Graph Homomorphisms with Complex Values: A Dichotomy Theorem

Abstract

Graph homomorphism has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined aswhere 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 [13], Bulatov and Grohe [4], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [19], 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.Recently Thurley [31] announced a complexity dichotomy theorem for Z A (·), where A is any complex Hermitian matrix. The polynomial-time tractability result of the present paper (in Section 12) is used in [31]. In [12], Dyer, Goldberg, and Paterson proved a dichotomy theorem for Z H (·) with H being a directed acyclic graph. Cai and Chen proved a dichotomy theorem for Z A (·), where A is a nonnegative matrix [5]. A dichotomy theorem is also proved [1, 2] for the more general counting constraint satisfaction problem, when the constraint functions take values in {0, 1} (with an alternative proof given in [11] which also proves the decidability of the dichotomy criterion), when the functions take non-negative and rational values [3], and when they are non-negative and algebraic [6]. OrganizationDue to the complexity of the proof of Theorem 1.1, both in terms of its overall structure and in terms of technical difficulty, we first give a high level description of the proof for the bipartite case in Section Z → C,D (G, u) = ξ∈Ξ 1 wt C,D (ξ) and Z ← C,D (G, u) = ξ∈Ξ 2 wt C,D (ξ). It then follows from the definition of Z C,D , Z → C,D and Z ← C,D that Lemma 2.2. For any graph G and vertex u ∈ G, Z C,D (G) = Z → C,D (G, u) + Z ← C,D (G, u). X d = Z A (G * , w * , i), where i ∈ A a and a ∈ S d .This number X d is well-defined, and is independent of the choices of a ∈ S d and i ∈ A a . Moreover, the definition of the equivalence relation ∼ * implies thatNext, let G be an undirected graph and w be a vertex. We show that, by querying EVAL(A, S) as an oracle, one can compute Z A (G, w, S d ) efficiently for all d.

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

“…To prove the lemma, let us fix k ≤ m and show how to compute the coefficients a H,k , assuming that we have already computed and listed the coefficients a H,k ′ for all k ′ < k. Let us fix H ∈ C αk (G). By (13), it suffices to compute the coefficient of ind(H, ·) in p k−i e i for i = 1, . .…”

“…Since e 1 = −p 1 , this implies that the same holds for p 1 . By induction, (29) and (13) (using that e 0 = 1) we have that for each ℓ…”

“…Let us fix F ∈ C ′ ℓ (G). By the identities (13), it suffices to compute the coefficient of ind * (F, ·) in e i p ℓ−i for i = 1, . .…”

The Complexity of Boolean Holant Problems with Nonnegative Weights

“…One natural framework is the counting Constraint Satisfaction Problem (#CSP) [19,2,20,4,23,3,8,7,1]. Another is Graph Homomorphism (GH) [31,28,22,5,21,26,6,9], which can be seen as a special case of #CSP. Such frameworks express a large class of counting problems in the Sum-of-Product form.…”

On the complexity of generalized chromatic polynomials

“…By a classical result of R. Ladner, and its generalization by K. Ambos-Spies, [Lad75,AS87], there are infinitely many degrees (via polynomial time reducibility) between P and NP, and between P and #P, provided P = NP. In contrast to this, the complexity of evaluating partition functions or counting graph homomorphisms satisfies a dichotomy theorem: either evaluation is in P or it is #P-complete, [DG00, BG05,CCL13]. For the definition of the complexity class #P, see [GJ79] or [Pap94].…”