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A cut varepsilon-sparsifier of a weighted graph G is a reweighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of varepsilon. Since their introduction by Benczur and Karger [Approximating s-t minimum cuts in O~ (n2) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC96), 1996, pp. 47-55], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIAM J. Discrete Math., 31 (2017), pp. 1263--1276] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains. [ABSTRACT FROM AUTHOR]

We characterize obstruction sets in caterpillar dualities in terms of regular languages and give a construction of the dual of a regular family of caterpillars. In particular, we prove that every monadic linear Datalog program with at most one extensional database per rule defines the complement of a contraint satisfaction problem. [ABSTRACT FROM AUTHOR]

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k ≥ 3: The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity. [ABSTRACT FROM AUTHOR]

Monotone monadic strict NP (MMSNP) is a class of computational problems that is closely related to the class of constraint satisfaction problems for constraint languages over finite domains. It is known that one of those classes has a complexity dichotomy if and only if the other class has. Whereas the dichotomy conjecture has been verified for several subclasses of constraint satisfaction problems, little is known about the the computational complexity for subclasses of MMSNP. In this paper we completely classify the complexity of MMSNP for the case where the obstructions are monochromatic and where loops in the input are forbidden. That is, we determine the computational complexity of natural partition problems of the following type. For fixed sets of finite structures S1,…, Sk, decide whether a given loopless structure can be vertex-partitioned into k parts such that for each i ≤ k none of the structures in Si is homomorphic to the ith part. [ABSTRACT FROM AUTHOR]