On Sketching Approximations for Symmetric Boolean CSPs

Authors Joanna Boyland, Michael Hwang, Tarun Prasad , Noah Singer , Santhoshini Velusamy



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Author Details

Joanna Boyland
  • Harvard College, Harvard University, Cambridge, MA, USA
Michael Hwang
  • Harvard College, Harvard University, Cambridge, MA, USA
Tarun Prasad
  • Harvard College, Harvard University, Cambridge, MA, USA
Noah Singer
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
  • Harvard College, Harvard University, Cambridge, MA, USA
Santhoshini Velusamy
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Acknowledgements

All authors of this paper are or were supervised by Madhu Sudan. We would like to thank him for advice and guidance on both technical and organizational aspects of this project, as well as for many helpful comments on earlier drafts of this paper.

Cite AsGet BibTex

Joanna Boyland, Michael Hwang, Tarun Prasad, Noah Singer, and Santhoshini Velusamy. On Sketching Approximations for Symmetric Boolean CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.38

Abstract

A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a predicate f:{-1,1}^k → {0,1}. An n-variable instance of Max-CSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ n-space streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ n-space sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, Max-CSP(f) is (α(f)-ε)-approximable by an O(log n)-space linear sketching algorithm, but (α(f)+ε)-approximation sketching algorithms require Ω(√n) space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2-o(1))2^{-k}-approximated by O(log n)-space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{-k}-approximations require Ω(n) space! We also resolve the ratio for the "at-least-(k-1)-1’s" function for all even k; the "exactly-(k+1)/2-1’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closed-form expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddle-point" properties of this dichotomy. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ n-space streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})-ε)-approximations in o(√ n) space.

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 algorithms
  • constraint satisfaction problems
  • approximability

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