A Complete Dichotomy Rises from the Capture of Vanishing Signatures
Abstract
We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued 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 expressible in terms of holographic transformations. A Holant problem defined by a set of constraint functions F is solvable in polynomial time if it satisfies this tractability criterion, and is #Phard 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 as both a proof technique and in the statement of the tractability criterion.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.6445
 Bibcode:
 2012arXiv1204.6445C
 Keywords:

 Computer Science  Computational Complexity;
 68Q17;
 F.1.3;
 G.2.1
 EPrint:
 Author accepted manuscript