Tight Hardness Results for Maximum Weight Rectangles

@article{Backurs2016TightHR,
  title={Tight Hardness Results for Maximum Weight Rectangles},
  author={Arturs Backurs and Nishanth Dikkala and Christos Tzamos},
  journal={ArXiv},
  year={2016},
  volume={abs/1602.05837}
}
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… 

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References

SHOWING 1-10 OF 53 REFERENCES
D S ] 3 M ar 2 01 6 Tight Hardness Results for Maximum Weight Rectangles
TLDR
This paper provides conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as M AXIMUM SUBARRAY problem) as well as for other related problems, based on assumptions that the best own algorithms for the ALL -PAIRS SHORTEST PATHS problem (APSP) and for the M AX -WEIGHT k-CLIQUE problem in edge-weighted graphs are essentially optimal.
Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
TLDR
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.
On a class of O(n2) problems in computational geometry
Klee's Measure Problem Made Easy
  • Timothy M. Chan
  • 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 the
Faster all-pairs shortest paths via circuit complexity
TLDR
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.
Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems
TLDR
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.
On Dynamic Shortest Paths Problems
TLDR
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.
...
...