Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems

TL;DR

The paper tackles the unweighted hitting set problem for disks whose centers lie on a line, a line-constrained variant of a classic NP-hard problem. It introduces a line-separable single-intersection framework that generalizes the line-constrained case and develops an algorithm running in $O(m\log^2 n+(n+m)\log(n+m))$ time, significantly improving prior weighted-time bounds and nearly linear in the worst case. The core idea is to prune “useless” points to obtain a reduced instance $P^*$, then reduce the hitting set problem to a 1D interval stabbing problem on projected points, solvable in $O((n+m)\log(n+m))$ time. The correctness proof hinges on a careful inductive pruning argument and a transfer to 1D, with an algebraic decision tree variant achieving $O((n+m)\log(n+m))$ comparisons. The work also covers unit-disk and half-plane special cases and outlines efficient implementation details, including advanced data structures and a constructive proof of correctness for the pruning steps.

Abstract

Given a set $P$ of $n$ points and a set $S$ of $m$ disks in the plane, the disk hitting set problem asks for a smallest subset of $P$ such that every disk of $S$ contains at least one point in the subset. The problem is NP-hard. In this paper, we consider a line-constrained version in which all disks have their centers on a line. We present an $O(m\log^2n+(n+m)\log(n+m))$ time algorithm for the problem. This improves the previously best result of $O(m^2\log m+(n+m)\log(n+m))$ time for the weighted case of the problem where every point of $P$ has a weight and the objective is to minimize the total weight of the hitting set. Our algorithm actually solves a more general line-separable problem with a single intersection property: The points of $P$ and the disk centers are separated by a line $\ell$ and the boundary of every two disks intersect at most once on the side of $\ell$ containing $P$.

Figures (8)

• Figure 1: Illustrating the line-separable single-intersection case: Centers of all disks are below $\ell$.
• Figure 2: Illustrating the proof of Observation \ref{['obser:hitprune']}.
• Figure 3: Illustrating the Observation \ref{['obser:fifo10']}. Left: $s\prec s'$. Right: $s'\prec s$.
• Figure 4: Illustrating the Observation \ref{['obser:diskS_star']}: If $x(\hat{p}_{j+1}) < x(\hat{p}_j)$, then $s^* \prec \hat{s}_j$.
• Figure 5: Illustrating the case $k=i-1$.

Theorems & Definitions (20)

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