License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.3
URN: urn:nbn:de:0030-drops-153869
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15386/
An, Haozhe ;
Gurumukhani, Mohit ;
Impagliazzo, Russell ;
Jaber, Michael ;
Künnemann, Marvin ;
Nina, Maria Paula Parga
The Fine-Grained Complexity of Multi-Dimensional Ordering Properties
Abstract
We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points.
Focusing on constant dimension d, we show that any k-quantifier, d-dimensional such problem is solvable in O(n^{k-1} log^{d-1} n) time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, (3k-3)-dimensional problem in this class that requires time Ω(n^{k-1-o(1)}).
Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time O(nlog^{d-1} n), and k-quantifier problems with k > 3 reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination VCND_d (Given three sets of vectors X,Y and Z of dimension d,d and 2d, respectively, is there an x ∈ X and a y ∈ Y so that their concatenation x∘y is not dominated by any z ∈ Z, where vector u is dominated by vector v if u_i ≤ v_i for each coordinate 1 ≤ i ≤ d), and determine it as the "unique" candidate to be complete for this class (under fine-grained assumptions).
BibTeX - Entry
@InProceedings{an_et_al:LIPIcs.IPEC.2021.3,
author = {An, Haozhe and Gurumukhani, Mohit and Impagliazzo, Russell and Jaber, Michael and K\"{u}nnemann, Marvin and Nina, Maria Paula Parga},
title = {{The Fine-Grained Complexity of Multi-Dimensional Ordering Properties}},
booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-216-7},
ISSN = {1868-8969},
year = {2021},
volume = {214},
editor = {Golovach, Petr A. and Zehavi, Meirav},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15386},
URN = {urn:nbn:de:0030-drops-153869},
doi = {10.4230/LIPIcs.IPEC.2021.3},
annote = {Keywords: Fine-grained complexity, First-order logic, Orthogonal vectors}
}
|
Keywords: |
|
Fine-grained complexity, First-order logic, Orthogonal vectors |
|
Collection: |
|
16th International Symposium on Parameterized and Exact Computation (IPEC 2021) |
|
Issue Date: |
|
2021 |
|
Date of publication: |
|
22.11.2021 |
