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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

The average properties of the well-known Subset Sum Problem can be studied by means of its randomised version, where we are given a target value z, random variables X_1, …, X_n, and an error parameter ε > 0, and we seek a subset of the X_is whose sum approximates z up to error ε. In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size 𝒪(log(1/ε)) suffices to obtain, with high probability, approximations for all values in [-1/2, 1/2]. Recently, this result has been rediscovered outside the algorithms community, enabling meaningful progress in other fields. In this work, we present an alternative proof for this theorem, with a more direct approach and resourcing to more elementary tools.

Arthur Carvalho Walraven Da Cunha, Francesco d'Amore, Frédéric Giroire, Hicham Lesfari, Emanuele Natale, and Laurent Viennot. Revisiting the Random Subset Sum Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 37:1-37:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dacunha_et_al:LIPIcs.ESA.2023.37, author = {Da Cunha, Arthur Carvalho Walraven and d'Amore, Francesco and Giroire, Fr\'{e}d\'{e}ric and Lesfari, Hicham and Natale, Emanuele and Viennot, Laurent}, title = {{Revisiting the Random Subset Sum Problem}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {37:1--37:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.37}, URN = {urn:nbn:de:0030-drops-186905}, doi = {10.4230/LIPIcs.ESA.2023.37}, annote = {Keywords: Random subset sum, Randomised method, Subset-sum, Combinatorics} }

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Complete Volume

**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

LIPIcs, Volume 190, SEA 2021, Complete Volume

19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 1-434, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{coudert_et_al:LIPIcs.SEA.2021, title = {{LIPIcs, Volume 190, SEA 2021, Complete Volume}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {1--434}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021}, URN = {urn:nbn:de:0030-drops-137716}, doi = {10.4230/LIPIcs.SEA.2021}, annote = {Keywords: LIPIcs, Volume 190, SEA 2021, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Front Matter, Table of Contents, Preface, Conference Organization

19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{coudert_et_al:LIPIcs.SEA.2021.0, author = {Coudert, David and Natale, Emanuele}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.0}, URN = {urn:nbn:de:0030-drops-137722}, doi = {10.4230/LIPIcs.SEA.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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Extended Abstract

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

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast?
In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent.
We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise).
We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, and Luca Trevisan. Consensus vs Broadcast, with and Without Noise (Extended Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clementi_et_al:LIPIcs.ITCS.2020.42, author = {Clementi, Andrea and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo and Trevisan, Luca}, title = {{Consensus vs Broadcast, with and Without Noise}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {42:1--42:13}, 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.2020.42}, URN = {urn:nbn:de:0030-drops-117277}, doi = {10.4230/LIPIcs.ITCS.2020.42}, annote = {Keywords: Distributed Computing, Consensus, Broadcast, Gossip Models, Noisy Communication Channels} }

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

The Undecided-State Dynamics is a well-known protocol for distributed consensus. We analyze it in the parallel PULL communication model on the complete graph with n nodes for the binary case (every node can either support one of two possible colors, or be in the undecided state).
An interesting open question is whether this dynamics is an efficient Self-Stabilizing protocol, namely, starting from an arbitrary initial configuration, it reaches consensus quickly (i.e., within a polylogarithmic number of rounds). Previous work in this setting only considers initial color configurations with no undecided nodes and a large bias (i.e., Theta(n)) towards the majority color.
In this paper we present an unconditional analysis of the Undecided-State Dynamics that answers to the above question in the affirmative. We prove that, starting from any initial configuration, the process reaches a monochromatic configuration within O(log n) rounds, with high probability. This bound turns out to be tight. Our analysis also shows that, if the initial configuration has bias Omega(sqrt(n log n)), then the dynamics converges toward the initial majority color, with high probability.

Andrea Clementi, Mohsen Ghaffari, Luciano Gualà, Emanuele Natale, Francesco Pasquale, and Giacomo Scornavacca. A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{clementi_et_al:LIPIcs.MFCS.2018.28, author = {Clementi, Andrea and Ghaffari, Mohsen and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo}, title = {{A Tight Analysis of the Parallel Undecided-State Dynamics with Two Colors}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {28:1--28:15}, 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.2018.28}, URN = {urn:nbn:de:0030-drops-96107}, doi = {10.4230/LIPIcs.MFCS.2018.28}, annote = {Keywords: Distributed Consensus, Self-Stabilization, PULL Model, Markov Chains} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say +/- 1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average.
We provide a rigorous analysis of the above process showing that, if G exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process. Our analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.
We then exploit our analysis to design the first opportunistic protocols that approximately recover community structure using only logarithmic (or polylogarithmic, depending on the sparsity of the cut) work per node.

Luca Becchetti, Andrea Clementi, Pasin Manurangsi, Emanuele Natale, Francesco Pasquale, Prasad Raghavendra, and Luca Trevisan. Average Whenever You Meet: Opportunistic Protocols for Community Detection. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{becchetti_et_al:LIPIcs.ESA.2018.7, author = {Becchetti, Luca and Clementi, Andrea and Manurangsi, Pasin and Natale, Emanuele and Pasquale, Francesco and Raghavendra, Prasad and Trevisan, Luca}, title = {{Average Whenever You Meet: Opportunistic Protocols for Community Detection}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.7}, URN = {urn:nbn:de:0030-drops-94705}, doi = {10.4230/LIPIcs.ESA.2018.7}, annote = {Keywords: Community Detection, Random Processes, Spectral Analysis} }

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

Biological systems can share and collectively process information to yield emergent effects, despite inherent noise in communication. While man-made systems often employ intricate structural solutions to overcome noise, the structure of many biological systems is more amorphous. It is not well understood how communication noise may affect the computational repertoire of such groups. To approach this question we consider the basic collective task of rumor spreading, in which information from few knowledgeable sources must reliably flow into the rest of the population.
In order to study the effect of communication noise on the ability of groups that lack stable structures to efficiently solve this task, we consider a noisy version of the uniform PULL model. We prove a lower bound which implies that, in the presence of even moderate levels of noise that affect all facets of the communication, no scheme can significantly outperform the trivial one in which agents have to wait until directly interacting with the sources. Our results thus show an exponential separation between the uniform PUSH and PULL communication models in the presence of noise. Such separation may be interpreted as suggesting that, in order to achieve efficient rumor spreading, a system must exhibit either some degree of structural stability or, alternatively, some facet of the communication which is immune to noise.
We corroborate our theoretical findings with a new analysis of experimental data regarding recruitment in Cataglyphis Niger desert ants.

Lucas Boczkowski, Ofer Feinerman, Amos Korman, and Emanuele Natale. Limits for Rumor Spreading in Stochastic Populations. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{boczkowski_et_al:LIPIcs.ITCS.2018.49, author = {Boczkowski, Lucas and Feinerman, Ofer and Korman, Amos and Natale, Emanuele}, title = {{Limits for Rumor Spreading in Stochastic Populations}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {49:1--49:21}, 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.2018.49}, URN = {urn:nbn:de:0030-drops-83207}, doi = {10.4230/LIPIcs.ITCS.2018.49}, annote = {Keywords: Noisy communication, Passive communication, Ant recruitment, Hypothesis testing} }

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

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, each node adopts the majority opinion among its neighbors. It is known that this process always converges in O(|E|) rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the O(|E|) bound on the convergence time of the deterministic binary majority process is even for dense graphs tight. However, in many graphs such as the complete graph the process converges in just
a constant number of rounds from any initial opinion assignment.
We show that it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than k rounds to converge to the two-periodic stable state, for a given integer k. We then give a new upper bound on the voting time of the deterministic binary majority process. Our bound can be computed in linear time by carefully exploiting the structure of the potential function by Goles and Olivos. We identify certain modules of a graph G to obtain a new graph G^Delta. This new graph G^Delta has the property that the worst-case convergence time of G^Delta is an upper bound on that of G. Our new bounds asymptotically improve the best known bounds for various graph classes.

Dominik Kaaser, Frederik Mallmann-Trenn, and Emanuele Natale. On the Voting Time of the Deterministic Majority Process. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kaaser_et_al:LIPIcs.MFCS.2016.55, author = {Kaaser, Dominik and Mallmann-Trenn, Frederik and Natale, Emanuele}, title = {{On the Voting Time of the Deterministic Majority Process}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {55:1--55:15}, 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.2016.55}, URN = {urn:nbn:de:0030-drops-64675}, doi = {10.4230/LIPIcs.MFCS.2016.55}, annote = {Keywords: distributed voting, majority rule} }

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

We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks.
The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest.
The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E|^{1/2+o(1)} with high probability, obtaining a significant speedup with respect to the Theta(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well.
The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well.

Michele Borassi and Emanuele Natale. KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{borassi_et_al:LIPIcs.ESA.2016.20, author = {Borassi, Michele and Natale, Emanuele}, title = {{KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {20:1--20:18}, 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.2016.20}, URN = {urn:nbn:de:0030-drops-63712}, doi = {10.4230/LIPIcs.ESA.2016.20}, annote = {Keywords: Betweenness centrality, shortest path algorithm, graph mining, sampling, network analysis} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000).

Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso. Large Peg-Army Maneuvers. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{guala_et_al:LIPIcs.FUN.2016.18, author = {Gual\`{a}, Luciano and Leucci, Stefano and Natale, Emanuele and Tauraso, Roberto}, title = {{Large Peg-Army Maneuvers}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.18}, URN = {urn:nbn:de:0030-drops-58709}, doi = {10.4230/LIPIcs.FUN.2016.18}, annote = {Keywords: Complexity of Games, Solitaire Army} }

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