Tight Hardness Results for Maximum Weight Rectangles

TL;DR

This paper establishes tight conditional lower bounds for the maximum weight rectangle problem in high dimensions, matching the best known algorithms and confirming conjectures about its computational complexity.

Contribution

It provides the first matching lower bounds for the problem, confirming the conjecture that the $O(n^d)$ algorithm is optimal under certain complexity assumptions.

Findings

Matching conditional lower bounds for the maximum weight rectangle problem

Lower bounds for grid-structured points and related problems

Results based on assumptions about APSP and Max-Weight k-Clique problems

Abstract

Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axis-aligned box which maximizes the total weight of the points it contains? The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in time $O(n^d)$. It was conjectured [Barbay et al., CCCG'13] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as Maximum Subarray problem) as well as for other related problems. All our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially optimal.