Holographic algorithms by Fibonacci gates

Abstract

We introduce Fibonacci gates as a polynomial time computable primitive, and develop a theory of holographic algorithms based on these gates. The Fibonacci gates play the role of matchgates in Valiant's theory (Valiant (2008) [19]). They give rise to polynomial time computable counting problems on general graphs, while matchgates mainly work over planar graphs only. We develop a signature theory and characterize all realizable signatures for Fibonacci gates. For bases of arbitrary dimensions we prove a basis collapse theorem. We apply this theory to give new polynomial time algorithms for certain counting problems. We also use this framework to prove that some slight variations of these counting problems are #P-hard. Holographic algorithms with Fibonacci gates prove to be useful as a general tool for classification results of counting problems (dichotomy theorems [7])).

Holographic reduction, interpolation and hardness

“…, where the recurrences are f i+2 = 0 · f i+1 + f i for the first two, and f i+2 = (−1) · f i+1 + f i for the last two. It is proved in [5] that f i+2 = m · f i+1 + f i can all be reduced by holographic reductions to the basic Fibonacci gates f i+2 = f i+1 + f i . The #P-hardness for Problem PM is proved in [17].…”

“…This implies that |α i β j | < |γ i+j |, a contradiction. Thus when f is irreducible and it has three distinct real roots, there are no non-trivial solutions to (5)…”

“…The second ingredient is to use linear algebra to create exponential sums of perfect matchings in a "holographic mix", and achieve exponential cancelations in the process. In [5] we have introduced another family of P-time computable primitives called Fibonacci gates. Holographic transformations with Fibonacci gates also create exponential cancelations to yield P-time algorithms.…”

A complete dichotomy rises from the capture of vanishing signatures

“…Another is called Constraint Satisfaction Problems (#CSP) [4,3,2,25,1,15,7,11,28,31,12,6]. Recently, inspired by Valiant's holographic algorithms [49,48], a further refined framework called Holant problems [21,20,15,17] was proposed. They all describe classes of counting problems that can be expressed as a sum-of-product computation, specified by a set of local constraint functions F , also called signatures.…”

“…We give a brief description of the Holant framework here [21,20,15,17]. A signature grid Ω = (G, F , π) is a tuple, where G = (V, E) is a graph, π labels each v ∈ V with a function f v ∈ F , and f v maps {0, 1} deg(v) to C. We consider all 0-1 edge…”