New Lower Bound and Algorithms for Online Geometric Hitting Set Problem

TL;DR

This work presents an almost tight randomized algorithm with a competitive ratio $O(d^2\log M)$ that significantly improves the best-known competitive ratio of $25^d\log M$.

Abstract

The hitting set problem is one of the fundamental problems in combinatorial optimization and is well-studied in offline setup. We consider the online hitting set problem, where only the set of points is known in advance, and objects are introduced one by one. Our objective is to maintain a minimum-sized hitting set by making irrevocable decisions. Here, we present the study of two variants of the online hitting set problem depending on the point set. In the first variant, we consider the point set to be the entire $\mathbb{Z}^d$, while in the second variant, we consider the point set to be a finite subset of $\mathbb{R}^2$. If you use points in $\mathbb{Z}^d$ to hit homothetic hypercubes in $\mathbb{R}^d$ with side lengths in $[1,M]$, we show that the competitive ratio of any algorithm is $Ω(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $Ω(\log M)$ by a factor of $d$. Then, we present an almost tight randomized algorithm with a competitive ratio $O(d^2\log M)$ that significantly improves the best-known competitive ratio of $25^d\log M$. Next, we propose a simple deterministic ${\lfloor\frac{2}α+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$α$-fat objects} in $\mathbb{R}^d$ having diameters in the range $[1, M]$ using points in $\mathbb{Z}^d$. This improves the current best-known upper bound by a factor of at least $5^d$. Finally, we consider the hitting set problem when the point set consists of $n$ points in $\mathbb{R}^2$, and the objects are homothetic regular $k$-gons having diameter in the range $[1, M]$. We present an $O(\log n\log M)$ competitive randomized algorithm for that. Whereas no result was known even for squares. In particular, our results answer some of the open questions raised by Khan et al. (SoCG'23) and Alefkhani et al. (WAOA'23).

Figures (9)

• Figure 1: Construction of the tree ${\mathcal{T}}$ and ${\mathcal{T}}_B$ when $d=2$ and $M=16$. The red and blue colors represent the $1$st and $2$nd level block nodes, respectively. Here, $c^{1,1}_{1,1}=(0, 0)$, $c^{1,1}_{2,1}=(16, 0)$, $c^{2,1}_{1,1}=(8, 8)$ and so on.
• Figure 2: Illustration in plane to determine the core $cr(\sigma)$ of a hypercube $\sigma\in L_k$. Here, the black colored points denote the integer points in $\mathbb{Z}_k$, where $k\in\mathbb{Z}^{+}\cup\{0\}$. Here, the orange-colored square denotes $\sigma$. The red-colored square denotes the core $cr(\sigma)$, centered at $c$, of $\sigma$. The purple-colored rectangle denotes the convex hull $C$, centered at $c$, of integer points within ${\cal Q}_{s(k)}(\sigma)$. When $\sigma$ contains (a) 4 (b) 6 (c) 9 (both $C$ and $cr(\sigma)$ are the same) points from $\mathbb{Z}_k$.
• Figure 3: (a) Super-square $S$ of a tile $s$ of the partition $P_{\sigma}$, where $\sigma$ is a unit square. (b) Here, the point $v$ is an extreme point i.e. $v\in V_{s,1}$ due to the existence of a unit square $\sigma_v\in {\cal D}_{s,1}$. However, the point $u$ does not belong to $V_{s,1}$ because any unit square containing the points $u$ and $o_1$ must contain the point $v$. (c) Illustration of the shrunk set for $\sigma$, when $\sigma$ is not an integral power of $2$.
• Figure 4: (a) The continuous black-colored regular pentagon is $\sigma$ and the dashed-dotted-dashed black-colored regular pentagon $-\sigma$ is the reflection of $\sigma$ through the center $o\in\sigma$. (b) Illustration of Observation \ref{['Obs:relation']}. (c) Illustration of Observation \ref{['obs:0']}.
• Figure 5: Illustration of Lemma \ref{['lemma:regular']}. (a) Case 1 and (b) Case 2. The red regions denote the two connected components of $\partial\sigma\cap\partial\sigma'$.

Theorems & Definitions (22)

• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Lemma 6
• Claim 7
• Claim 8
• Theorem 9
• Claim 10
• Lemma 11