Holant Problems for Regular Graphs with Complex Edge Functions

In this paper we consider the following subclass of Holant Problems [FOCS08, TAMC]. An input regular graph $G=(V,E)$ is given, where every $e\in E$ is labeled with a (symmetric) edge function $g$. The function $g$ takes 0-1 inputs from its incident nodes and outputs arbitrary values in $\mathbb{C}$. The problem is to compute the quantity ${\rm Holant}(G)=\sum_{\sigma:V\rightarrow\{0,1\}}\prod_{\{u,v\}\in E}g(\{\sigma(u),\sigma(v)\})$.

Holant Problems are a natural class of counting problems. As introduced in [FOCS08, TAMC], the general Holant Problem framework can encode all Counting Constraint Satisfaction Problems (#CSP). This includes special cases such as weighted Vertex Cover, Graph Colorings, Matchings, and Perfect Matchings. The subclass of Holant Problems in this paper can also be considered as (weighted) $H$-homomorphism (or $H$-coloring) problems [BulatovG05, Homomorphisms, acyclic, DyerG00, Goldberg-4, Hell] with an arbitrary $2\times 2$ symmetric complex matrix $H$, however restricted to regular graphs $G$ as input. E.g., Vertex Cover is the case when $H={\left[\begin{array}[]{cc}0&1\\ 1&1\end{array}\right]}$. When the matrix $H$ is a 0-1 matrix, it is called unweighted. Dichotomy theorems (i.e., the problem is either in ${\rm{P}}$ or #P-hard, depending on $H$) for unweighted $H$-homomorphisms with undirected graphs $H$ and directed acyclic graphs $H$ are given in [DyerG00] and [acyclic] respectively. A dichotomy theorem for any symmetric matrix $H$ with non-negative real entries is proved in [BulatovG05]. Goldberg et al. [Goldberg-4] proved a dichotomy theorem for all real symmetric matrices $H$. Finally, Cai, Chen, and Lu have proved a dichotomy theorem for all complex symmetric matrices $H$ [Homomorphisms].

The crucial difference between Holant Problems and #CSP is that in #CSP, Equality functions of arbitrary arity are presumed to be present. In terms of $H$-homomorphism problems, this means that the input graph is allowed to have vertices of arbitrarily high degrees. This may appear to be a minor distinction; in fact it has a major impact on complexity. It turns out that if Equality gates of arbitrary arity are freely available in possible inputs then it is technically easier to prove #P-hardness. Proofs of previous dichotomy theorems make extensive use of constructions called thickening and stretching. These constructions require the availability of Equality gates of arbitrary arity (equivalently, vertices of arbitrarily high degrees) to carry out. Proving #P-hardness becomes more challenging in the degree restricted case. Furthermore there are indeed cases within this class of counting problems where the problem is #P-hard for general graphs, but solvable in ${\rm{P}}$ when restricted to 3-regular graphs.

We denote the (symmetric) edge function $g$ by $[x,y,z]$, where $x=g(0,0)$, $y=g(0,1)=g(1,0)$ and $z=g(1,1)$. Functions will also be called gates or signatures. (For Vertex Cover, the function corresponding to $H$ is the Or gate, and is denoted by the signature $[0,1,1]$.) In this paper we give a dichotomy theorem for the complexity of Holant Problems on 3-regular graphs with arbitrary signature $g=[x,y,z]$, where $x,y,z\in\mathbb{C}$. First, if $y=0$, the Holant Problem is easily solvable in ${\rm{P}}$. Assuming $y\not=0$ we may normalize $g$ and assume $y=1$. Our main theorem is as follows:

These results can be extended to $k$-regular graphs (we detail how this is accomplished in a forthcoming work). One can also use holographic reductions [HA_FOCS] to extend this theorem to more general Holant Problems.

In order to achieve this result, some new proof techniques are introduced. To discuss this we first take a look at some previous results. Valiant [Valiant79b, Valiant:sharpP] introduced the powerful technique of interpolation, which was further developed by many others. In [FOCS08] a dichotomy theorem is proved for the case when $g$ is a Boolean function. The technique from [FOCS08] is to provide certain algebraic criteria which ensure that interpolation succeeds, and then apply these criteria to prove that (a large number yet) finitely many individual problems are #P-hard. This involves (a small number of) gadget constructions, and the algebraic criteria are powerful enough to show that they succeed in each case. Nonetheless this involves a case-by-case verification. In [TAMC] this theorem is extended to all real-valued $a$ and $b$, and we have to deal with infinitely many problems. So instead of focusing on one problem, we devised (a large number of) recursive gadgets and analyzed the regions of $(a,b)\in\mathbb{R}^{2}$ where they fail to prove #P-hardness. The algebraic criteria from [FOCS08] are not suitable (Galois theoretic) for general $a$ and $b$, and so we formulated weaker but simpler criteria. Using these criteria, the analysis of the failure set becomes expressible as containment of semi-algebraic sets. As semi-algebraic sets are decidable, this offers the ultimate possibility that if we found enough gadgets to prove #P-hardness, then there is a computational proof (of computational intractability) in a finite number of steps. However this turned out to be a tremendous undertaking in symbolic computation, and many additional ideas were needed to finally carry out this plan. In particular, it would seem hopeless to extend that approach to all complex $a$ and $b$.

In this paper, we introduce three new ideas. (1) We introduce a method to construct gadgets that carry out iterations at a higher dimension, and then collapse to a lower dimension for the purpose of constructing unary signatures. This involves a starter gadget, a recursive iteration gadget, and a finisher gadget. We prove a lemma that guarantees that among polynomially many iterations, some subset of them satisfies properties sufficient for interpolation to succeed (it may not be known a priori which subset worked, but that does not matter). (2) Eigenvalue Shifted Pairs are coupled pairs of gadgets whose transition matrices differ by $\delta I$ where $\delta\neq 0$. They have shifted eigenvalues, and by analyzing their failure conditions, we can show that except on very rare points, one or the other gadget succeeds. (3) Algebraic symmetrization. We derive a new expression of the Holant polynomial over 3-regular graphs, with a crucially reduced degree. This simplification of the Holant and related polynomials condenses the problem of proving ${\#\rm{P}}$-hardness to the point where all remaining cases can be handled by symbolic computation. We also use the same expression to prove tractability.

The rest of this paper is organized as follows. In Section 2 we discuss notation and background information. In Section 3 we cover interpolation techniques, including how to collapse higher dimensional iterations to interpolate unary signatures. In Section LABEL:complexSignatures we show how to perform algebraic symmetrization of the Holant, and introduce Eigenvalue Shifted Pairs (ESP) of gadgets. Then we combine the new techniques to prove Theorem 1.1.

We state the counting framework more formally. A signature grid $\Omega=(G,{\mathcal{F}},\pi)$ consists of a labeled graph $G=(V,E)$ where $\pi$ labels each vertex $v\in V$ with a function $f_{v}\in{\mathcal{F}}$. We consider all edge assignments $\xi:E\rightarrow\{0,1\}$; $f_{v}$ takes inputs from its incident edges $E(v)$ at $v$ and outputs values in $\mathbb{C}$. The counting problem on the instance $\Omega$ is to compute²²2The term Holant was first introduced by Valiant in [HA_FOCS] to denote a related exponential sum.

Suppose $G$ is a bipartite graph $(U,V,E)$ such that each $u\in U$ has degree 2. Furthermore suppose each $v\in V$ is labeled by an Equality gate $=_{k}$ where $k={\rm deg}(v)$. Then any non-zero term in ${\rm Holant}_{\Omega}$ corresponds to a 0-1 assignment $\sigma:V\rightarrow\{0,1\}$. In fact, we can merge the two incident edges at $u\in U$ into one edge $e_{u}$, and label this edge $e_{u}$ by the function $f_{u}$. This gives an edge-labeled graph $(V,E^{\prime})$ where $E^{\prime}=\{e_{u}:u\in U\}$. For an edge-labeled graph $(V,E^{\prime})$ where $e\in E^{\prime}$ has label $g_{e}$, ${\rm Holant}_{\Omega}=\sum_{\sigma:V\rightarrow\{0,1\}}\prod_{e=(v,w)\in E^{\prime}}g_{e}(\sigma(v),\sigma(w))$. If each $g_{e}$ is the same function $g$ (but assignments $\sigma:V\rightarrow[q]$ take values in a finite set $[q]$) this is exactly the $H$-coloring problem (for undirected graphs $g$ is a symmetric function). In particular, if $(U,V,E)$ is a $(2,k)$-regular bipartite graph, equivalently $G^{\prime}=(V,E^{\prime})$ is a $k$-regular graph, then this is the $H$-coloring problem restricted to $k$-regular graphs. In this paper we will discuss 3-regular graphs, where each $g_{e}$ is the same symmetric complex-valued function. We also remark that for general bipartite graphs $(U,V,E)$, giving Equality (of various arities) to all vertices on one side $V$ defines #CSP as a special case of Holant Problems. But whether Equality of various arities are present has a major impact on complexity, thus Holant Problems are a refinement of #CSP.

A symmetric function $g:\{0,1\}^{k}\rightarrow\mathbb{C}$ can be denoted as $[g_{0},g_{1},\ldots,g_{k}]$, where $g_{i}$ is the value of $g$ on inputs of Hamming weight $i$. They are also called signatures. Frequently we will revert back to the bipartite view: for $(2,3)$-regular bipartite graphs $(U,V,E)$, if every $u\in U$ is labeled $g=[g_{0},g_{1},g_{2}]$ and every $v\in V$ is labeled $r=[r_{0},r_{1},r_{2},r_{3}]$, then we also use $\#[g_{0},g_{1},g_{2}]\mid[r_{0},r_{1},r_{2},r_{3}]$ to denote the Holant Problem. Note that $[1,0,1]$ and $[1,0,0,1]$ are Equality gates $=_{2}$ and $=_{3}$ respectively, and the main dichotomy theorem in this paper is about $\#[x,y,z]\mid[1,0,0,1]$, for all $x,y,z\in\mathbb{C}$. We will also denote $\mathrm{Hol}(a,b)=\#[a,1,b]\mid[1,0,0,1]$. More generally, If ${\mathcal{G}}$ and ${\mathcal{R}}$ are sets of signatures, and vertices of $U$ (resp. $V$) are labeled by signatures from ${\mathcal{G}}$ (resp. ${\mathcal{R}}$), then we also use $\#{\mathcal{G}}\mid{\mathcal{R}}$ to denote the bipartite Holant Problem. Signatures in ${\mathcal{G}}$ are called generators and signatures in ${\mathcal{R}}$ are called recognizers. This notation is particularly convenient when we perform holographic transformations. Throughout this paper, all $(2,3)$-regular bipartite graphs are arranged with generators on the degree 2 side and recognizers on the degree 3 side.

We use ${\mathrm{Arg}}$ to denote the principal value of the complex argument; i.e., ${\mathrm{Arg}}(c)\in(-\pi,\pi]$ for all nonzero $c\in\mathbb{C}$.