Combinatorial Optimization Algorithms via Polymorphisms

An elegant characterization of the complexity of constraint satisfaction problems has emerged in the form of the the algebraic dichotomy conjecture of [BKJ00]. Roughly speaking, the characterization asserts that a CSP Λ is tractable if and only if there exist certain non-trivial operations known as polymorphisms to combine solutions to Λ to create new ones. In an entirely separate line of work, the unique games conjecture yields a characterization of approximability of MaxCSPs. Surprisingly, this characterization for Max-CSPs can also be reformulated in the language of polymorphisms. In this work, we study whether existence of non-trivial polymorphisms implies tractability beyond the realm of constraint satisfaction problems, namely in the value-oracle model. Specifically, given a function f in the value-oracle model along with an appropriate operation that never increases the value of f , we design algorithms to minimize f . In particular, we design a randomized algorithm to minimize a function f : [q]n → R that admits a fractional polymorphism which is measure preserving and has a transitive symmetry. We also reinterpret known results on MaxCSPs and thereby reformulate the unique games conjecture as a characterization of approximability of max-CSPs in terms of their approximate polymorphisms. Supported by NSF Career Award and Alfred. P. Sloan Fellowship ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 7 (2015)