Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, October 28, 2021, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Julian Portmann

Simple and Optimal Pebble Guided Treasure Hunt

We study the problem of finding a treasure in an unweighted undirected graph by a mobile agent with advice given in the form of pebbles. This variant of the treasure hunting problem was introduced by Gorain et al. (SIROCCO '21). The vertices are anonymous and the edges incident to a vertex v are labeled from 1 to deg(v). Without prior knowledge of the graph and starting from an arbitrary source vertex s, the agent must find a treasure located on an arbitrary unknown target vertex t in unknown distance D from s.

To speed up the exploration, an oracle that has complete knowledge of the graph, s, and t, is allowed to give advice to the agent in the form of placing pebbles on the vertices of the graph, at most one per vertex. The goal is to design such an oracle and a corresponding treasure hunting algorithm that minimizes the time needed, measured as the number of edge traversals, for the agent to find the treasure. Gorain et al. showed that Ω(D log ∆) steps are necessary and O(D log ∆ + log³ ∆) steps suffice, where ∆ is the maximum degree in the graph. In this work, we show that their lower bound is tight by providing an algorithm that yields a matching upper bound: treasure hunting can be performed in O(D log ∆) steps using O(D log ∆) pebbles as advice.

Joint work with Sebastian Brandt, Davin Choo, Christoph Grunau, and Václav Rozhoň.

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