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

The price of anarchy quantifies the degradation of social welfare in games due to the lack of a centralized authority that can enforce the optimal outcome. It is known that, in certain games, such effects can be ameliorated via tolls or taxes. This leads to a natural, but largely unexplored, question: what is the effect of such transfers on social inequality? We study this question in nonatomic congestion games, arguably one of the most thoroughly studied settings from the perspective of the price of anarchy. We introduce a new model that incorporates the income distribution of the population and captures the income elasticity of travel time (i.e., how does loss of time translate to lost income). This allows us to argue about the equality of wealth distribution both before and after employing a mechanism. We establish that, under reasonable assumptions, tolls always increase inequality in symmetric congestion games under any reasonable metric of inequality such as the Gini index. We introduce the inequity index, a novel measure for quantifying the magnitude of these forces towards a more unbalanced wealth distribution and show it has good normative properties (robustness to scaling of income, no-regret learning). We analyze inequity both in theoretical settings (Pigou’s network under various wealth distributions) as well as experimental ones (based on a large scale field experiment in Singapore). Finally, we provide an algorithm for computing optimal tolls for any point of the trade-off of relative importance of efficiency and equality. We conclude with a discussion of our findings in the context of theories of justice as developed in contemporary social sciences and present several directions for future research.

Kurtuluş Gemici, Elias Koutsoupias, Barnabé Monnot, Christos H. Papadimitriou, and Georgios Piliouras. Wealth Inequality and the Price of Anarchy. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gemici_et_al:LIPIcs.STACS.2019.31, author = {Gemici, Kurtulu\c{s} and Koutsoupias, Elias and Monnot, Barnab\'{e} and Papadimitriou, Christos H. and Piliouras, Georgios}, title = {{Wealth Inequality and the Price of Anarchy}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {31:1--31:16}, 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.2019.31}, URN = {urn:nbn:de:0030-drops-102707}, doi = {10.4230/LIPIcs.STACS.2019.31}, annote = {Keywords: congestion games, inequality} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

It has been recently shown via simulations [Dasgupta et al., 2017] that random projection followed by a cap operation (setting to one the k largest elements of a vector and everything else to zero), a map believed to be an important part of the insect olfactory system, has strong locality sensitivity properties. We calculate the asymptotic law whereby the overlap in the input vectors is conserved, verifying mathematically this empirical finding. We then focus on the far more complex homologous operation in the mammalian brain, the creation through successive projections and caps of an assembly (roughly, a set of excitatory neurons representing a memory or concept) in the presence of recurrent synapses and plasticity. After providing a careful definition of assemblies, we prove that the operation of assembly projection converges with high probability, over the randomness of synaptic connectivity, even if plasticity is relatively small (previous proofs relied on high plasticity). We also show that assembly projection has itself some locality preservation properties. Finally, we propose a large repertoire of assembly operations, including associate, merge, reciprocal project, and append, each of them both biologically plausible and consistent with what we know from experiments, and show that this computational system is capable of simulating, again with high probability, arbitrary computation in a quite natural way. We hope that this novel way of looking at brain computation, open-ended and based on reasonably mainstream ideas in neuroscience, may prove an attractive entry point for computer scientists to work on understanding the brain.

Christos H. Papadimitriou and Santosh S. Vempala. Random Projection in the Brain and Computation with Assemblies of Neurons. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{papadimitriou_et_al:LIPIcs.ITCS.2019.57, author = {Papadimitriou, Christos H. and Vempala, Santosh S.}, title = {{Random Projection in the Brain and Computation with Assemblies of Neurons}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {57:1--57:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.57}, URN = {urn:nbn:de:0030-drops-101506}, doi = {10.4230/LIPIcs.ITCS.2019.57}, annote = {Keywords: Brain computation, random projection, assemblies, plasticity, memory, association} }

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

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.

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} }

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

In a recent experiment, a cell in the human medial temporal lobe (MTL) encoding one sensory stimulus starts to also respond to a second stimulus following a combined experience associating the two. We develop a theoretical model predicting that an assembly of cells with exceptionally high synaptic intraconnectivity can emerge, in response to a particular sensory experience, to encode and abstract that experience. We also show that two such assemblies are modified to increase their intersection after a sensory event that associates the two corresponding stimuli. The main technical tools employed are random graph theory, and Bernoulli approximations. Assembly creation must overcome a computational challenge akin to the Densest K-Subgraph problem, namely selecting, from a large population of randomly and sparsely interconnected cells, a subset with exceptionally high density of interconnections. We identify three mechanisms that help achieve this feat in our model: (1) a simple two-stage randomized algorithm, and (2) the "triangle completion bias" in synaptic connectivity and a "birthday paradox", while (3) the strength of these connections is enhanced through Hebbian plasticity.

Robert Legenstein, Wolfgang Maass, Christos H. Papadimitriou, and Santosh S. Vempala. Long Term Memory and the Densest K-Subgraph Problem. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{legenstein_et_al:LIPIcs.ITCS.2018.57, author = {Legenstein, Robert and Maass, Wolfgang and Papadimitriou, Christos H. and Vempala, Santosh S.}, title = {{Long Term Memory and the Densest K-Subgraph Problem}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {57:1--57:15}, 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.2018.57}, URN = {urn:nbn:de:0030-drops-83593}, doi = {10.4230/LIPIcs.ITCS.2018.57}, annote = {Keywords: Brain computation, long term memory, assemblies, association} }

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Complete Volume

**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

LIPIcs, Volume 67, ITCS'17, Complete Volume

8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Proceedings{papadimitriou:LIPIcs.ITCS.2017, title = {{LIPIcs, Volume 67, ITCS'17, Complete Volume}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017}, URN = {urn:nbn:de:0030-drops-82066}, doi = {10.4230/LIPIcs.ITCS.2017}, annote = {Keywords: Theory of Computation, Mathematics of Computing} }

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Front Matter

**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Front Matter, Table of Contents, Preface, Conference Organization

8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{papadimitriou:LIPIcs.ITCS.2017.0, author = {Papadimitriou, Christos H.}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {0:i--0:x}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.0}, URN = {urn:nbn:de:0030-drops-82025}, doi = {10.4230/LIPIcs.ITCS.2017.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

It was recently shown [Sæther, Telle, and Vatshelle, JAIR 54, 2015] that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within each partite set). Here we relax this condition in several directions: First, we show an FPT algorithm parameterized by k for k-interval bigraphs, bipartite graphs which can be converted to interval bipartite graphs by adding to each node of one side at most k edges; the same result holds for the counting and the weighted maximization version of satisfiability. Second, given two linear orders, one for the variables and one for the clauses, we show how to find, in polynomial time, the smallest k such that there is a k-interval bigraph compatible with these two orders. On the negative side we prove that, barring complexity collapses, no such extensions are possible for CSPs more general than satisfiability. We also show NP-hardness of recognizing 1-interval
bigraphs.

Serge Gaspers, Christos H. Papadimitriou, Sigve Hortemo Sæther, and Jan Arne Telle. On Satisfiability Problems with a Linear Structure. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gaspers_et_al:LIPIcs.IPEC.2016.14, author = {Gaspers, Serge and Papadimitriou, Christos H. and S{\ae}ther, Sigve Hortemo and Telle, Jan Arne}, title = {{On Satisfiability Problems with a Linear Structure}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.14}, URN = {urn:nbn:de:0030-drops-69412}, doi = {10.4230/LIPIcs.IPEC.2016.14}, annote = {Keywords: Satisfiability, interval graphs, FPT algorithms} }

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Invited Talk

**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Computation as a mechanical reality is young - almost exactly seventy years of age - and yet the spirit of computation can be traced several millennia back. Any moderately advanced civilization depends on calculation (for inventory, taxation, navigation, land partition, among many others) - our civilization is the first one that is conscious of this reliance.
Computation has also been central to science for centuries. This is most immediately apparent in the case of mathematics: the idea of the algorithm as a mathematical object of some significance was pioneered by Euclid in the 4th century BC, and advanced by Archimedes a century later. But computation plays an important role in virtually all sciences: natural, life, or social. Implicit algorithmic processes are present in the great objects of scientific inquiry - the cell, the universe, the market, the brain - as well as in the models developed by scientists over the centuries for studying them. This brings about a very recent - merely a few decades old - mode of scientific inquiry, which is sometime referred to as the lens of computation: When students of computation revisit central problems in science from the computational viewpoint, often unexpected progress results. This has happened in statistical physics through the study of phase transitions in terms of the convergence of Markov chain-Monte Carlo algorithms, and in quantum mechanics through quantum computing.
This talk will focus on three other manifestations of this phenomenon. Almost a decade ago, ideas and methodologies from computational complexity revealed a subtle conceptual flaw in the solution concept of Nash equilibrium, which lies at the foundations of modern economic thought. In the study of evolution, a new understanding of century-old questions has been achieved through surprisingly algorithmic ideas. Finally, current work in theoretical neuroscience suggests that the algorithmic point of view may be invaluable in the central scientific question of our era, namely understanding how behavior and cognition emerge from the structure and activity of neurons and synapses.

Christos H. Papadimitriou. Computation as a Scientific Weltanschauung (Invited Talk). In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, p. 33:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{papadimitriou:LIPIcs.SWAT.2016.33, author = {Papadimitriou, Christos H.}, title = {{Computation as a Scientific Weltanschauung}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {33:1--33:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.33}, URN = {urn:nbn:de:0030-drops-60558}, doi = {10.4230/LIPIcs.SWAT.2016.33}, annote = {Keywords: Lens of computation, Nash equilibrium, neuroscience} }

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Invited Talk

**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Even the most seasoned students of evolution, starting with Darwin himself, have occasionally expressed amazement at the fact that the mechanism of natural selection has produced the whole of Life as we see it around us. From a computational perspective, it is natural to marvel at evolution's solution to the problems of robotics, vision and theorem proving! What, then, is the complexity of evolution, viewed as an algorithm? One answer to this question is 10^{12}, roughly the number of sequential steps or generations from the earliest single celled creatures to today's Homo Sapiens. To put this into perspective, the processor of a modern cell phone can perform 10^{12} steps in less than an hour. Another answer is 10^30, the degree of parallelism, roughly the maximum number of organisms living on the Earth at any time. Perhaps the answer should be the product of the two numbers, roughly 10^42, to reflect the total work done by evolution, viewed as a parallel algorithm.
Here we argue, interpreting our recently published paper, that none of the above answers is really correct. Viewing evolution as an algorithm poses an additional challenge: recombination. Even if evolution succeeds in producing a particularly good solution (a highly fit individual), its offspring would only inherit half its genes, and therefore appear unlikely to be a good solution. This is the core of the problem of explaining the role of sex in evolution, known as the "queen of problems in evolutionary biology".
The starting point is the diffusion-equation-based approach of theoretical population geneticists, who analyze the changing allele frequencies (over the generations) in the gene pool, consisting of the aggregate of the genetic variants (or "alleles") over all genes (or "loci") and over all individuals in a species. Taking this viewpoint to its logical conclusion, rather than acting on individuals or species or genes, evolution acts on this gene pool, or genetic soup, by making it more "potent", in the sense that it increases the expected fitness of genotype drawn randomly from this soup. Moreover, for much genetic variation, this soup may be assumed to be in the regime of weak selection, a regime where the probability of occurrence of a certain genotype involving various alleles at different loci is simply the product of the probabilities of each of its alleles. In this regime, we show that evolution in the regime of weak selection can be formulated as a game, where the recombining loci are the players, the alleles in those loci are possible moves or actions of each player, and the expected payoff of each player-locus is precisely the organism's expected fitness across the genotypes that are present in the population. Moreover, the dynamics specified by the diffusion equations of theoretical population geneticists is closely approximated by the dynamics of multiplicative weight updates (MWUA).
The algorithmic connection to MWUA brings with it new insights for evolutionary biology, specifically, into the question of how genetic diversity is maintained in the presence of natural selection. For this it is useful to consider a dual view of MWUA, which expresses "what each gene is optimizing" as it plays the game. Remarkably this turns out to be a particular convex combination of the entropy of its distribution over alleles and cumulative expected fitness. This sheds new light on the maintenance of diversity in evolution.
All of this suggests that the complexity of evolution should indeed be viewed as 10^12, but for a subtle reason. It is the number of steps of multiplicative weight updates carried out on allele frequencies in the genetic soup. A closer examination of this reveals further that the accurate tracking of allele frequencies over the generations requires the simulation of a quadratic dynamical system (two parents for each offspring). Moreover the simulation of even simple quadratic dynamical systems is known to be PSPACE-hard. This suggests that the tracking of allele frequencies might require large population sizes for each species, putting into perspective the number 10^30. Finally, it is worth noting that in this view there is a primacy to recombination or sex, which serve to provide robustness to the mechanism of evolution, as well as the framework within which MWUA operates.

Erick Chastain, Adi Livnat, Christos H. Papadimitriou, and Umesh V. Vazirani. Algorithms, Games, and Evolution (Invited Talk). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 45-46, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chastain_et_al:LIPIcs.FSTTCS.2014.45, author = {Chastain, Erick and Livnat, Adi and Papadimitriou, Christos H. and Vazirani, Umesh V.}, title = {{Algorithms, Games, and Evolution}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {45--46}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.45}, URN = {urn:nbn:de:0030-drops-48310}, doi = {10.4230/LIPIcs.FSTTCS.2014.45}, annote = {Keywords: evolution, recombination, coordination games, multiplicative weight updates} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Marek Karpinski, Christos H. Papadimitriou, and Vijay V. Vazirani. Algorithmic Game Theory and the Internet (Dagstuhl Seminar 03291). Dagstuhl Seminar Report 386, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2003)

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@TechReport{karpinski_et_al:DagSemRep.386, author = {Karpinski, Marek and Papadimitriou, Christos H. and Vazirani, Vijay V.}, title = {{Algorithmic Game Theory and the Internet (Dagstuhl Seminar 03291)}}, pages = {1--9}, ISSN = {1619-0203}, year = {2003}, type = {Dagstuhl Seminar Report}, number = {386}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.386}, URN = {urn:nbn:de:0030-drops-152660}, doi = {10.4230/DagSemRep.386}, }

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

We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey’s theorem, the Sunflower theorem, the Erdős-Ko-Rado lemma, and König’s lemma. Whether the first two of these four problems are PLC-complete is an important open question which we pursue; in contrast, we show that the latter two are PPP-complete. Finally, we reframe PPP as an optimization problem, and define a hierarchy of such problems related to Turàn’s theorem.

Amol Pasarkar, Christos Papadimitriou, and Mihalis Yannakakis. Extremal Combinatorics, Iterated Pigeonhole Arguments and Generalizations of PPP. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 88:1-88:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{pasarkar_et_al:LIPIcs.ITCS.2023.88, author = {Pasarkar, Amol and Papadimitriou, Christos and Yannakakis, Mihalis}, title = {{Extremal Combinatorics, Iterated Pigeonhole Arguments and Generalizations of PPP}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {88:1--88:20}, 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.2023.88}, URN = {urn:nbn:de:0030-drops-175913}, doi = {10.4230/LIPIcs.ITCS.2023.88}, annote = {Keywords: Total Complexity, Extremal Combinatorics, Pigeonhole Principle} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FP^NP|poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it, as well as the problem of finding a king in a tournament (a vertex k such that all other vertices are defeated by k, or by somebody k defeated).

Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou. Total Functions in the Polynomial Hierarchy. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kleinberg_et_al:LIPIcs.ITCS.2021.44, author = {Kleinberg, Robert and Korten, Oliver and Mitropolsky, Daniel and Papadimitriou, Christos}, title = {{Total Functions in the Polynomial Hierarchy}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {44:1--44:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.44}, URN = {urn:nbn:de:0030-drops-135835}, doi = {10.4230/LIPIcs.ITCS.2021.44}, annote = {Keywords: total complexity, polynomial hierarchy, pigeonhole principle} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

The use of monotonicity and Tarski’s theorem in existence proofs of equilibria is very widespread in economics, while Tarski’s theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the d-dimensional grid with sides of length N, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time log^d N, and we show it requires at least log^2 N function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski’s theorem, however, requires d ⋅ N steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in O(log N) steps. We also show that computing (approximating) the value of Condon’s (Shapley’s) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a Ω(log^d N) lower bound for small fixed dimension d ≥ 3.

Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{etessami_et_al:LIPIcs.ITCS.2020.18, author = {Etessami, Kousha and Papadimitriou, Christos and Rubinstein, Aviad and Yannakakis, Mihalis}, title = {{Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {18:1--18:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.18}, URN = {urn:nbn:de:0030-drops-117037}, doi = {10.4230/LIPIcs.ITCS.2020.18}, annote = {Keywords: Tarski’s theorem, supermodular games, monotone functions, lattices, fixed points, Nash equilibria, computational complexity, PLS, PPAD, stochastic games, oracle model, lower bounds} }

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