Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic twospin
Abstract
We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rationalvalued functions of Boolean variables. A complete classification is known in the conservative case, where $\mathcal{F}$ is assumed to contain arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and adding only those to $\mathcal{F}$: this is weak conservativity. The resulting classification is employed to characterise the complexity of a wide range of twospin problems, fully classifying the ferromagnetic case. In a further weakening of conservativity, we also consider what happens if only the pinning functions are assumed to be in $\mathcal{F}$ (instead of the two permissive unaries). We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a nontrivial example of a set of functions satisfying the monotonicity condition.
 Publication:

arXiv eprints
 Pub Date:
 April 2018
 arXiv:
 arXiv:1804.04993
 Bibcode:
 2018arXiv180404993B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics
 EPrint:
 37 pages