Graph Homomorphisms with Complex Values: A Dichotomy Graph Homomorphisms with Complex Values: A...

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    Graph Homomorphisms with Complex Values:

    A Dichotomy Theorem

    Jin-Yi Cai∗ Xi Chen† Pinyan Lu‡

    Abstract

    Graph homomorphism problem has been studied intensively. Given an m×m symmetric matrix A, the graph homomorphism function is defined as

    ZA(G) = ∑

    ξ:V →[m]

    (u,v)∈E

    Aξ(u),ξ(v),

    where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of ZA(G) for arbitrary complex valued symmetric matrices A. Building on work by Dyer and Greenhill [6], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [10], we prove a complete dichotomy theorem for this problem.

    ∗University of Wisconsin-Madison: jyc@cs.wisc.edu †Princeton University: csxichen@gmail.com ‡Microsoft Research Asia: pinyanl@microsoft.com

    1

    http://arXiv.org/abs/0903.4728v1

  • Contents

    1 Introduction 4

    2 Preliminaries 7

    2.1 Definitions of EVAL(A) and EVAL(C,D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Basic #P-Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    3 A High Level Description of the Proof 10

    4 Pinning Lemmas and Preliminary Reductions 12

    4.1 A Pinning Lemma for EVAL(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 A Pinning Lemma for EVAL(C,D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Reduction to Connected Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    5 Proof Outline of the Case: A is Bipartite 16

    5.1 Step 1: Purification of Matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Step 2: Reduction to Discrete Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Step 3: Canonical Form of C, F and D . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    5.3.1 Step 3.1: Entries of D[r] are either 0 or Powers of ωN . . . . . . . . . . . . . . . 19 5.3.2 Step 3.2: Fourier Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3.3 Step 3.3: Affine Support for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3.4 Step 3.4: Quadratic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    5.4 Tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    6 Proof Outline of the Case: A is not Bipartite 22

    6.1 Step 1: Purification of Matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.2 Step 2: Reduction to Discrete Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . 23 6.3 Step 3: Canonical Form of F and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    6.3.1 Step 3.1: Entries of D[r] are either 0 or Powers of ωN . . . . . . . . . . . . . . . 23 6.3.2 Step 3.2: Fourier Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3.3 Step 3.3: Affine Support for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.3.4 Step 3.4: Quadratic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    6.4 Tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    7 Proofs of Theorem 5.1 and Theorem 6.1 27

    7.1 Equivalence between EVAL(A) and COUNT(A) . . . . . . . . . . . . . . . . . . . . . . . 27 7.2 Step 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.3 Step 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    8 Proof of Theorem 5.2 32

    8.1 Cyclotomic Reduction and Inverse Cyclotomic Reduction . . . . . . . . . . . . . . . . . 32 8.2 Step 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.3 Step 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.4 Step 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    8.4.1 The Vanishing Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8.4.2 Proof of Lemma 8.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    8.5 Step 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.6 Step 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    2

  • 9 Proofs of Theorem 5.3 and Theorem 5.4 63

    9.1 The Group Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 9.2 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 9.3 Decomposing F into Fourier Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    10 Proof of Theorem 5.5 74

    10.1 Proof of Lemma 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 10.2 Some Corollaries of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    11 Proof of Theorem 5.6 79

    12 Tractability: Proof of Theorem 5.7 86

    12.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 12.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.3 Proof of Theorem 12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    13 Proof of Theorem 6.2 98

    13.1 Step 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 13.2 Steps 2.2 and 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 13.3 Step 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 13.4 Step 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    14 Proofs of Theorem 6.3 and Theorem 6.4 102

    14.1 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 14.2 Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    15 Proofs of Theorem 6.5 and Theorem 6.6 105

    16 Tractability: Proof of Theorem 6.7 107

    16.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 16.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

    3

  • 1 Introduction

    Graph homomorphism has been studied intensively over the years [16, 12, 6, 9, 2, 5]. Given two graphs G and H, a graph homomorphism from G to H is a map f from 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 [16] proved that H and H ′ are isomorphic iff for all G, the number of homomorphisms from G to H and from G to H ′ are the same.

    In this paper all graphs considered are undirected. We follow standard definitions: G is allowed to have multiple edges but no loops; H can have loops, multiple edges, and more generally, edge weights. Formally, let A be an m × m symmetric matrix with entries (Ai,j), i, j ∈ [m] = {1, 2, . . . ,m}. For any undirected graph G = (V,E), we define

    ZA(G) = ∑

    ξ:V→[m]

    (u,v)∈E Aξ(u),ξ(v). (1)

    This is also called the partition function from statistical physics.

    Graph homomorphisms can express many natural graph properties. For example, if we take H to be a graph on 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 on k vertices (without self loops), then the problem is exactly the k-Coloring problem for G. Many additional graph invariants can be expressed as ZA(G) for appropriate A. Consider the Hadamard matrix

    H =

    ( 1 1 1 −1

    ) , (2)

    where we index the rows and columns by 0, 1. In ZH(G), every product

    (u,v)∈E Hξ(u),ξ(v) = ±1,

    and is −1 precisely when the induced subgraph of G on ξ−1(1) has an odd number of edges. Therefore (2n − ZH(G))/2 is the number of induced subgraphs with an odd number of edges. Also expressible as ZA(G) are S-flows where S is a subset of a finite Abelian group closed under inversion [9]. If we take

    A =

    ( 1 −1 −1 1

    )

    then ZA(G) = 2 n if G is Eulerian and 0 otherwise. Further examples include (a scaled version of) the

    Tutte polynomial T̂ (x, y) when (x−1)(y−1) is a positive integer. In [9], Freedman et. al. characterized what graph functions can be expressed as ZA(G).

    In this paper, we study the computational complexity of ZA(G), where A ∈ Cm×m is an arbitrary fixed symmetric matrix over the complex numbers and G is an input graph. The complexity question of ZA(G) has also been intensively studied. Hell and Nešeťril [11, 12] first studied the computational complexity of the H-coloring problem (that is, given an undirected graph G, decide whether there exists a graph homomorphism from G to H) and proved th