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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.15
URN: urn:nbn:de:0030-drops-188409
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18840/
Singer, Noah G.
Oblivious Algorithms for the Max-kAND Problem
Abstract
Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-kAND problem. This is a class of simple, combinatorial algorithms which round each variable with probability depending only on a quantity called the variable’s bias. Our definition generalizes a class of algorithms defined by Feige and Jozeph (Algorithmica '15) for Max-DICUT, a special case of Max-2AND.
For each oblivious algorithm, we design a so-called factor-revealing linear program (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all k, oblivious algorithms for Max-kAND provably outperform a special subclass of algorithms we call "superoblivious" algorithms.
Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every k, O(log n)-space sketching algorithms for Max-kAND known to be optimal in o(√n)-space can be beaten in (a) O(log n)-space under a random-ordering assumption, and (b) O(n^{1-1/k} D^{1/k}) space under a maximum-degree-D assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.
BibTeX - Entry
@InProceedings{singer:LIPIcs.APPROX/RANDOM.2023.15,
author = {Singer, Noah G.},
title = {{Oblivious Algorithms for the Max-kAND Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
pages = {15:1--15:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-296-9},
ISSN = {1868-8969},
year = {2023},
volume = {275},
editor = {Megow, Nicole and Smith, Adam},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18840},
URN = {urn:nbn:de:0030-drops-188409},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.15},
annote = {Keywords: streaming algorithm, approximation algorithm, constraint satisfaction problem (CSP), factor-revealing linear program}
}
