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Oblivious Algorithms for the Max-kAND Problem

Author Noah G. Singer

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Noah G. Singer
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE2140739. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Acknowledgements

I would like to thank Madhu Sudan and Santhoshini Velusamy for generous feedback and comments on the manuscript; Pravesh Kothari and Peter Manohar for helpful discussions; and anonymous reviewers at APPROX whose feedback helped improve the presentation in the paper.

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Noah G. Singer. Oblivious Algorithms for the Max-kAND Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.15

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Discrete optimization
Keywords
  • streaming algorithm
  • approximation algorithm
  • constraint satisfaction problem (CSP)
  • factor-revealing linear program

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References

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