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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds.
Compared to the best previous lower bounds by Hao and Heydenreich [Hao and Heydenreich, 2023], our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds of [Hao and Heydenreich, 2023] for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the k-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this k-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.

Kostas Lakis, Johannes Lengler, Kalina Petrova, and Leon Schiller. Improved Bounds for Graph Distances in Scale Free Percolation and Related Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 74:1-74:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lakis_et_al:LIPIcs.APPROX/RANDOM.2024.74, author = {Lakis, Kostas and Lengler, Johannes and Petrova, Kalina and Schiller, Leon}, title = {{Improved Bounds for Graph Distances in Scale Free Percolation and Related Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {74:1--74:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.74}, URN = {urn:nbn:de:0030-drops-210676}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.74}, annote = {Keywords: Mathematics, Probability Theory, Combinatorics, Random Graphs, Random Metric Spaces} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability?
As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.

Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl. On Connectivity in Random Graph Models with Limited Dependencies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lengler_et_al:LIPIcs.APPROX/RANDOM.2023.30, author = {Lengler, Johannes and Martinsson, Anders and Petrova, Kalina and Schnider, Patrick and Steiner, Raphael and Weber, Simon and Welzl, Emo}, title = {{On Connectivity in Random Graph Models with Limited Dependencies}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {30:1--30:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.30}, URN = {urn:nbn:de:0030-drops-188556}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.30}, annote = {Keywords: Random Graphs, Independence, Dependency, Connectivity, Threshold, Probabilistic Method} }

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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)

A classical multi-agent fence patrolling problem asks: What is the maximum length L of a line fence that k agents with maximum speeds v_1,..., v_k can patrol if each point on the line needs to be visited at least once every unit of time. It is easy to see that L = alpha sum_{i=1}^k v_i for some efficiency alpha in [1/2,1). After a series of works [Czyzowicz et al., 2011; Dumitrescu et al., 2014; Kawamura and Kobayashi, 2015; Kawamura and Soejima, 2015] giving better and better efficiencies, it was conjectured by Kawamura and Soejima [Kawamura and Soejima, 2015] that the best possible efficiency approaches 2/3. No upper bounds on the efficiency below 1 were known.
We prove the first such upper bounds and tightly bound the optimal efficiency in terms of the minimum speed ratio s = {v_{max}}/{v_{min}} and the number of agents k. Our bounds of alpha <= 1/{1 + 1/s} and alpha <= 1 - 1/(sqrt{k)+1} imply that in order to achieve efficiency 1 - epsilon, at least k >= Omega(epsilon^{-2}) agents with a speed ratio of s >= Omega(epsilon^{-1}) are necessary. Guided by our upper bounds, we construct a scheme whose efficiency approaches 1, disproving the conjecture stated above. Our scheme asymptotically matches our upper bounds in terms of the maximal speed difference and the number of agents used.
A variation of the fence patrolling problem considers a circular fence instead and asks for its circumference to be maximized. We consider the unidirectional case of this variation, where all agents are only allowed to move in one direction, say clockwise. At first, a strategy yielding L = max_{r in [k]} r * v_r where v_1 >= v_2 >= ... >= v_k was conjectured to be optimal by Czyzowicz et al. [Czyzowicz et al., 2011] This was proven not to be the case by giving constructions for only specific numbers of agents with marginal improvements of L. We give a general construction that yields L = 1/{33 log_e log_2(k)} sum_{i=1}^k v_i for any set of agents, which in particular for the case 1, 1/2, ..., 1/k diverges as k - > infty, thus resolving a conjecture by Kawamura and Soejima [Kawamura and Soejima, 2015] affirmatively.

Bernhard Haeupler, Fabian Kuhn, Anders Martinsson, Kalina Petrova, and Pascal Pfister. Optimal Strategies for Patrolling Fences. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 144:1-144:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haeupler_et_al:LIPIcs.ICALP.2019.144, author = {Haeupler, Bernhard and Kuhn, Fabian and Martinsson, Anders and Petrova, Kalina and Pfister, Pascal}, title = {{Optimal Strategies for Patrolling Fences}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {144:1--144:13}, 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.2019.144}, URN = {urn:nbn:de:0030-drops-107202}, doi = {10.4230/LIPIcs.ICALP.2019.144}, annote = {Keywords: multi-agent systems, patrolling algorithms} }

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