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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [Tsaknakis and Spirakis, 2008], with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+δ)-Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [Tsaknakis and Spirakis, 2008], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.

Argyrios Deligkas, Michail Fasoulakis, and Evangelos Markakis. A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deligkas_et_al:LIPIcs.ESA.2022.41, author = {Deligkas, Argyrios and Fasoulakis, Michail and Markakis, Evangelos}, title = {{A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.41}, URN = {urn:nbn:de:0030-drops-169790}, doi = {10.4230/LIPIcs.ESA.2022.41}, annote = {Keywords: bimatrix games, approximate Nash equilibria} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The computational complexity of pairwise energy minimisation of N points in real space is a long-standing open problem. The idea of the potential intractability of the problem was supported by a lack of progress in finding efficient algorithms, even when restricted the integer grid approximation. In this paper we provide a firm answer to the problem on ℤ^d by showing that for a large class of pairwise energy functions the problem of periodic energy minimisation is NP-hard if the size of the period (known as a unit cell) is fixed, and is undecidable otherwise. We do so by introducing an abstraction of pairwise average energy minimisation as a mathematical problem, which covers many existing models. The most influential aspects of this work are showing for the first time: 1) undecidability of average pairwise energy minimisation in general 2) computational hardness for the most natural model with periodic boundary conditions, and 3) novel reductions for a large class of generic pairwise energy functions covering many physical abstractions at once. In particular, we develop a new tool of overlapping digital rhombuses to incorporate the properties of the physical force fields, and we connect it with classical tiling problems. Moreover, we illustrate the power of such reductions by incorporating more physical properties such as charge neutrality, and we show an inapproximability result for the extreme case of the 1D average energy minimisation problem.

Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, and Igor Potapov. The Complexity of Periodic Energy Minimisation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{adamson_et_al:LIPIcs.MFCS.2022.8, author = {Adamson, Duncan and Deligkas, Argyrios and Gusev, Vladimir V. and Potapov, Igor}, title = {{The Complexity of Periodic Energy Minimisation}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.8}, URN = {urn:nbn:de:0030-drops-168065}, doi = {10.4230/LIPIcs.MFCS.2022.8}, annote = {Keywords: Optimisation of periodic structures, tiling, undecidability, NP-hardness} }

Document

**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k²⋅ n⁴), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm.

Duncan Adamson, Vladimir V. Gusev, Igor Potapov, and Argyrios Deligkas. Ranking Bracelets in Polynomial Time. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{adamson_et_al:LIPIcs.CPM.2021.4, author = {Adamson, Duncan and Gusev, Vladimir V. and Potapov, Igor and Deligkas, Argyrios}, title = {{Ranking Bracelets in Polynomial Time}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {4:1--4:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.4}, URN = {urn:nbn:de:0030-drops-139554}, doi = {10.4230/LIPIcs.CPM.2021.4}, annote = {Keywords: Bracelets, Ranking, Necklaces} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph G and a Hamiltonian cycle C₀ of G, how can we compute a second Hamiltonian cycle C₁ ≠ C₀ of G? Cedric Smith and William Tutte proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is a deterministic algorithm which computes the second Hamiltonian cycle in O(n⋅2^0.299862744n) = O(1.23103ⁿ) time and in linear space, thus improving the state of the art running time of O*(2^0.3n) = O(1.2312ⁿ) for solving this problem (among deterministic algorithms running in polynomial space). Whenever the input graph G does not contain any induced cycle C₆ on 6 vertices, the running time becomes O(n⋅ 2^0.2971925n) = O(1.22876ⁿ). Our algorithm is based on a fundamental structural property of Thomason’s lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a (not necessarily cubic) Hamiltonian graph G with a given Hamiltonian cycle C₀ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least n - 4α (√n+2α)+8, where α = (Δ-2)/(δ-2) and δ,Δ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.

Argyrios Deligkas, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2020.27, author = {Deligkas, Argyrios and Mertzios, George B. and Spirakis, Paul G. and Zamaraev, Viktor}, title = {{Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {27:1--27:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.27}, URN = {urn:nbn:de:0030-drops-126953}, doi = {10.4230/LIPIcs.MFCS.2020.27}, annote = {Keywords: Hamiltonian cycle, cubic graph, exact algorithm, approximation algorithm} }

Document

Track A: Algorithms, Complexity and Games

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

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.

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-dev.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

**Published in:** LIPIcs, Volume 160, 18th International Symposium on Experimental Algorithms (SEA 2020)

We study Crystal Structure Prediction, one of the major problems in computational chemistry. This is essentially a continuous optimization problem, where many different, simple and sophisticated, methods have been proposed and applied. The simple searching techniques are easy to understand, usually easy to implement, but they can be slow in practice. On the other hand, the more sophisticated approaches perform well in general, however almost all of them have a large number of parameters that require fine tuning and, in the majority of the cases, chemical expertise is needed in order to properly set them up. In addition, due to the chemical expertise involved in the parameter-tuning, these approaches can be biased towards previously-known crystal structures. Our contribution is twofold. Firstly, we formalize the Crystal Structure Prediction problem, alongside several other intermediate problems, from a theoretical computer science perspective. Secondly, we propose an oblivious algorithm for Crystal Structure Prediction that is based on local search. Oblivious means that our algorithm requires minimal knowledge about the composition we are trying to compute a crystal structure for. In addition, our algorithm can be used as an intermediate step by any method. Our experiments show that our algorithms outperform the standard basin hopping, a well studied algorithm for the problem.

Dmytro Antypov, Argyrios Deligkas, Vladimir Gusev, Matthew J. Rosseinsky, Paul G. Spirakis, and Michail Theofilatos. Crystal Structure Prediction via Oblivious Local Search. In 18th International Symposium on Experimental Algorithms (SEA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 160, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{antypov_et_al:LIPIcs.SEA.2020.21, author = {Antypov, Dmytro and Deligkas, Argyrios and Gusev, Vladimir and Rosseinsky, Matthew J. and Spirakis, Paul G. and Theofilatos, Michail}, title = {{Crystal Structure Prediction via Oblivious Local Search}}, booktitle = {18th International Symposium on Experimental Algorithms (SEA 2020)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-148-1}, ISSN = {1868-8969}, year = {2020}, volume = {160}, editor = {Faro, Simone and Cantone, Domenico}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2020.21}, URN = {urn:nbn:de:0030-drops-120950}, doi = {10.4230/LIPIcs.SEA.2020.21}, annote = {Keywords: crystal structure prediction, local search, combinatorial neighborhood} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

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

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.

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-dev.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} }

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

Graph minors are a primary tool in understanding the structure of undirected graphs, with many conceptual and algorithmic implications. We propose new variants of directed graph minors and directed graph embeddings, by modifying familiar definitions. For the class of 2-terminal directed acyclic graphs (TDAGs) our two definitions coincide, and the class is closed under both operations. The usefulness of our directed minor operations is demonstrated by characterizing all TDAGs with serial-parallel width at most k; a class of networks known to guarantee bounded negative externality in nonatomic routing games. Our characterization implies that a TDAG has serial-parallel width of 1 if and only if it is a directed series-parallel graph. We also study the computational complexity of finding a directed minor and computing the serial-parallel width.

Argyrios Deligkas and Reshef Meir. Directed Graph Minors and Serial-Parallel Width. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2018.44, author = {Deligkas, Argyrios and Meir, Reshef}, title = {{Directed Graph Minors and Serial-Parallel Width}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {44:1--44:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.44}, URN = {urn:nbn:de:0030-drops-96261}, doi = {10.4230/LIPIcs.MFCS.2018.44}, annote = {Keywords: directed minors, pathwidth} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex– the "median"–which minimises a potential \Phi among the vertices of the candidates' set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates' set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential Γ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential \Gamma that allows querying approximate medians.

Argyrios Deligkas, George B. Mertzios, and Paul G. Spirakis. Binary Search in Graphs Revisited. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deligkas_et_al:LIPIcs.MFCS.2017.20, author = {Deligkas, Argyrios and Mertzios, George B. and Spirakis, Paul G.}, title = {{Binary Search in Graphs Revisited}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.20}, URN = {urn:nbn:de:0030-drops-80589}, doi = {10.4230/LIPIcs.MFCS.2017.20}, annote = {Keywords: binary search, graph, approximate query, probabilistic algorithm, lower bound.} }

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