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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover (Mult-MSSC), a natural and intriguing generalization of the classical List Update problem. In Mult-MSSC, we maintain a sequence of permutations (π⁰, π¹, …, π^T) on n elements, based on a sequence of requests ℛ = (R¹, …, R^T). We aim to minimize the total cost of updating π^{t-1} to π^{t}, quantified by the Kendall tau distance d_{KT}(π^{t-1}, π^t), plus the total cost of covering each request R^t with the current permutation π^t, quantified by the position of the first element of R^t in π^t.
Using a reduction from Set Cover, we show that Mult-MSSC does not admit an O(1)-approximation, unless P = NP, and that any o(log n) (resp. o(r)) approximation to Mult-MSSC implies a sublogarithmic (resp. o(r)) approximation to Set Cover (resp. where each element appears at most r times). Our main technical contribution is to show that Mult-MSSC can be approximated in polynomial-time within a factor of O(log² n) in general instances, by randomized rounding, and within a factor of O(r²), if all requests have cardinality at most r, by deterministic rounding.

Dimitris Fotakis, Panagiotis Kostopanagiotis, Vasileios Nakos, Georgios Piliouras, and Stratis Skoulakis. On the Approximability of Multistage Min-Sum Set Cover. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 65:1-65:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fotakis_et_al:LIPIcs.ICALP.2021.65, author = {Fotakis, Dimitris and Kostopanagiotis, Panagiotis and Nakos, Vasileios and Piliouras, Georgios and Skoulakis, Stratis}, title = {{On the Approximability of Multistage Min-Sum Set Cover}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {65:1--65:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.65}, URN = {urn:nbn:de:0030-drops-141341}, doi = {10.4230/LIPIcs.ICALP.2021.65}, annote = {Keywords: Approximation Algorithms, Multistage Min-Sum Set Cover, Multistage Optimization Problems} }

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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 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We analyze a stylized model of co-evolution between any two purely competing species (e.g., host and parasite), both sexually reproducing. Similarly to a recent model [Livnat et al. FOCS'14] the fitness of an individual depends on whether the truth assignments on n variables that reproduce through recombination satisfy a particular Boolean function. Whereas in the original model a satisfying assignment always confers a small evolutionary advantage, in our model the two species are in an evolutionary race with the parasite enjoying the advantage if the value of its Boolean function matches its host, and the host wishing to mismatch its parasite. Surprisingly, this model makes a simple and robust behavioral prediction. The typical system behavior is periodic. These cycles stay bounded away from the boundary and thus, learning-dynamics competition between sexual species can provide an explanation for genetic diversity. This explanation is due solely to the natural selection process. No mutations, environmental changes, etc., need be invoked.
The game played at the gene level may have many Nash equilibria with widely diverse fitness levels. Nevertheless, sexual evolution leads to gene coordination that implements an optimal strategy, i.e., an optimal population mixture, at the species level. Namely, the play of the many "selfish genes" implements a time-averaged correlated equilibrium where the average fitness of each species is exactly equal to its value in the two species zero-sum competition.
Our analysis combines tools from game theory, dynamical systems and Boolean functions to establish a novel class of conservative dynamical systems.

Georgios Piliouras and Leonard J. Schulman. Learning Dynamics and the Co-Evolution of Competing Sexual Species. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 59:1-59:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{piliouras_et_al:LIPIcs.ITCS.2018.59, author = {Piliouras, Georgios and Schulman, Leonard J.}, title = {{Learning Dynamics and the Co-Evolution of Competing Sexual Species}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {59:1--59:3}, 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.2018.59}, URN = {urn:nbn:de:0030-drops-83637}, doi = {10.4230/LIPIcs.ITCS.2018.59}, annote = {Keywords: Dynamical Systems, Potential Game, Team Zero-Sum Game, Boolean Functions, Replicator Dynamics} }

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

Given a twice continuously differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue, has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT 2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.

Ioannis Panageas and Georgios Piliouras. Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 2:1-2:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{panageas_et_al:LIPIcs.ITCS.2017.2, author = {Panageas, Ioannis and Piliouras, Georgios}, title = {{Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {2:1--2:12}, 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.2017.2}, URN = {urn:nbn:de:0030-drops-81640}, doi = {10.4230/LIPIcs.ITCS.2017.2}, annote = {Keywords: Gradient Descent, Center-stable manifold, Saddle points, Hessian} }

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

A new approach to understanding evolution [Valiant, JACM 2009], namely viewing it through the lens of computation,
has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted
as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [Chastain, Livnat, Papadimitriou, Vazirani, PNAS 2014]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [Wolf, Vazirani, Arkin, J. Theor. Biology], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability.
On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game.
Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation.

Ruta Mehta, Ioannis Panageas, Georgios Piliouras, Prasad Tetali, and Vijay V. Vazirani. Mutation, Sexual Reproduction and Survival in Dynamic Environments. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 16:1-16:29, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mehta_et_al:LIPIcs.ITCS.2017.16, author = {Mehta, Ruta and Panageas, Ioannis and Piliouras, Georgios and Tetali, Prasad and Vazirani, Vijay V.}, title = {{Mutation, Sexual Reproduction and Survival in Dynamic Environments}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {16:1--16:29}, 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.2017.16}, URN = {urn:nbn:de:0030-drops-81655}, doi = {10.4230/LIPIcs.ITCS.2017.16}, annote = {Keywords: Evolution, Non-linear dynamics, Mutation} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition, or whether a single dominant genotype emerges. Classic work by [Kalmus, J. og Genetics, 1945] has established that even in simple diploid species (species with chromosome pairs) diversity can be guaranteed as long as the heterozygous (having different alleles for a gene on two chromosomes) individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g., human blood types) predicting diversity (and its implications) is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus (gene) diploid models, predicting whether diversity survives or not given its fitness landscape is algorithmically intractable.
Our hardness results are structurally robust along several dimensions, e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes. Technically, our results exploit connections between game theory, nonlinear dynamical systems, and complexity theory and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games; finding one Nash equilibrium is easy in these games. In the process we characterize stable fixed points of these dynamics using the notions of Nash equilibrium and negative semidefiniteness. This as well as hardness results for decision problems in coordination games may be of independent interest. Finally, we complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. The full version of this paper is available at http://arxiv.org/abs/1411.6322.

Ruta Mehta, Ioannis Panageas, Georgios Piliouras, and Sadra Yazdanbod. The Computational Complexity of Genetic Diversity. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 65:1-65:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mehta_et_al:LIPIcs.ESA.2016.65, author = {Mehta, Ruta and Panageas, Ioannis and Piliouras, Georgios and Yazdanbod, Sadra}, title = {{The Computational Complexity of Genetic Diversity}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {65:1--65:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.65}, URN = {urn:nbn:de:0030-drops-64073}, doi = {10.4230/LIPIcs.ESA.2016.65}, annote = {Keywords: Dynamical Systems, Stability, Complexity, Optimization, Equilibrium} }

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