Chasing Small Sets Optimally Against Adaptive Adversaries

2026-05-11

AI Summary

This work addresses the online problem of chasing sets of size at most $k$ in metric spaces—equivalently, layered graph traversal with width $k$. By generalizing the classical doubling strategy, it presents the first deterministic online algorithm achieving an $O(2^k)$ competitive ratio against adaptive adversaries. The paper establishes the tight deterministic competitive ratio for this problem as $\Theta(2^k)$, demonstrates that the generalized Work Function Algorithm is suboptimal in this setting, and introduces a novel recursive lower-bound construction $D_k$. Notably, matching upper and lower bounds are provided for the case $k=3$, leading to improved bounds for related problems such as distributed asynchronous tree exploration and the $k$-taxi problem.

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 $Ω(2^k)\cap O(k2^k)$ 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.

Problem

• metrical service systems
• online algorithms
• competitive ratio
• layered graph traversal
• adaptive adversaries

Innovation

• metrical service systems
• competitive analysis
• online algorithms
• adaptive adversaries
• doubling strategy