Tight Hardness Results for Maximum Weight Rectangles

@article{Backurs2016TightHR,
title={Tight Hardness Results for Maximum Weight Rectangles},
author={A. Backurs and Nishanth Dikkala and Christos Tzamos},
journal={ArXiv},
year={2016},
volume={abs/1602.05837}
}
• Published 2016
• Mathematics, Computer Science
• ArXiv
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… Expand
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References

SHOWING 1-10 OF 54 REFERENCES
D S ] 3 M ar 2 01 6 Tight Hardness Results for Maximum Weight Rectangles
Givenn 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 isExpand
Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
• Mathematics, Computer Science
• STOC
• 2015
Novel reductions from 3-SUM, APSP, and CNF-SAT are designed, and interesting consequences of this very plausible conjecture are derived, including tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs and new conditional lower bound for the Single-Source-Max-Flow problem. Expand
On a class of O(n2) problems in computational geometry
• Computer Science
• Comput. Geom.
• 1995
A large class of problems is described for which it is proved that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0. Expand
Klee's Measure Problem Made Easy
• Timothy M. Chan
• Mathematics, Computer Science
• 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
We present a new algorithm for a classic problem in computational geometry, Klee's measure problem: given a set of n axis-parallel boxes in d-dimensional space, compute the volume of the union of theExpand
Faster all-pairs shortest paths via circuit complexity
A new randomized method for computing the min-plus product of two n × n matrices is presented, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights. Expand
Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems
• Mathematics, Computer Science
• 2014 IEEE 55th Annual Symposium on Foundations of Computer Science
• 2014
It is proved that sufficient progress would imply a breakthrough on one of five major open problems in the theory of algorithms, including dynamic versions of bipartite perfect matching, bipartites maximum weight matching, single source reachability, single sources shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems. Expand
Maximum-Weight Planar Boxes in O(n2) Time (and Better)
• Mathematics, Computer Science
• CCCG
• 2013
Algorithms for the Maximum-Weight Box problem are described in two dimensions that run in the worst case inO(n 2 ) time, and much less on more specic classes of instances. Expand
Subcubic Equivalences between Path, Matrix and Triangle Problems
• Mathematics, Computer Science
• 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
Generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure are shown. Expand
On Dynamic Shortest Paths Problems
• Mathematics, Computer Science
• Algorithmica
• 2010
Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest- Paths problem. Expand
Improved Algorithms for the k Maximum-Sums Problems
• Computer Science, Mathematics
• ISAAC
• 2005
An O(n+k log(min{n, k}))-time algorithm is proposed which is superior to Bengtsson and Chen's when k is o(nlog n), and the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k ≤ n is given. Expand