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 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 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 #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 as both a proof technique and in the statement of the tractability criterion.

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

“…These partition functions can be seen as Holant problems; see e.g. [15,16,14]. They can also be seen as tensor network contractions.…”

“…Just as for partition functions for spin models much work has been done to establish a complexity dichotomy result for exactly computing Holant problems; see [15,16,14]. Not much is known about the complexity of approximating partition functions of edge-coloring models except for a few special cases.…”

“…Often the pair (G, H) is called a signature grid, cf. [15,16,14]. Then we can extend the definition of the partition function of an edge-coloring model as follows:…”

Dichotomy for Real Holantc Problems

“…A dichotomy for #CSP c 2 is somewhat unavoidable to get a dichotomy for Holant. This is not only logically true in the sense that a dichotomy for Holant will imply a dichotomy for #CSP c 2 , but also true in the sense that one usually proves a dichotomy #CSP c 2 as a major step toward a dichotomy of Holant [22,6]. Previously one could only prove dichotomy for #CSP c 2 for symmetric functions.…”

“…Consequently, it is also much more challenging to prove dichotomy theorems in the Holant framework. After a great deal of work [9,7,19,22,6], a dichotomy for Holant problems was proved for symmetric constraint functions. But obviously symmetric functions are only a tiny fraction of all constraint functions.…”