Holant clones and the approximability of conservative holant problems

Abstract

We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set $\mathcal{F}$ of functions is "universal in the conservative case", which means that all functions are contained in the holant clone generated by $\mathcal{F}$ together with all unary functions. When $\mathcal{F}$ is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when $\mathcal{F}$ is universal in the conservative case.