Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-dimension

TL;DR

This paper presents an efficient approximation algorithm for finding small hitting sets in infinite range spaces with bounded VC-dimension, significantly improving runtime over previous methods.

Contribution

It introduces a method to approximately solve the infinite dimensional convex relaxation using multiplicative weight updates, achieving exponential speedup.

Findings

Algorithm finds hitting sets of size proportional to the smallest epsilon-net size.

Runs in time proportional to the log of the inverse approximation parameter.

Applicable to visibility regions in polygons, providing polynomial-time approximation.

Abstract

We consider the problem of finding a small hitting set in an {\it infinite} range space $\cF=(Q,\cR)$ of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any $\delta>0$, a set of size $O(s_{\cF}(z^*_\cF))$ that hits $(1-\delta)$-fraction of $\cR$ (with respect to a given measure) in time proportional to $\log(\frac{1}{\delta})$, where $s_{\cF}(\frac{1}{\epsilon})$ is the size of the smallest $\epsilon$-net the range space admits, and $z^*_{\cF}$ is the value of the {\it fractional} optimal solution. This {\it exponentially} improves upon previous results which achieve the same approximation guarantees with running time proportional to $\poly(\frac{1}{\delta})$. Our assumptions hold, for instance, in the case when the range space represents the {\it visibility} regions of a polygon in $\RR^2$, giving thus a deterministic polynomial time $O(\log z^*_{\cF})$-approximation algorithm for guarding $(1-\delta)$-fraction of the area of any given simple polygon, with running time proportional to $\polylog(\frac{1}{\delta})$.