Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT

2025-04-26

AI Summary

This paper studies the Min-NAE-3-SAT problem within *complete* $k$-CSPs ($k geq 3$): given a complete instance on $n$ variables—where every 3-tuple of variables incurs exactly one nontrivial constraint—the goal is to minimize the number of 3-clauses satisfied *monochromatically* (i.e., all true or all false). This NP-hard problem is addressed for the first time on complete instances with a $mathrm{polylog}(n)$ approximation ratio. The core method introduces a novel Sherali–Adams hierarchy-based iterative rounding framework, which seamlessly integrates *conditioning-based rounding* and *threshold-based rounding*, supported by refined combinatorial analysis. This yields a quasipolynomial-time $mathrm{polylog}(n)$-approximation algorithm. Furthermore, the framework accelerates the decision algorithm for *any* complete Boolean $k$-CSP to polynomial time—significantly extending prior approximation guarantees, which were limited to 2-CSPs or sparse instances.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper studies complete $k$-Constraint Satisfaction Problems (CSPs), where an $n$-variable instance has exactly one nontrivial constraint for each subset of $k$ variables, i.e., it has $inom{n}{k}$ constraints. A recent work started a systematic study of complete $k$-CSPs [Anand, Lee, Sharma, SODA'25], and showed a quasi-polynomial time algorithm that decides if there is an assignment satisfying all the constraints of any complete Boolean-alphabet $k$-CSP, algorithmically separating complete instances from dense instances. The tractability of this decision problem is necessary for any nontrivial (multiplicative) approximation for the minimization version, whose goal is to minimize the number of violated constraints. The same paper raised the question of whether it is possible to obtain nontrivial approximation algorithms for complete Min-$k$-CSPs with $k geq 3$. In this work, we make progress in this direction and show a quasi-polynomial time $ ext{polylog}(n)$-approximation to Min-NAE-3-SAT on complete instances, which asks to minimize the number of $3$-clauses where all the three literals equal the same bit. To the best of our knowledge, this is the first known example of a CSP whose decision version is NP-Hard in general (and dense) instances while admitting a $ ext{polylog}(n)$-approximation in complete instances. Our algorithm presents a new iterative framework for rounding a solution from the Sherali-Adams hierarchy, where each iteration interleaves the two well-known rounding tools: the conditioning procedure, in order to almost fix many variables, and the thresholding procedure, in order to completely fix them. Finally, we improve the running time of the decision algorithms of Anand, Lee, and Sharma and show a simple algorithm that decides any complete Boolean-alphabet $k$-CSP in polynomial time.

Problem

• Develops polylogarithmic approximation for Min-NAE-3-SAT on complete instances.
• Addresses tractability of complete Min-k-CSPs for k ≥ 3.
• Improves decision algorithms for complete Boolean-alphabet k-CSPs.

Innovation

• Quasi-polynomial time polylog(n)-approximation for Min-NAE-3-SAT
• New iterative Sherali-Adams rounding framework
• Improved polynomial-time decision algorithm for complete k-CSPs