New Algorithms and Hardness Results for Robust Satisfiability of (Promise) CSPs

TL;DR

The paper advances the theory of robust satisfiability for Promise CSPs by (i) proving under UGC that Alternating Threshold-based robustness incurs a near-optimal exponential loss, (ii) delivering asymptotically tight, √ε-type robustness for PCSPs with Majority (and extending to Plurality/separable families), and (iii) showing that robustness is preserved under equality gadget reductions, enabling robust algebraic techniques to extend to robust PCSPs. The authors introduce a correlated SDP-rounding framework, including a δ-smoothing REQ that preserves robustness and enables equality-preserving rounding, and they adapt Brown-Cohen–Raghavendra methods to the PCSP setting. The work unifies algorithmic and hardness perspectives, providing a flexible toolkit (ρ-separability, δ-smoothing, and COR rounding) for robust SDP rounding across Boolean and non-Boolean domains, with significant implications for the design and analysis of robust PCSP algorithms under UGC. Overall, the results illuminate the precise robustness losses possible for key polymorphism families and establish a principled, gadget-compatible path to transferring robustness through equality constraints.

Abstract

In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025), and obtain the following results: For the PCSP 1-in-3-SAT vs NAE-SAT with negations, we prove that it is hard, under the Unique Games conjecture (UGC), to satisfy $1-Ω(1/\log (1/ε))$ constraints in a $(1-ε)$-satisfiable instance. This shows that the exponential loss incurred by the BGS algorithm for the case of Alternating-Threshold polymorphisms is necessary, in contrast to the polynomial loss achievable for Majority polymorphisms. For any Boolean PCSP that admits Majority polymorphisms, we give an algorithm satisfying $1-O(\sqrtε)$ fraction of the weaker constraints when promised the existence of an assignment satisfying $1-ε$ fraction of the stronger constraints. This significantly generalizes the Charikar--Makarychev--Makarychev algorithm for 2-SAT, and matches the optimal trade-off possible under the UGC. The algorithm also extends, with the loss of an extra $\log (1/ε)$ factor, to PCSPs on larger domains with a certain structural condition, which is implied by, e.g., a family of Plurality polymorphisms. We prove that assuming the UGC, robust satisfiability is preserved under the addition of equality constraints. As a consequence, we can extend the rich algebraic techniques for decision/search PCSPs to robust PCSPs. The methods involve the development of a correlated and robust version of the general SDP rounding algorithm for CSPs due to (Brown-Cohen, Raghavendra, ICALP 2016), which might be of independent interest.

Theorems & Definitions (117)

• Theorem 1.1: AT hardness, informal
• Theorem 1.2: MAJ robustness, informal
• Theorem 1.3: PLUR robustness, informal
• Theorem 1.5: Robustness of Equality, informal
• Theorem 1.6: Gadget Reductions, informal
• Proposition 2.1: e.g., vershynin2018high, BGS23:stoc
• Proposition 2.2: e.g., vershynin2018high, BGS23:stoc
• proof
• Proposition 2.3: laurent2000adaptive
• Definition 3.1