Problem
Research questions and friction points this paper is trying to address.
Semidefinite Programming
Promise Constraint Satisfaction Problems
Threshold Polymorphisms
SDPs and Robust Satisfiability of Promise CSP
🤖 AI Summary
This work investigates the robust satisfiability problem for Promise Constraint Satisfaction Problems (PCSPs), focusing on the expressive power and limitations of semidefinite programming (SDP) algorithms. For symmetric Boolean PCSPs, we establish a complete computational complexity classification. We prove that SDP achieves robust approximation precisely when the PCSP admits either a majority or an alternating threshold polymorphism; moreover, we provide the first algebraic necessary and sufficient condition—formulated via minion homomorphisms—for SDP feasibility to imply exact satisfiability. Innovatively, we introduce spherical Ramsey theory into PCSP analysis, revealing a deep connection between SDP integrality gaps and spherical coloring unsatisfiability. This yields the first algebraic-geometric method for proving SDP hardness gaps. Our results unify and extend robust satisfiability theory to the promise setting, delivering systematic criteria for SDP tractability in PCSPs and proposing a central conjecture governing its scope.
📝 Abstract
For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semi Definite Programming(SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our method involves a novel method to argue SDP gaps via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Semidefinite Programming
Promise Constraint Satisfaction Problems
Algebraic Techniques
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