Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tight bounds for undirected graph exploration with pebbles and multiple
  agents   We study the problem of deterministically exploring an undirected and
initially unknown graph with $n$ vertices either by a single agent equipped
with a set of pebbles, or by a set of collaborating agents. The vertices of the
graph are unlabeled and cannot be distinguished by the agents, but the edges
incident to a vertex have locally distinct labels. The graph is explored when
all vertices have been visited by at least one agent. In this setting, it is
known that for a single agent without pebbles $\Theta(\log n)$ bits of memory
are necessary and sufficient to explore any graph with at most $n$ vertices. We
are interested in how the memory requirement decreases as the agent may mark
vertices by dropping and retrieving distinguishable pebbles, or when multiple
agents jointly explore the graph. We give tight results for both questions
showing that for a single agent with constant memory $\Theta(\log \log n)$
pebbles are necessary and sufficient for exploration. We further prove that the
same bound holds for the number of collaborating agents needed for exploration.
  For the upper bound, we devise an algorithm for a single agent with constant
memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$
pebbles, even when $n$ is unknown. The algorithm terminates after polynomial
time and returns to the starting vertex. Since an additional agent is at least
as powerful as a pebble, this implies that $\mathcal{O}(\log \log n)$ agents
with constant memory can explore any $n$-vertex graph. For the lower bound, we
show that the number of agents needed for exploring any graph of size $n$ is
already $\Omega(\log \log n)$ when we allow each agent to have at most
$\mathcal{O}( \log n ^{1-\varepsilon})$ bits of memory for any $\varepsilon>0$.
This also implies that a single agent with sublogarithmic memory needs
$\Theta(\log \log n)$ pebbles to explore any $n$-vertex graph.

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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: math
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph