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

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 $P\cap Q$. 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 $(\frac{1}{2}-\epsilon)$-approximation with update time $O_\epsilon(\log n)$. 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 $\Omega(mn)$ time (via conjectured $(\min,+)$-convolution hardness), showing the trivial $O(mn\log n)$ 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 $P\cap Q$. Prior work only considered axis-aligned rectangles in $\mathbb{R}^2$. For $d$-balls, we give: (a) a randomized $(\frac{1}{2}-\epsilon)$-approximation in $O_\epsilon(n\log n)$ time (avoiding exponential dependence on $d$), and (b) in $\mathbb{R}^2$, a $(1-\epsilon)$-approximation in expected $O_\epsilon(n\log n)$ time. Both approximations hold with high probability. Our algorithms rely on two techniques of broader interest. The first yields $(\frac{1}{2}-\epsilon)$-approximations via a volume argument on $d$-balls and a randomized game. The second achieves $(1-\epsilon)$-approximations through an exact output-sensitive algorithm, which we speed up by random sampling on colors.