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Best-of-Both-Worlds Analysis of Online Search

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Abstract

In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the target’s position is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice. Motivated by the above observation, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.

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Notes

  1. In [18], online search refers to the problem of selling a specific item at the highest possible price, and is not related to the problem of searching for a target.

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Funding

This work was partially funded by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR), DIM RFSI DACM and Labex Mathématique Hadamard. This research also benefited from the support of the FMJH Program PGMO and from the support of EDF-Thales-Orange.

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Authors and Affiliations

  1. LIP6, Sorbonne Université, CNRS, 4 place Jussieu, Paris, 75252, France

    Spyros Angelopoulos & Christoph Dürr

  2. Murex Inc., 25A Bd de l’Amiral Bruix, Paris, 75016, France

    Shendan Jin

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  1. Spyros Angelopoulos
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  2. Christoph Dürr
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  3. Shendan Jin
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Correspondence to Spyros Angelopoulos.

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Angelopoulos, S., Dürr, C. & Jin, S. Best-of-Both-Worlds Analysis of Online Search. Algorithmica 85, 3766–3792 (2023). https://doi.org/10.1007/s00453-023-01165-5

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