The Complexity of the Distributed Constraint Satisfaction Problem

Weisfeiler-Leman Invariant Promise Valued CSPs

“…We also point out that our construction for this decomposition is much simpler than the construction used in [5] for the less general setting. The distributed algorithm that we design to prove (iii) ⇒ (i) is completely different from the one used for the CSP in [4]. The original algorithm relied on a deep theorem from the algebraic CSP theory [12] about the strength of a certain local propagation algorithm and designed a distributed version of that algorithm.…”

“…Given two σ-structures A and B, the Promise CSP over (A, B), denoted PCSP(A, B), is defined as follows: given a σ-structure I, output Yes if I is homomorphic to A, and output No if I is not homomorphic to B. 4 This problem makes sense iff the sets of Yes and No instances are disjoint. It is easy to see that this happens exactly when A is homomorphic to B.…”

“…The agents are anonymous, they all run the same deterministic algorithm, and the communication proceeds in synchronous rounds. For a more detailed discussion on the distributed set-up, we refer the reader to [4].…”

“…The class of CSPs decided by BLP (SA 1 ) has reappeared recently in [4], where it was shown that it coincides with the class of CSPs which can be solved on a distributed network. The distributed set-up here is based on the DCSP framework of Yooko et al [26]; informally, each constraint and each variable is controlled by an agent; the communication is only between a constraint and a variable that participates in it; and the agents are anonymous, they communicate in synchronous rounds, and they all run the same deterministic algorithm.…”

“…The papers [4,5] contribute another interesting characterization, by means of an equivalence akin to the 1-dimensional Weisfeiler-Leman graph isomorphism test [17]. For two CSP instances I, J we write I ≡ 1 J if, very roughly, they cannot be distinguished by considering their local structure around variables (number and type of constraints they participate in, number and type of constraints their adjacent variables participate in, and so on).…”