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Multi-agent Search-Type Problems on Polygons

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

We present several advancements in search-type problems for fleets of mobile agents operating in two dimensions under the wireless model. Potential hidden target locations are equidistant from a central point, forming either a unit-radius disk (infinite possible locations) or regular polygons (finite possible locations) inscribed in a unit-radius disk. Building and extending on the foundational disk evacuation problem [23], the disk priority evacuation problem with k Servants [21, 27], and the disk w-weighted search problem [49], we make improvements on several fronts. First, we establish new upper and lower bounds for the n-gon priority evacuation problem with 1 Servant for \(n \le 13\), and for \(n_k\)-gons with \(k=2, 3, 4\) Servants, where \(n_2 \le 11\), \(n_3 \le 9\), and \(n_4 \le 10\), offering tight or nearly tight bounds. The only previous results known were a tight upper bound for \(k=1\) and \(n=6\) in [27] and lower bounds for \(k=1\) and \(n \le 9\) in [49]. Second, our work improves the best lower bound known for the disk priority evacuation problem with \(k=1\) Servant from 4.46798 to 4.64666 and for \(k=2\) Servants from 3.6307 of [27] to 3.65332. Third, we improve the best lower bounds known for the disk w-weighted group search problem, significantly reducing the gap between the best upper and lower bounds for w values where the gap was largest. These improvements are based on nearly tight upper and lower bounds for the 11-gon and 12-gon w-weighted evacuation problems, while the previous study of [49] was limited only to lower bounds and only to 7-gons.

Research supported in part by NSERC Discovery grant.

A full version of the paper is available on arXiv [43].

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Notes

  1. 1.

    The possible target placements on a disk are uncountable many, whereas for every n, the vertices of an n-gon are finite many. However, for every \(\epsilon >0\), there is large enough n so that every target placement on the disk is no more than \(\epsilon \) away from a target placement on the n-gon.

  2. 2.

    Our positive and negative results provide only the first 5 digits of our computations, even though our numerical evaluations extend to at least 10 digits of accuracy. Often, we also have closed-form expressions, involving algebraic and trigonometric operations, that describe these numbers. However, for larger values of n, these expressions become too extensive to be informative, and hence we omit them.

  3. 3.

    The lower bounds were computed for values of w from 0 to 1 with step size 0.01.

  4. 4.

    The lower bounds were computed for values of w from 0 to 1 with step size 0.01. The upper bounds were computed for values of w from 0 to 1 with step size 0.02, hence we only depict them as points.

  5. 5.

    For our lower bound arguments, we only need that the the optimal value to \(\textsc {NLP}^n_k(s,\rho )\) is a lower bound to the cost of the optimal \((s,\rho )\)-algorithm.

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  1. Department of Mathematics, Toronto Metropolitan University, Toronto, ON, Canada

    Konstantinos Georgiou, Caleb Jones & Jesse Lucier

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  1. Konstantinos Georgiou
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Correspondence to Konstantinos Georgiou .

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  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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Georgiou, K., Jones, C., Lucier, J. (2025). Multi-agent Search-Type Problems on Polygons. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_23

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