A complete dichotomy rises from the capture of vanishing signatures: extended abstract

  • Authors:

  • Jin-Yi Cai;Heng Guo;Tyson Williams

  • Affiliations:

  • University of Wisconsin-Madison and Peking University, Madison, WI, USA;University of Wisconsin-Madison, Madison, WI, USA;University of Wisconsin-Madison, Madison, WI, USA

  • Venue:
  • Proceedings of the forty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions {F} on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion. A Holant problem defined by a set of constraint functions {F} is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set {F} is tractable if (1) every function in {F} has arity at most two, or (2) {F} is transformable to an affine type, or (3) {F} is transformable to a product type, or (4) {F} is vanishing, combined with the right type of binary functions, or (5) {F} belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean #CSP. Holographic transformations play an indispensable role, not only as a proof technique, but also in the statement of the dichotomy criterion.