MAJORITY-3SAT (and Related Problems) in Polynomial Time

• URL: http://arxiv.org/abs/2107.02748v1
• Date: Tue, 6 Jul 2021 17:24:04 GMT
• Title: MAJORITY-3SAT (and Related Problems) in Polynomial Time
• Authors: Shyan Akmal and Ryan Williams
• Abstract summary: We give an algorithm that can determine whether a given $k$-CNF has at least $ rho in (0,1)$ with bounded denominator. Our algorithms have interesting positive implications for counting complexity and the complexity of inference.
• Score: 4.035753155957698
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Majority-SAT is the problem of determining whether an input $n$-variable formula in conjunctive normal form (CNF) has at least $2^{n-1}$ satisfying assignments. Majority-SAT and related problems have been studied extensively in various AI communities interested in the complexity of probabilistic planning and inference. Although Majority-SAT has been known to be PP-complete for over 40 years, the complexity of a natural variant has remained open: Majority-$k$SAT, where the input CNF formula is restricted to have clause width at most $k$. We prove that for every $k$, Majority-$k$SAT is in P. In fact, for any positive integer $k$ and rational $\rho \in (0,1)$ with bounded denominator, we give an algorithm that can determine whether a given $k$-CNF has at least $\rho \cdot 2^n$ satisfying assignments, in deterministic linear time (whereas the previous best-known algorithm ran in exponential time). Our algorithms have interesting positive implications for counting complexity and the complexity of inference, significantly reducing the known complexities of related problems such as E-MAJ-$k$SAT and MAJ-MAJ-$k$SAT. At the heart of our approach is an efficient method for solving threshold counting problems by extracting sunflowers found in the corresponding set system of a $k$-CNF. We also show that the tractability of Majority-$k$SAT is somewhat fragile. For the closely related GtMajority-SAT problem (where we ask whether a given formula has greater than $2^{n-1}$ satisfying assignments) which is known to be PP-complete, we show that GtMajority-$k$SAT is in P for $k\le 3$, but becomes NP-complete for $k\geq 4$. These results are counterintuitive, because the ``natural'' classifications of these problems would have been PP-completeness, and because there is a stark difference in the complexity of GtMajority-$k$SAT and Majority-$k$SAT for all $k\ge 4$.

Related papers

• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Local Quantum Search Algorithm for Random $k$-SAT with $Ω(n^{1+ε})$ Clauses [0.0]
  We show that max-k-SSAT is the computational algorithm on average when $m=Omega(n2+delta+epsilon)$ based on the average-case complexity theory. We also prove that max-k-SSAT is on average when $m=Omega(n2+delta+epsilon)$ based on the average-case complexity theory.
  arXiv  Detail & Related papers  (2024-03-05T15:03:47Z)
• Decomposing Hard SAT Instances with Metaheuristic Optimization [52.03315747221343]
  We introduce the notion of decomposition hardness (d-hardness) We show that the d-hardness expresses an estimate of the hardness of $C$ w.r.t.
  arXiv  Detail & Related papers  (2023-12-16T12:44:36Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Pseudo Polynomial-Time Top-k Algorithms for d-DNNF Circuits [0.0]
  We present an algorithm that computes $k$ models of $C$ among those having the largest values w.r. $$. We also present a pseudo-time algorithm that transforms $C$ into a d-DNNF circuit $C'$ satisfied exactly by the models of $C$ having a value among the top-$k$ ones.
  arXiv  Detail & Related papers  (2022-02-11T23:53:43Z)
• The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials [18.00315760946089]
  It is known that a satisfying assignment exists with high probability at clause density $m/n 2k log 2 - frac12. We show that low degree algorithms can find a satisfying assignment at clause density $(1 - o_k(1)) 2k log k / k$.
  arXiv  Detail & Related papers  (2021-06-03T21:01:02Z)
• Learning from Survey Propagation: a Neural Network for MAX-E-$3$-SAT [0.0]
  This paper presents a new algorithm for computing approximate solutions in $Theta(N)$ for the Maximum Exact 3-Satisfiability (MAX-E-$3$-SAT) problem.
  arXiv  Detail & Related papers  (2020-12-10T07:59:54Z)
• List-Decodable Mean Estimation in Nearly-PCA Time [50.79691056481693]
  We study the fundamental task of list-decodable mean estimation in high dimensions. Our algorithm runs in time $widetildeO(ndk)$ for all $k = O(sqrtd) cup Omega(d)$, where $n$ is the size of the dataset. A variant of our algorithm has runtime $widetildeO(ndk)$ for all $k$, at the expense of an $O(sqrtlog k)$ factor in the recovery guarantee
  arXiv  Detail & Related papers  (2020-11-19T17:21:37Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.