SODA Conference 2022 Conference Paper
Combinatorial Gap Theorem and Reductions between Promise CSPs
- Libor Barto
- Marcin Kozik
A value of a CSP instance is typically defined as a fraction of constraints that can be simultaneously met. We propose an alternative definition of a value of an instance and show that, for purely combinatorial reasons, a value of an unsolvable instance is bounded away from one; we call this fact a gap theorem. We show that the gap theorem implies NP-hardness of a gap version of the Layered Label Cover Problem. The same result can be derived from the PCP Theorem, but a full, self-contained proof of our reduction is quite short and the result can still provide PCP–free NP–hardness proofs for numerous problems. The simplicity of our reasoning also suggests that weaker versions of Unique-Games-type conjectures, e. g. , the d -to-1 conjecture, might be accessible and serve as an intermediate step for proving these conjectures in their full strength. As the second, main application we provide a sufficient condition under which a fixed template Promise Constraint Satisfaction Problem (PCSP) reduces to another PCSP. The correctness of the reduction hinges on the gap theorem, but the reduction itself is very simple. As a consequence, we obtain that every CSP can be canonically reduced to most of the known NP-hard PCSPs, such as the approximate hypergraph coloring problem.