[Paper Review] DNF Sparsification and a Faster Deterministic Counting Algorithm

Parikshit Gopala, Raghu Meka+1 more|arXiv (Cornell University)|May 16, 2012

Complexity and Algorithms in Graphs28 references4 citations

TL;DR

This paper introduces a novel sparsification technique for DNF formulas, proving that any width-$w$ DNF can be $\varepsilon$-approximated by a width-$w$ DNF with only $(w\log(1/\varepsilon))^{O(w)}$ terms. This leads to a faster deterministic algorithm for approximating the fraction of satisfying assignments of a DNF, running in $n^{\tilde{O}(\log\log n)}$ time for $\varepsilon = 1/\mathrm{poly}(\log n)$, significantly improving upon the prior $n^{\exp(O(\sqrt{=\log\log n}))}$ bound.

ABSTRACT

Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be $ε$-approximated by a width $w$ DNF with at most $(w\log(1/ε))^{O(w)}$ terms. We combine our sparsification result with the work of Luby and Velikovic to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic $n^{ ilde{O}(\log \log(n))}$ time algorithm that computes an additive $ε$ approximation to the fraction of satisfying assignments of f for $ε= 1/\poly(\log n)$. The previous best result due to Luby and Velickovic from nearly two decades ago had a run-time of $n^{\exp(O(\sqrt{\log \log n}))}$.

Motivation & Objective

• To develop a method for sparsifying DNF formulas while preserving width and approximation quality.
• To improve the runtime of deterministic algorithms for approximating the number of satisfying assignments of a DNF formula.
• To establish the existence of $\varepsilon$-sandwiching approximators for DNFs using only width-$w$ DNFs of size depending only on $w$ and $\varepsilon$.
• To provide a stronger alternative to Friedgut’s junta theorem in the context of DNFs with bounded width.

Proposed method

• The authors use a sparsification procedure based on the notion of quasi-sunflowers from Rossman (2010) to construct approximating DNFs.
• They prove that any width-$w$ DNF admits $\varepsilon$-sandwiching approximators that are also width-$w$ DNFs of size at most $(w\log(1/\varepsilon))^{O(w)}$.
• The construction leverages a structural result on DNFs that allows for efficient approximation using only a small number of terms.
• The method combines this sparsification result with the framework of Luby and Velikovic (1991, 1996) to design a faster deterministic counting algorithm.
• The algorithm runs in $n^{\tilde{O}(\log\log n)}$ time for $\varepsilon = 1/\mathrm{poly}(\log n)$, improving over the previous $n^{\exp(O(\sqrt{\log\log n}))}$ bound.

Experimental results

Research questions

• RQ1 Can narrow DNF formulas be sparsified into significantly shorter DNFs while preserving approximation quality?
• RQ2 Is it possible to construct $\varepsilon$-sandwiching approximators for DNFs using only width-$w$ DNFs of size polynomial in $w$ and $\log(1/\varepsilon)$?
• RQ3 Can the deterministic counting algorithm for DNFs be accelerated beyond the $n^{\exp(O(\sqrt{\log\log n}))}$ runtime of Luby and Velikovic?
• RQ4 Does a stronger form of Mansour’s conjecture hold for DNFs of bounded width, with sparse Fourier approximations of size $2^{O(w\log(1/\varepsilon))}$?

Key findings

• Any width-$w$ DNF admits $\varepsilon$-sandwiching approximators that are also width-$w$ DNFs of size at most $(w\log(1/\varepsilon))^{O(w)}$.
• The deterministic DNF counting algorithm runs in $n^{\tilde{O}(\log\log n)}$ time for $\varepsilon = 1/\mathrm{poly}(\log n)$, improving over the prior $n^{\exp(O(\sqrt{\log\log n}))}$ bound.
• Every width-$w$ DNF is $\varepsilon$-approximated by a $(w\log(1/\varepsilon))^{O(w)}$-junta, providing a strong junta theorem for bounded-width DNFs.
• The result implies that every width-$O(\log n)$ DNF on $n$ variables is $n^{-O(1)}$-close to a DNF of size $n^{O(\log\log n)}$ and width $O\left(\log n\right)$.
• The sparsification result is tight up to the $O(w)$ exponent, and a conjecture is proposed that would improve the bound to singly exponential in $w$.