A combinatorial view of Holant problems on higher domains
Yin Liu
TL;DR
This work extends the tractability of Fibonacci gates from the Boolean domain to higher domains by defining generalized Fibonacci gates on domain sizes $3$ and $4$ and proving polynomial-time algorithms for Holant problems when the constraint sets consist of such gates. The approach relies on an orthogonal holographic transformation to represent domain-3 signatures as $g=α^{⊗3}+β^{⊗3}+γ^{⊗3}$ with mutually orthogonal vectors and a parameterized recurrence $(s,x,y,t)$ satisfying $sy+xt+1=x^2+y^2$, together with a contraction-based edge-merging procedure that preserves the generalized Fibonacci structure. For domain $4$, the method generalizes to $g=α^{⊗3}+β^{⊗3}+γ^{⊗3}+δ^{⊗3}$ with ten parameters $(a,b,c,d,e,f,h,i,j,p)$ constrained by equations (eq_d4q) and (eq_d4p, eq_d4r)$, and analogous depth-2 tetrahedral recurrences ensure the merge steps keep the gate within the generalized Fibonacci family. Consequently, Holant$( ilde{ or F})$ is computable in polynomial time for any finite set of such gates on both domain sizes $3$ and $4$, and the work motivates a conjecture that Fibonacci gates—and corresponding P-time algorithms—exist on all higher domains.
Abstract
On the Boolean domain, there is a class of symmetric signatures called ``Fibonacci gates'' for which a beautiful P-time combinatorial algorithm has been designed for the corresponding $\operatorname{Holant}$ problems. In this work, I give a combinatorial view for $\operatorname{Holant}(\mathcal{F})$ problems on a domain of size 3 where $\mathcal{F}$ is a set of arity 3 functions with inputs taking values on the domain of size 3 and the functions share some common properties. The combinatorial view can also be extended to the domain of size 4. Specifically, I extend the definition of "Fibonacci gates" to the domain of size 3 and the domain of size 4. Moreover, I give the corresponding combinatorial algorithms.