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.
We consider the problem of finding a homomorphism from an input digraph $G$ to a fixed digraph $H$. We show that if $H$ admits a weak-near-unanimity polymorphism $ϕ$ then deciding whether $G$ admits a homomorphism to $H$ (HOM($H$)) is polynomial time solvable? This gives a proof of the dichotomy conjecture (now dichotomy theorem) by Feder and Vardi [29]. Our approach is combinatorial, and it is simpler than the two algorithms found by Bulatov [9] and Zhuk [46] in 2017. We have implemented our algorithm and show some experimental results.
We prove a complexity dichotomy theorem for counting planar graph homomorphisms of domain size 3. Given any 3 by 3 real valued symmetric matrix $H$ defining a graph homomorphism from all planar graphs $G \mapsto Z_H(G)$, we completely classify the computational complexity of this problem according to the matrix $H$. We show that for every $H$, the problem is either polynomial time computable or \#P-hard. The P-time computable cases consist of precisely those that are P-time computable for general graphs (a complete classification is known) or computable by Valiant's holographic algorithm via m…
许多重要的图论概念可以编码为计数图同态问题，例如统计物理中的配分函数，特别是独立集和着色。在这篇文章中，我们研究了$\#_p\mathsf{H}$ OMS $\mathsf{T}$ O $H$的复杂性，从输入图到图$H$模素数$p$的图同态计数问题。Dyer和Greenhill证明了一个二分法，说明非模计数图同态的易处理性取决于输入图的结构。由于取消现象的普遍存在，非模计数中许多难以处理的情况在模计数中变得容易处理。然而，在随后的研究计数模$2$的影响，结构的$H$上的易处理性，被证明是持久的，产生类似的二分法。我们的主要结果表明，对于每一棵树$H$和每一个素数$p$，问题$\#_p\mathsf{H}$ OMS $\mathsf{T}$ O $H$是多项式时间可计算的或$\#_p\mathsf{P}$ -完全的.这解决了Faben和Jerrum在模2计数时对每个图$H$陈述这种二分法的猜想。为了证明这个结果，我们研究了同态的结构性质。作为一个重要的过渡，这项研究产生了一个二分法的问题的加权计数独立集的二分图模一些素数$p$。我们的结果是第一次表明，这样的二分法不仅适用于1位功能的模~2的情况下，但所有素数$p$的模计数功能。
Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy theorem for Holant problems for domain size $>2$. We discover unexpected tractable families of counting problems, by giving new polynomial time algorithms. This paper also initiates holographic reductions in domains of size $>2$. This is our main algorithmic technique, and is used for both tractable…
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped with a list $L(v) \subseteq V(H)$. We ask if there exists a homomorphism $f$ from $G$ to $H$, in which $f(v) \in L(v)$ for every $v \in V(G)$. Feder, Hell, and Huang [JGT~2003] proved that LHom($H$) is polynomial time-solvable if $H$ is a bi-arc-graph, and NP-complete otherwise. We are interested in the complexity of the LHom($H$) problem in graphs excluding…
A dichotomy theorem for conservative general-valued CSPs
研究了有值约束满足问题（VCSP）的复杂性.一个问题，从VCSP的特点是一个\emph{constraint language}，一个固定的成本函数在有限域。问题的一个实例由语言中的成本函数之和指定，目标是最小化该和。我们考虑的情况下，所谓的\emph{conservative}语言，也就是说，语言包含所有的一元成本函数，从而允许任意限制域的变量。在这种情况下，我们证明了一个类似Schaefer的二分法定理：如果语言中的所有代价函数都满足一定的条件（由\emph{STP and MJN multimorphisms}的互补组合指定），那么任何实例都可以在多项式时间内通过Kolmogorov和Zivny的算法（arXiv：1008.3104v1）求解，否则语言是NP-难的。这推广了Takhanov（STACS'10）和Kolmogorov和Zivny（arXiv：1008.1555v1）最近的结果，Takhanov（STACS'10）认为$\{0,\infty\}$值语言包含额外的所有有限值一元代价函数，Kolmogorov和Zivny（arXiv：1008.1555v1）认为\emph{finite-valued}保守语言。
We prove a complexity dichotomy theorem for a class of Holant problems on 3-regular bipartite graphs. Given an arbitrary nonnegative weighted symmetric constraint function $f = [x_0, x_1, x_2, x_3]$, we prove that the bipartite Holant problem $\operatorname{Holant} \left( f \mid \left( =_3 \right) \right)$ is \emph{either} computable in polynomial time \emph{or} $\#$P-hard. The dichotomy criterion on $f$ is explicit.
PCSP是由两个相似模板定义的两个CSP的组合;计算问题是区分第一个的YES实例和第二个的NO实例。许多PCSP的计算复杂性仍然是未知的。即使是布尔模板的情况下（解决了CSP的舍费尔[STOC'78]）仍然是开放的。Brakensiek和Guruswami [SODA'18]的主要结果表明，当“所有子句都是对称的并且允许变量的否定”时，布尔PCSP表现出二分法（PTIME vs. NPC）。本文从定理中去掉了“允许变量否定”的假设。“对称”假设意味着改变约束中变量的顺序不会改变其可满足性。“变量求反”意味着两个模板共享一个关系，该关系可用于有效地求反布尔变量。本文的主要结果建立了所有对称布尔模板的二分法。我们定理的易处理情形与Brakensiek和Guruswami定理几乎相同。主要的区别，这项工作的主要贡献，是新的原因，硬度和推理证明分裂。
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by \textsc{LHom}($H$), the instance is a graph $G$, whose every vertex is equipped with a subset of $V(H)$, called list. We ask whether there exists a homomorphism from $G$ to $H$, such that every vertex from $G$ is mapped to a vertex from its list. We study the complexity of the \textsc{LHom}($H$) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the q…
We prove a complexity dichotomy theorem for a class of Holant problems on planar 3-regular bipartite graphs. The complexity dichotomy states that for every weighted constraint function $f$ defining the problem (the weights can even be negative), the problem is either computable in polynomial time if $f$ satisfies a tractability criterion, or \#P-hard otherwise. One particular problem in this problem space is a long-standing open problem of Moore and Robson on counting Cubic Planar X3C. The dichotomy resolves this problem by showing that it is \numP-hard. Our proof relies on the machinery of si…
In a non-uniform Constraint Satisfaction problem CSP(G), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from G. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language G the problem CSP(G) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal 1993 paper. In this paper we confirm the Dichotomy Conjecture.
Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and the pattern $H$. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of $H$,…
