Dichotomy for approximating satisfiable CSPs

Show a dichotomy for approximating satisfiable CSPs: for all k ∈ N, finite alphabets E, and collections P of k-ary predicates over E with P-CSP NP-hard, determine an s ∈ (0,1) such that (1) there is a polynomial-time algorithm for gap-P-CSP[1, s], and (2) for all ε > 0, gap-P-CSP[1, s + ε] is NP-hard (possibly assuming the Rich 2-to-1 Games Conjecture); alternatively, construct a dictatorship test for P with completeness 1 and soundness s + ε.

Background

Raghavendra’s dichotomy characterizes approximability in the almost-satisfiable regime (c < 1), but the satisfiable regime (c = 1) remains poorly understood. The paper’s analytic machinery yields progress for certain predicate classes, yet a general dichotomy remains elusive.

Resolving this would unify algorithmic and hardness results for satisfiable CSPs, potentially via new dictatorship tests and structural analyses tied to k-wise correlations.

References

We finish this article by mentioning a few open problems for future research. PROBLEM 6.4. Show that the following dichotomy result: for all k E N, finite alphabet E and P collection of k-ary predicates over E such that P-CSP is NP-hard, there is an s € (0, 1) such that:

  1. Algorithm: there is a polynomial time algorithm for gap-P-CSP[1, s].
  2. Hardness: for all 8, the problem gap-P-CSP[1, s +8] is NP-hard, possibly assuming a conjecture such as the Rich 2-to-1. Games Conjecture [17]. A bit less ambitiously, show a dictatorship test for P with completeness 1 and soundness s + 8.
— The Lens of Abelian Embeddings  (2602.22183 - Minzer, 25 Feb 2026) in Problem 6.4, Section 6 (Open Problems)