Promise Constraint Satisfaction: Algebraic Structure and a Symmetric
Boolean Dichotomy
release_xi7bthoilvavdmiwx4abxho4qm
by
Joshua Brakensiek, Venkatesan Guruswami
2017
Abstract
A classic result due to Schaefer (1978) classifies all constraint
satisfaction problems (CSPs) over the Boolean domain as being either in
P or NP-hard. This paper considers a promise-problem
variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints
Γ consists of a pair (Ψ_P, Ψ_Q) of CSPs with the same set of
variables such that for every (P, Q) ∈Γ, P(x_i_1, ..., x_i_k)
is a clause of Ψ_P if and only if Q(x_i_1, ..., x_i_k) is a clause
of Ψ_Q. The promise problem PCSP(Γ) is to
distinguish, given (Ψ_P, Ψ_Q), between the cases Ψ_P is
satisfiable and Ψ_Q is unsatisfiable. Many natural problems including
approximate graph and hypergraph coloring can be placed in this framework.
This paper is motivated by the pursuit of understanding the computational
complexity of Boolean promise CSPs. As our main result, we show that
PCSP(Γ) exhibits a dichotomy (it is either polynomial
time solvable or NP-hard) when the relations in Γ are
symmetric and allow for negations of variables. We achieve our dichotomy
theorem by extending the weak polymorphism framework of Austrin, Guruswami, and
Hå stad [FOCS '14] which itself is a generalization of the algebraic approach
to study CSPs. In both the algorithm and hardness portions of our proof, we
incorporate new ideas and techniques not utilized in the CSP case.
Furthermore, we show that the computational complexity of any promise CSP
(over arbitrary finite domains) is captured entirely by its weak polymorphisms,
a feature known as Galois correspondence, as well as give necessary and
sufficient conditions for the structure of this set of weak polymorphisms. Such
insights call us to question the existence of a general dichotomy for Boolean
PCSPs.
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