Combinatorial Optimization Algorithms via Polymorphisms

TL;DR

This paper explores how polymorphisms, algebraic operations that combine solutions, relate to the complexity and approximability of constraint satisfaction problems and their generalizations.

Contribution

It extends the concept of polymorphisms to the value-oracle model, providing algorithms for minimization and connecting algebraic properties to problem tractability.

Findings

Designed a randomized algorithm for minimizing functions with certain polymorphisms.

Reformulated the unique games conjecture using approximate polymorphisms.

Established a link between polymorphisms and problem approximability.

Abstract

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 {\Lambda} is tractable if and only if there exist certain non-trivial operations known as polymorphisms to combine solutions to {\Lambda} to create new ones. In an entirely separate line of work, the unique games conjecture yields a characterization of approximability of Max-CSPs. 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 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.