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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

In this paper we initiate the study of populations of agents with very limited capabilities that are globally able to compute order statistics of their arithmetic input values via pair-wise meetings.
To this extent, we introduce the Arithmetic Population Protocol (APP) model, embarking from the well known Population Protocol (PP) model and inspired by two recent papers in which states are treated as integer numbers. In the APP model, every agent has a state from a set Q of states, as well as a fixed number of registers (independent of the size of the population), each of which can store an element from a totally ordered set S of samples. Whenever two agents interact with each other, they update their states and the values stored in their registers according to a joint transition function. This transition function is also restricted; it only allows (a) comparisons and (b) copy / paste operations for the sample values that are stored in the registers of the two interacting agents.
Agents can only meet in pairs via a fair scheduler and are required to eventually converge to the same output value of the function that the protocol globally and stably computes.
We present two different APPs for stably computing the median of the input values, initially stored on the agents of the population.
Our first APP, in which every agent has 3 registers and no states, stably computes (with probability 1)
the median under any fair scheduler in any strongly connected directed (or connected undirected) interaction graph.
Under the probabilistic scheduler, we show that our protocol stably computes the median in O(n^6) number of interactions in a connected undirected interaction graph of n agents.
Our second APP, in which every agent has 2 registers and O(n^2 log{n}) states, computes to the correct median of the input with high probability in O(n^3 log{n}) interactions, assuming the probabilistic scheduler and the complete interaction graph. Finally we present a third APP which, for any k, stably computes the k-th smallest element of the input of the population under any fair scheduler and in any strongly connected directed (or connected undirected) interaction graph. In this APP every agent has 2 registers and n states. Upon convergence every agent has a different state; all these states provide a total ordering of the agents with respect to their input values.

George B. Mertzios, Sotiris E. Nikoletseas, Christoforos L. Raptopoulos, and Paul G. Spirakis. Stably Computing Order Statistics with Arithmetic Population Protocols. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2016.68, author = {Mertzios, George B. and Nikoletseas, Sotiris E. and Raptopoulos, Christoforos L. and Spirakis, Paul G.}, title = {{Stably Computing Order Statistics with Arithmetic Population Protocols}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {68:1--68:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.68}, URN = {urn:nbn:de:0030-drops-64805}, doi = {10.4230/LIPIcs.MFCS.2016.68}, annote = {Keywords: arithmetic population protocols, order statistics, median, k-minimum element} }

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Brief Announcement

**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

We consider random simple temporal graphs in which every edge of the complete graph K_n appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately (2 log n)/(log 1/δ) with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n²) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdős-Rényi random graphs model, the size of the maximum δ-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-Temporal Clique is unlikely to have very large probability of success.

George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 27:1-27:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mertzios_et_al:LIPIcs.SAND.2024.27, author = {Mertzios, George B. and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul G.}, title = {{Brief Announcement: On the Existence of \delta-Temporal Cliques in Random Simple Temporal Graphs}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {27:1--27:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.27}, URN = {urn:nbn:de:0030-drops-199056}, doi = {10.4230/LIPIcs.SAND.2024.27}, annote = {Keywords: Simple random temporal graph, \delta-temporal clique, probabilistic method} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Let V be a set of n vertices, M a set of m labels, and let 𝐑 be an m × n matrix of independent Bernoulli random variables with probability of success p; columns of 𝐑 are incidence vectors of label sets assigned to vertices. A random instance G(V, E, 𝐑^T 𝐑) of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely [𝐑^T 𝐑]_{v,u}) between any two vertices u, v for which this weight is strictly larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given G(V, E, 𝐑^T 𝐑) we wish to find a partition of V into two sets so that the total weight of the edges having exactly one endpoint in each set is maximized.
In particular, we initially prove that the weight of a maximum cut of G(V, E, 𝐑^T 𝐑) is concentrated around its expected value, and then show that, when the number of labels is much smaller than the number of vertices (in particular, m = n^α, α < 1), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case n = m and constant average degree (i.e. p = Θ(1)/n), we show that with high probability, a majority type randomized algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we formally prove a connection between the computational problem of finding a (weighted) maximum cut in G(V, E, 𝐑^T 𝐑) and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system Σ with incidence matrix 𝐑 (i.e. minimum imbalance over all sets in Σ). We exploit this connection by proposing a (weak) bipartization algorithm for the case m = n, p = Θ(1)/n that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in a set system with incidence matrix 𝐑. In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in G(V, E, 𝐑^T 𝐑). Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when p = c/n, c < 1.

Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul Spirakis. MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nikoletseas_et_al:LIPIcs.ISAAC.2021.28, author = {Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul}, title = {{MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {28:1--28:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.28}, URN = {urn:nbn:de:0030-drops-154612}, doi = {10.4230/LIPIcs.ISAAC.2021.28}, annote = {Keywords: Random Intersection Graphs, Maximum Cut, Discrepancy} }

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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)

Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature, in the sense that the network structure undergoes discrete changes over time. Given a static underlying graph G=(V,E), a temporal graph on G is a sequence of snapshots {G_t=(V,E_t) subseteq G: t in N}, one for each time step t >= 1. In this paper we study stochastic temporal graphs, i.e. stochastic processes G={G_t subseteq G: t in N} whose random variables are the snapshots of a temporal graph on G. A natural feature of stochastic temporal graphs which can be observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; that is, the probability an edge e in E appears at time step t depends on its appearance (or absence) at the previous k steps. In this paper we study the hierarchy of models memory-k, k >= 0, which address this memory effect in an edge-centric network evolution: every edge of G has its own probability distribution for its appearance over time, independently of all other edges. Clearly, for every k >= 1, memory-(k-1) is a special case of memory-k. However, in this paper we make a clear distinction between the values k=0 ("no memory") and k >= 1 ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior for these values of k, as our results indicate. For every k >= 0 we investigate the computational complexity of two naturally related, but fundamentally different, temporal path (or journey) problems: {Minimum Arrival} and {Best Policy}. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices {s},{y}. In the second one we are looking for the expected arrival time of the best policy for actually choosing a particular {s}-{y} journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory k. Among other results we prove that, surprisingly, {Minimum Arrival} is strictly harder than {Best Policy}; in fact, for k=0, {Minimum Arrival} is #P-hard while {Best Policy} is solvable in O(n^2) time.

Eleni C. Akrida, George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis, and Viktor Zamaraev. How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 131:1-131:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akrida_et_al:LIPIcs.ICALP.2019.131, author = {Akrida, Eleni C. and Mertzios, George B. and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul G. and Zamaraev, Viktor}, title = {{How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {131:1--131:14}, 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.2019.131}, URN = {urn:nbn:de:0030-drops-107071}, doi = {10.4230/LIPIcs.ICALP.2019.131}, annote = {Keywords: Temporal network, stochastic temporal graph, temporal path, #P-hard problem, polynomial-time approximation scheme} }

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

In this paper, we study contention resolution protocols from a game-theoretic perspective. We focus on acknowledgment-based protocols, where a user gets feedback from the channel only when she attempts transmission. In this case she will learn whether her transmission was successful or not. Users that do not transmit will not receive any feedback. We are interested in equilibrium protocols, where no player has an incentive to deviate.
The limited feedback makes the design of equilibrium protocols a hard task as best response policies usually have to be modeled as Partially Observable Markov Decision Processes, which are hard to analyze. Nevertheless, we show how to circumvent this for the case of two players and present an equilibrium protocol. For many players, we give impossibility results for a large class of acknowledgment-based protocols, namely age-based and backoff protocols with finite expected finishing time. Finally, we provide an age-based equilibrium protocol, which has infinite expected finishing time, but every player finishes in linear time with high probability.

George Christodoulou, Martin Gairing, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul Spirakis. Strategic Contention Resolution with Limited Feedback. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{christodoulou_et_al:LIPIcs.ESA.2016.30, author = {Christodoulou, George and Gairing, Martin and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul}, title = {{Strategic Contention Resolution with Limited Feedback}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {30:1--30:16}, 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.2016.30}, URN = {urn:nbn:de:0030-drops-63813}, doi = {10.4230/LIPIcs.ESA.2016.30}, annote = {Keywords: contention resolution, acknowledgment-based protocols, game theory} }

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