Graph homomorphism has been studied intensively. Given an m x m symmetric matrix A, the graph homomorphism function is defined as \[Z_A (G) = \sum_{f:V->[m]} \prod_{(u,v)\in E} A_{f(u),f(v)}, \] where G = (V,E) is any undirected graph. The function Z_A can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of Z_A for arbitrary symmetric matrices A with algebraic complex values. Building on work by Dyer and Greenhill, Bulatov and Grohe, and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley, we prove a complete dichotomy theorem for this problem. We show that Z_A is either computable in polynomial-time or #P-hard, depending explicitly on the matrix A. We further prove that the tractability criterion on A is polynomial-time decidable.
Deep Dive into Graph Homomorphisms with Complex Values: A Dichotomy Theorem.
Graph homomorphism has been studied intensively. Given an m x m symmetric matrix A, the graph homomorphism function is defined as [Z_A (G) = \sum_{f:V->[m]} \prod_{(u,v)\in E} A_{f(u),f(v)}, ] where G = (V,E) is any undirected graph. The function Z_A can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of Z_A for arbitrary symmetric matrices A with algebraic complex values. Building on work by Dyer and Greenhill, Bulatov and Grohe, and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley, we prove a complete dichotomy theorem for this problem. We show that Z_A is either computable in polynomial-time or #P-hard, depending explicitly on the matrix A. We further prove that the tractability criterion on A is polynomial-time decidable.
(Pinning) p.
(U 1 ) -(U 4 ) p.
(U 5 ) p.
(R 1 ) -(R 3 ) p.
(L 1 ) -(L 3 ) p.
(D 1 ) -(D 4 ) p.
(U ′ 1 ) -(U ′ 4 ) p.
(U ′ 5 ) p.
(R ′ 1 ) -(R ′ 3 ) p.
(L ′ 1 ) -(L ′ 2 ) p.
(D ′ 1 ) -(D ′ 2 ) p.
(T 1 ) -(T 3 ) p.
(S 1 ) p.
(S 2 ) -(S 3 ) p.
(Shape 1 ) -(Shape 5 ) p.
(
(GC) p.
(F 1 ) -(F 4 ) p.
(S ′ 1 ) -(S ′ 2 ) p.
(Shape ′ 1 ) -(Shape ′ 6 ) p.
(F ′ 1 ) -(F ′ 4 ) p.
Z A (G) and EVAL(A) p.
Z C,D (G) and EVAL(C, D) p.
Z A (G, w, k) and EVALP(A) p.
Z q (f ) and EVAL(q) p.
Z A (G, w, S) and EVAL(A, S) p.
Graph homomorphism has been studied intensively over the years [26,21,13,16,4,12,19]. Given two graphs G and H, a graph homomorphism from G to H is a map f from the vertex set V (G) to V (H) such that, whenever (u, v) is an edge in G, (f (u), f (v)) is an edge in H. The counting problem for graph homomorphism is to compute the number of homomorphisms from G to H. For a fixed graph H, this problem is also known as the #H-coloring problem. In 1967, Lovász [26] proved that H and H ′ are isomorphic if and only if for all G, the number of homomorphisms from G to H and from G to H ′ are the same. Graph homomorphism and the associated partition function defined below provide an elegant and general notion of graph properties [21].
In this paper, all graphs considered are undirected. We follow standard definitions: G is allowed to have multiple edges; H can have loops, multiple edges, and more generally, edge weights. (The standard definition of graph homomorphism does not allow self-loops for G. However, our result is stronger: We prove polynomial-time tractability even for input graphs G with self-loops; and at the same time, our hardness results hold for the more restricted case of G with no self-loops.) Formally, we use A to denote an m × m symmetric matrix with entries (A i,j ), i, j ∈ [m] = {1, 2, . . . , m}. Given any undirected graph G = (V, E), we define the graph homomorphism function
This is also called the partition function from statistical physics. Graph homomorphism can express many natural graph properties. For example, if we take H to be the graph over two vertices {0, 1} with an edge (0, 1) and a loop at 1, then a graph homomorphism from G to H corresponds to a Vertex Cover of G, and the counting problem simply counts the number of vertex covers. As another example, if H is the complete graph over k vertices (without self-loops), then the problem is exactly the k-Coloring problem for G. Many additional graph invariants can be expressed as Z A (G) for appropriate A. Consider the Hadamard matrix
where we index the rows and columns by {0, 1}. In Z H (G), every product
and is -1 precisely when the induced subgraph of G on ξ -1 (1) has an odd number of edges. Therefore,
is the number of induced subgraphs of G with an odd number of edges. Also expressible as Z A (•) are S-flows where S is a subset of a finite Abelian group closed under inversion [16], and (a scaled version of) the Tutte polynomial T (x, y) where (x -1)(y -1) is a positive integer. In [16], Freedman, Lovász and Schrijver characterized what graph functions can be expressed as Z A (•).
In this paper, we study the complexity of the partition function Z A (•) where A is an arbitrary fixed symmetric matrix over the algebraic complex numbers. Throughout the paper, we let C denote the set of algebraic complex numbers, and refer to them simply as complex numbers when it is clear from the context. More discussion on the model of computation can be found in Section 2.2.
The complexity question of Z A (•) has been intensively studied. Hell and Nešetřil first studied the H-coloring problem [20,21] (that is, given an undirected graph G, decide whether there exists a graph homomorphism from G to H) and proved that for any fixed undirected graph H, the problem is either in polynomial time or NP-complete. Results of this type are called complexity dichotomy theorems. They state that every member of the class of problems concerned is either tractable (i.e., solvable in P) or intractable (i.e., NP-hard or #P-hard depending on whether it is a decision or counting problem). This includes the well-known Schaefer’s theorem [29] and more generally the study on constraint satisfaction problems (CSP in short) [10]. In particular, the famous dichotomy conjecture by Vardi and Feder [14] on decision CSP motivated much of subsequent work.
In [13] Dyer and Greenhill studied the counting version of the H-coloring problem. They proved that for any fixed symmetric {0, 1}-matrix A, computing Z A (•) is either in P or #P-hard. Bulatov and Grohe [4] then gave a sweeping generalization of this theorem to all non-negative symmetric matrices A (see Theorem 2.1 for the precise statement). They obtained an elegant dichotomy theorem, which basically says that Z A (•) is computable in P if each block of A has rank at most one, and is #P-hard otherwise. More precisely, decompose A as a direct sum of A i which correspond to the connected components H i
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