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TR22-066 | 4th May 2022 23:44

On sketching approximations for symmetric Boolean CSPs



A Boolean maximum constraint satisfaction problem, Max-CSP\((f)\), is specified by a predicate \(f:\{-1,1\}^k\to\{0,1\}\). 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 [CGV20, FOCS 2020] obtained explicit ratios characterizing the \(\sqrt n\)-space streaming approximability of every predicate. For \(k \geq 3\), Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general dichotomy theorem for \(\sqrt n\)-space sketching algorithms: For every \(f\), there exists \(\alpha(f)\in (0,1]\) such that for every \(\epsilon>0\), Max-CSP\((f)\) is \((\alpha(f)-\epsilon)\)-approximable by an \(O(\log n)\)-space linear sketching algorithm, but \((\alpha(f)+\epsilon)\)-approximation sketching algorithms require \(\Omega(\sqrt{n})\) space.

In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting \(\alpha'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}\), we show that for odd \(k \geq 3\), \(\alpha(k\)AND\()\, = \alpha'_k\), and for even \(k \geq 2\), \(\alpha(k\)AND\()\, = 2\alpha'_{k+1}\). Thus, for every \(k\), \(k\)AND 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 [STOC 2022] implying that streaming \((2+\epsilon)\cdot2^{-k}\)-approximations require \(\Omega(n)\) space! We also resolve the ratio for the "at-least-\((k-1)\)-\(1\)'s" function for all even \(k\); the "exactly-\(\frac{k+1}2\)-\(1\)'s" function for odd \(k \in \{3,\ldots,51\}\); and fifteen other functions. We stress here that for general \(f\), the dichotomy theorem in [CGSV21] only implies that \(\alpha(f)\) can be computed to arbitrary precision in \(\textbf{PSPACE}\), 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 [CGV20] while simplifying [CGSV21]. Finally, we investigate the \(\sqrt n\)-space streaming lower bounds in [CGSV21], and show that they are incomplete for \(3\)AND, i.e., they fail to rule out \((\alpha(3\)AND\()-\epsilon)\)-approximations in \(o(\sqrt n)\) space.

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