of our proposed data structure, called gfkit, which is 2.1 to 11.2 (median 4.0) times faster than the state-of-the-art tool on empirical and simulated datasets at computing important population genetics statistics such as the Allele Frequency Spectrum, Patterson’s f, the Fixation Index, Tajima’s D, pairwise Lowest Common Ancestors, and others. On Lowest Common Ancestor queries with more than two samples as input, gfkit scales asymptotically better than the state-of-the-art, and is thus up to 990 times faster. In conclusion, our proposed data structure compresses genealogical trees by storing shared subtrees only once, thereby enabling straight-forward memoization of intermediate results, yielding a substantial runtime reduction and a potentially more intuitive data representation over the state-of-the-art. Our improvements will boost the development of novel analyses and models in the field of population genetics and increases scalability to ever-growing genomic datasets.
Cite as

Lukas Hübner and Alexandros Stamatakis. Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hubner_et_al:LIPIcs.WABI.2024.5,
  author =	{H\"{u}bner, Lukas and Stamatakis, Alexandros},
  title =	{{Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests}},
  booktitle =	{24th International Workshop on Algorithms in Bioinformatics (WABI 2024)},
  pages =	{5:1--5:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-340-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{312},
  editor =	{Pissis, Solon P. and Sung, Wing-Kin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.5},
  URN =		{urn:nbn:de:0030-drops-206499},
  doi =		{10.4230/LIPIcs.WABI.2024.5},
  annote =	{Keywords: bioinformatics, population genetics, algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.57
Consensus Division in an Arbitrary Ratio

Authors: Paul Goldberg and Jiawei Li

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

Abstract
We consider the problem of partitioning a line segment into two subsets, so that n finite measures all have the same ratio of values for the subsets. Letting α ∈ [0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which α = 1/2. Stromquist and Woodall [Stromquist and Woodall, 1985] showed that for any α, there exists a solution using 2n cuts of the segment. They also showed that if α is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values α. For α = 𝓁/k, we show a lower bound of (k-1)/k ⋅ 2n - O(1) cuts; we also obtain almost matching upper bounds for a large subset of rational α. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1) when using the minimal number of cuts for each instance is required, the problem is NP-hard for any α; 2) for a large subset of rational α = 𝓁/k, when (k-1)/k ⋅ 2n cuts are available, the problem is in PPA-k under Turing reduction; 3) when 2n cuts are allowed, the problem belongs to PPA for any α; more generally, the problem belong to PPA-p for any prime p if 2(p-1)⋅⌈p/2⌉/⌊p/2⌋ ⋅ n cuts are available.
Cite as

Paul Goldberg and Jiawei Li. Consensus Division in an Arbitrary Ratio. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goldberg_et_al:LIPIcs.ITCS.2023.57,
  author =	{Goldberg, Paul and Li, Jiawei},
  title =	{{Consensus Division in an Arbitrary Ratio}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{57:1--57:18},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.57},
  URN =		{urn:nbn:de:0030-drops-175606},
  doi =		{10.4230/LIPIcs.ITCS.2023.57},
  annote =	{Keywords: Consensus Halving, TFNP, PPA-k, Necklace Splitting}
}
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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.65
The Hairy Ball Problem is PPAD-Complete

Authors: Paul W. Goldberg and Alexandros Hollender

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

Abstract
The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem. In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of End-of-Line, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source End-of-Line problem. If the domain is changed to be the torus of genus g >= 2 instead (where the Hairy Ball Theorem also holds), then the problem reduces to a 2(g-1)-source End-of-Line problem. These multiple-source End-of-Line results are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.
Cite as

Paul W. Goldberg and Alexandros Hollender. The Hairy Ball Problem is PPAD-Complete. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goldberg_et_al:LIPIcs.ICALP.2019.65,
  author =	{Goldberg, Paul W. and Hollender, Alexandros},
  title =	{{The Hairy Ball Problem is PPAD-Complete}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{65:1--65:14},
  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.2019.65},
  URN =		{urn:nbn:de:0030-drops-106416},
  doi =		{10.4230/LIPIcs.ICALP.2019.65},
  annote =	{Keywords: Computational Complexity, TFNP, PPAD, End-of-Line}
}
Document
Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management
DOI: 10.4230/LIPIcs.ICALP.2019.138
Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem

Authors: Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis

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

Abstract
We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete [Filos-Ratsikas and Goldberg, 2018; Filos-Ratsikas and Goldberg, 2018], we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP-hard, and deciding whether there exists a solution with fewer than n cuts is ETR-complete. We also give a QPTAS for the case where each agent’s valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP subseteq BU subseteq TFETR and that LinearBU = PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.
Cite as

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis. Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 138:1-138:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{deligkas_et_al:LIPIcs.ICALP.2019.138,
  author =	{Deligkas, Argyrios and Fearnley, John and Melissourgos, Themistoklis and Spirakis, Paul G.},
  title =	{{Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{138:1--138:14},
  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.2019.138},
  URN =		{urn:nbn:de:0030-drops-107141},
  doi =		{10.4230/LIPIcs.ICALP.2019.138},
  annote =	{Keywords: PPA, FIXP, ETR, consensus halving, circuit, reduction, complexity class}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2019.1
Computational Complexity and Partition Functions (Invited Talk)

Authors: Leslie Ann Goldberg

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
This paper is an extended abstract of my STACS 2019 talk "Computational Complexity and Partition Functions".
Cite as

Leslie Ann Goldberg. Computational Complexity and Partition Functions (Invited Talk). In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 1:1-1:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goldberg:LIPIcs.STACS.2019.1,
  author =	{Goldberg, Leslie Ann},
  title =	{{Computational Complexity and Partition Functions}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{1:1--1:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.1},
  URN =		{urn:nbn:de:0030-drops-102405},
  doi =		{10.4230/LIPIcs.STACS.2019.1},
  annote =	{Keywords: partition functions, approximation}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.24
Hardness Results for Consensus-Halving

Authors: Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard.
Cite as

Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang. Hardness Results for Consensus-Halving. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filosratsikas_et_al:LIPIcs.MFCS.2018.24,
  author =	{Filos-Ratsikas, Aris and Frederiksen, S{\o}ren Kristoffer Stiil and Goldberg, Paul W. and Zhang, Jie},
  title =	{{Hardness Results for Consensus-Halving}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.24},
  URN =		{urn:nbn:de:0030-drops-96069},
  doi =		{10.4230/LIPIcs.MFCS.2018.24},
  annote =	{Keywords: PPAD, PPA, consensus halving, generalized-circuit, reduction}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.37
Towards a Unified Complexity Theory of Total Functions

Authors: Paul W. Goldberg and Christos H. Papadimitriou

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
The class TFNP, of NP search problems where all instances have solutions, appears not to have complete problems. However, TFNP contains various syntactic subclasses and important problems. We introduce a syntactic class of problems that contains these known subclasses, for the purpose of understanding and classifying TFNP problems. This class is defined in terms of the search for an error in a concisely-represented formal proof. Finally, the known complexity subclasses are based on existence theorems that hold for finite structures; from Herbrand's Theorem, we note that such theorems must apply specifically to finite structures, and not infinite ones.
Cite as

Paul W. Goldberg and Christos H. Papadimitriou. Towards a Unified Complexity Theory of Total Functions. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goldberg_et_al:LIPIcs.ITCS.2018.37,
  author =	{Goldberg, Paul W. and Papadimitriou, Christos H.},
  title =	{{Towards a Unified Complexity Theory of Total Functions}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.37},
  URN =		{urn:nbn:de:0030-drops-83403},
  doi =		{10.4230/LIPIcs.ITCS.2018.37},
  annote =	{Keywords: Computational complexity, first-order logic, proof system, NP search functions, TFNP}
}
Document
DOI: 10.4230/DagRep.7.6.68
Game Theory Meets Computational Learning Theory (Dagstuhl Seminar 17251)

Authors: Paul W. Goldberg, Yishay Mansour, and Paul Dütting

Published in: Dagstuhl Reports, Volume 7, Issue 6 (2018)

Abstract
his report documents the program and the outcomes of Dagstuhl Seminar 17251 "Game Theory Meets Computational Learning Theory". While there have been many Dagstuhl seminars on various aspects of Algorithmic Game Theory, this was the first one to focus on the emerging field of its intersection with computational learning theory.
Cite as

Paul W. Goldberg, Yishay Mansour, and Paul Dütting. Game Theory Meets Computational Learning Theory (Dagstuhl Seminar 17251). In Dagstuhl Reports, Volume 7, Issue 6, pp. 68-85, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{goldberg_et_al:DagRep.7.6.68,
  author =	{Goldberg, Paul W. and Mansour, Yishay and D\"{u}tting, Paul},
  title =	{{Game Theory Meets Computational Learning Theory (Dagstuhl Seminar 17251)}},
  pages =	{68--85},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{7},
  number =	{6},
  editor =	{Goldberg, Paul W. and Mansour, Yishay and D\"{u}tting, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.6.68},
  URN =		{urn:nbn:de:0030-drops-82876},
  doi =		{10.4230/DagRep.7.6.68},
  annote =	{Keywords: Algorithmic Game Theory, Computational Learning Theory, Economics}
}
Document
Tutorial
DOI: 10.4230/LIPIcs.STACS.2015.20
Algorithmic Game Theory (Tutorial)

Authors: Paul W. Goldberg

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
Game theory studies mathematical models of interactions amongst self-interested entities. A "solution concept" means a description of the outcome of a game, and it is important that it should be defined in such a way that a solution always exists (every game should have an outcome). Nash's famous theorem that mixed-strategy equilibria are guaranteed to exist, resulted in Nash equilibrium being the most prominent solution concept in game theory. As a result, computational challenges of the form "given a game, find a solution", have the property that we are searching for something whose existence is guaranteed (they are total search problems). Moreover, these solutions belong to the complexity class NP, since it is usually straightforward to check whether a proposed solution is correct (an incorrect one will admit a profitable deviation by one or more of the players, and this is usually easy to find). However, in versions of the problem that appear to be computationally hard, we cannot apply NP-completeness, due to a result of Megiddo saying that total search problems cannot be NP-complete unless NP is equal to co-NP. In this tutorial, which is intended for people familiar with NP-completeness, I give an overview of the alternative notions of computational hardness that apply to game-theoretic solution concepts. I discuss the complexity class PPAD (introduced by Papadimitriou) which captures the computational complexity of various classes of games that don’t seem to be solvable in polynomial time. I also mention the complexity classes PLS and FIXP, and the kinds of games that they apply to. Suppose, alternatively, that we have a polynomial-time algorithm that applies to some given class of games. A follow-up question is whether there exist algorithms that find a solution via processes that reflect decentralised selfish behaviour. This is because a solution concept arguably remains unrealistic if it can be efficiently computed, but only using a highly centralised algorithm. In the second half of the tutorial I present some results on learning dynamics for equilibrium computation, and mention recent work on communication complexity and query complexity. I discuss some research directions and open problems, such as the following. What are the prospects for proving that PPAD is as hard as NP? How about algorithms that find improved approximate Nash equilibria? 2-player games are easy to solve in practice, using the Lemke-Howson algorithm, so is there a satisfying mathematical sense in which 2-player games are easy to solve? (For example, a sense in which Lemke-Howson works "most of the time"?)
Cite as

Paul W. Goldberg. Algorithmic Game Theory (Tutorial). In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, p. 20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goldberg:LIPIcs.STACS.2015.20,
  author =	{Goldberg, Paul W.},
  title =	{{Algorithmic Game Theory}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{20--20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.20},
  URN =		{urn:nbn:de:0030-drops-49606},
  doi =		{10.4230/LIPIcs.STACS.2015.20},
  annote =	{Keywords: equilibrium, non-cooperative games, computational complexity}
}
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