A Bouquet of Results on Maximum Range Sum: General Techniques and Hardness Reductions

2025-09-19

AI Summary

This paper studies three important variants of the Maximum Range Sum (MaxRS) problem. (1) Dynamic MaxRS: supports insertions and deletions of points; we propose the first $(1/2-varepsilon)$-approximation algorithm with $O_varepsilon(log n)$ update time. (2) Batched MaxRS: we establish a conditional lower bound on computational complexity and provide a matching, efficient algorithm. (3) Colored MaxRS: generalizes to $d$-dimensional balls while avoiding the curse of dimensionality; we introduce color-aware random sampling and an output-sensitive exact algorithm, achieving $(1-varepsilon)$-approximation in expected $O_varepsilon(n log n)$ time for the 2D case. All results hold with high probability. Collectively, these contributions significantly advance the state of the art in MaxRS research—extending its applicability to dynamic settings, establishing foundational theoretical lower bounds, and enabling multi-attribute (color-based) coverage in higher dimensions.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the maximum range sum (MaxRS) problem: given a set $P$ of $n$ weighted points in $mathbb{R}^d$ and a range $Q$ (typically axis-aligned $d$-box or $d$-ball), the goal is to place $Q$ to maximize the total weight of points in $Pcap Q$. We study three natural variations: (1) Dynamic MaxRS: The goal is to update the placement of a $d$-ball under point insertions and deletions. We give a randomized $(frac{1}{2}-ε)$-approximation with update time $O_ε(log n)$. The approximation factor holds with high probability. To the best of our knowledge, this is the first result on dynamic MaxRS. (2) Batched MaxRS: In $mathbb{R}^1$, along with $P$ we are given $m$ intervals of varying lengths. We prove a conditional lower bound of $Ω(mn)$ time (via conjectured $(min,+)$-convolution hardness), showing the trivial $O(mnlog n)$ upper bound in $mathbb{R}^2$ is essentially tight. We also establish a similar bound for a related problem of batched smallest $k$-enclosing interval. (3) Colored MaxRS: Each point has a color from $[m]$, and the goal is to place $Q$ to maximize the number of uniquely colored points in $Pcap Q$. Prior work only considered axis-aligned rectangles in $mathbb{R}^2$. For $d$-balls, we give: (a) a randomized $(frac{1}{2}-ε)$-approximation in $O_ε(nlog n)$ time (avoiding exponential dependence on $d$), and (b) in $mathbb{R}^2$, a $(1-ε)$-approximation in expected $O_ε(nlog n)$ time. Both approximations hold with high probability. Our algorithms rely on two techniques of broader interest. The first yields $(frac{1}{2}-ε)$-approximations via a volume argument on $d$-balls and a randomized game. The second achieves $(1-ε)$-approximations through an exact output-sensitive algorithm, which we speed up by random sampling on colors.

Problem

• Dynamic MaxRS: updating d-ball placement under point insertions and deletions
• Batched MaxRS: proving conditional lower bounds for interval queries in R1
• Colored MaxRS: maximizing uniquely colored points in d-balls with approximations

Innovation

• Randomized approximation with volume arguments
• Conditional lower bounds via convolution hardness
• Output-sensitive algorithm with color sampling