Chasing Small Sets Optimally Against Adaptive Adversaries

TL;DR

This paper presents an optimal deterministic algorithm with an $O(2^k)$ competitive ratio for chasing small sets in metric spaces, resolving a long-standing open problem and improving bounds for related problems.

Contribution

The authors introduce an $O(2^k)$-competitive deterministic algorithm that is optimal against adaptive adversaries, and improve lower bounds for the problem.

Findings

Established an $O(2^k)$-competitive algorithm for small set chasing.

Improved the deterministic lower bound to a recursive sequence $D_k$.

Provided a matching upper bound for the case $k=3$, confirming the bounds' tightness.

Abstract

We study deterministic online algorithms for the problem of chasing sets of cardinality at most $k$ in a metric space, also known as metrical service systems and equivalent to width-$k$ layered graph traversal. We resolve the 30-year-old gap of $\Omega(2^k)\cap O(k2^k)$ on the competitive ratio of this problem by giving an $O(2^k)$-competitive deterministic algorithm. This bound is optimal even among randomized algorithms against adaptive adversaries. We also (slightly) improve the deterministic lower bound to $D_k$, defined recursively by $D_1=1$ and $D_{k+1}=2D_k+\sqrt{8+8D_k}+3$, which we conjecture to be exactly tight. For $k=3$, we provide a matching upper bound of $D_3$. Our results imply slightly improved upper and lower bounds for distributed asynchronous collective tree exploration and for the $k$-taxi problem, respectively. Our algorithm generalizes the classical doubling strategy, previously known to be optimal for $k=2$. The previous best bound for general $k$ was achieved by the generalized work function algorithm (WFA), and was known to be tight for WFA. Our improved bound therefore implies that WFA is sub-optimal for chasing small sets.