Low-Rank Semidefinite Programming for the MAX2SAT Problem

Wang, Po-Wei; Kolter, J. Zico

doi:10.1609/aaai.v33i01.33011641

Cited by 9 publications

(13 citation statements)

References 24 publications

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“…It minimizes J sat w (4) by Newton's method using (5). In addition to Set-E and Set-F, we use another test set Set-G from SATLIB [14] in the experiment 29 containing 100 instances of 5-SAT with m = 3100 clauses in n = 500 variables. The experimental result is summarized in Table 4.…”

Section: Weighted Variables and Clausesmentioning

confidence: 99%

“…Wang et. al built a MAXSAT solver based on the combination of semidefinite programming relaxation and branch-and-bound strategy [29]. Selsam et al proposed a neural net classifier NeuroSAT for SAT problems that learn embeddings of a SAT problem through three perceptrons and two LSTMs so that the system predicts one bit, i.e., the satisfiability of the problem [27].…”

Section: Introductionmentioning

confidence: 99%

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MatSat: a matrix-based differentiable SAT solver

Sato,

Kojima

2021

Preprint

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We propose a new approach to SAT solving which solves SAT problems in vector spaces as cost minimization of a differentiable cost function J sat . In our approach, a solution, satisfying assignment, of a SAT problem in n variables is represented by a binary vector u ∈ {0, 1} n such that J sat (u) = 0. We search for such u in a vector space R n by cost minimization, i.e., starting from an initial u 0 , we minimize J sat to zero while iteratively updating u by Newton's method. We implemented our approach as an incomplete matrix-based differentiable SAT solver MatSat. Although existing main-stream SAT solvers decide each bit of a solution assignment one by one, be they of conflict driven clause learning (CDCL) type or of stochastic local search (SLS) type, MatSat fundamentally differs from them in that it updates all variables at once and continuously approaches a solution in a vector space. We conducted experiments to measure the scalability of MatSat with random 3-SAT problems. In these experiments, for example, we showed that MatSat implemented on GPU can solve the problem with n = 3 × 10 5 variables, demonstrating the feasibility of hardware acceleration by GPU for matrix-based solvers like MatSat. We also compared MatSat with nine state-of-theart CDCL and SLS SAT solvers in terms of execution time by conducting experiments with several random and non-random data sets. In the case of easy random SAT, the performance of MatSat comes between the SLS solvers and the CDCL solvers whereas it is ranked 1st on the difficult one. On the other hand, MatSat showed poor performance on non-random SAT problems. To improve its poor performance, we introduced weighted variables and clauses and confirmed the effectiveness of the weighted version of MatSat on non-random SAT.

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Section: Weighted Variables and Clausesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

MatSat: a matrix-based differentiable SAT solver

Sato,

Kojima

2021

Preprint

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show abstract

“…Besides, several efforts have been put on relaxing logical constraints such as soft constraint learning [36]. [43] use semidefinite programming (SDP) [42] to relax the maxSAT problem and cooperate with deep learning. Another way to relax the logical constraints is to relax the Boolean variables to be continuous variables [23,22], or continuous random variables like probabilistic soft logic [19,1,13].…”

Section: Related Workmentioning

confidence: 99%

An Integer Linear Programming Framework for Mining Constraints from Data

Meng¹,

Chang²

2020

Preprint

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Various structured output prediction problems (e.g., sequential tagging) involve constraints over the output space. By identifying these constraints, we can filter out infeasible solutions and build an accountable model. To this end, we present a general integer linear programming (ILP) framework for mining constraints from data. We model the inference of structured output prediction as an ILP problem. Then, given the coefficients of the objective function and the corresponding solution, we mine the underlying constraints by estimating the outer and inner polytopes of the feasible set. We verify the proposed constraint mining algorithm in various synthetic and real-world applications and demonstrate that the proposed approach successfully identifies the feasible set at scale. In particular, we show that our approach can learn to solve 9x9 Sudoku puzzles and minimal spanning tree problems from examples without providing the underlying rules. We also demonstrate results on hierarchical multi-label classification and conduct a theoretical analysis on how close the mined constraints are from the ground truth.Preprint. Under review.

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“…More recent work, e.g. Wang et al (2017); Wang & Kolter (2019), has developed low-rank SDP solvers for general MAXSAT problems. We extend the work of Wang et al (2017) to create a differentiable optimization-based MAXSAT solver that can be employed in the loop of deep learning.…”

Section: Related Workmentioning

confidence: 99%

SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver

Wang¹,

Donti²,

Wilder³

et al. 2019

Preprint

Self Cite

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Integrating logical reasoning within deep learning architectures has been a major goal of modern AI systems. In this paper, we propose a new direction toward this goal by introducing a differentiable (smoothed) maximum satisfiability (MAXSAT) solver that can be integrated into the loop of larger deep learning systems. Our (approximate) solver is based upon a fast coordinate descent approach to solving the semidefinite program (SDP) associated with the MAXSAT problem. We show how to analytically differentiate through the solution to this SDP and efficiently solve the associated backward pass. We demonstrate that by integrating this solver into end-to-end learning systems, we can learn the logical structure of challenging problems in a minimally supervised fashion. In particular, we show that we can learn the parity function using single-bit supervision (a traditionally hard task for deep networks) and learn how to play 9 × 9 Sudoku solely from examples. We also solve a "visual Sudoku" problem that maps images of Sudoku puzzles to their associated logical solutions by combining our MAXSAT solver with a traditional convolutional architecture. Our approach thus shows promise in integrating logical structures within deep learning.

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Low-Rank Semidefinite Programming for the MAX2SAT Problem

Cited by 9 publications

References 24 publications

MatSat: a matrix-based differentiable SAT solver

MatSat: a matrix-based differentiable SAT solver

An Integer Linear Programming Framework for Mining Constraints from Data

SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver

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