Hamed Hatami, Michael Molloy
We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction problems with domain size 3.
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We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction problems with domain size three.
The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov's classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs; the technique will apply in a broader context. The key concept we introduce is that of a digr…
Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+δ$. Going beyond the Erdős-Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochast…
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of pr…
We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between k-colorings, which was shown to be in P for k<=3 and PSPACE-complete otherwise by Cereceda et al. 2011. We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H…
A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and $(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical graph homomorphism problems. We consider two settings. First, when the tropical graph $(H,c)$ is fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring of $H$ is part of the input; the associated decision problem is called $H$-…
Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$
该研究从Johnson图是超立方体切片的视角出发，构造了泛Racah代数到$U(\mathfrak{sl}_2)$的新代数同态，刻画了其核，证明拉回得到的有限维$\Re$-模完全可约且生成元作用为Leonard三元组，还给出了Leonard三元组的具体构造方式。
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of gr…
A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with $\E_p[f] = \eps$ and $\E_q[f] = 1-\eps$ then $q-p \to 0$ as $n \to \infty$. Here $\E_p$ denotes the product probability measure on $\{0,1\}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well…
This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the $(k,q)$-colourings of a graph $G$ can be obtained by successively recolouring a single vertex provided $k/q\geq 2col(G)$ along the lines of Cereceda, van den Heuvel and Johnson's result for $k$-colourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for…
Hardness of 4-Colourings G-Colourable Graphs
本文在G和H均为4-可着色图的条件下，证实了Brakensiek和Guruswami关于承诺图同态问题NP难度的猜想，统一推广了此前两项结果，方法上结合了代数、拓扑组合学与等变阻碍理论。
The Fine-Grained Complexity of Graph Homomorphism Problems: Towards the Okrasa and Rzążewski Conjecture
本文研究图同态（H-着色）问题的细粒度复杂度，通过分析二元对称关系的部分多态性表达能力，证明相关结构性质，刻画多态性与射影核，最终验证了顶点数≤7的图满足Okrasa与Rzążewski猜想。
Many NP-complete constraint satisfaction problems appear to undergo a "phase transition'' from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that …
Interpolating chromatic and homomorphism thresholds
本文通过VC维理论定义广义阈值插值染色与同态阈值，证明奇环膨胀阈值为1/(2k-1)，回答Schacht问题并揭示膨胀阈值聚点特性，统一并强化相关领域结果。
Beyond chromatic threshold via the $(p,q)$-theorem, and a sharp blow-up phenomenon
本文将极值组合学的色数阈值问题与离散几何的$(p,q)$-定理建立联系，证明超极大$K_r$-free图的$χ$-有界性，并揭示同态阈值问题的决定性因素是顶点共邻域的团密度条件，得出最优的单指数爆炸界。
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connecte…
Maximizing subgraph counts in regular graphs
研究正则图中最大化子图密度的问题，通过图的特征值连续优化方法，针对不同类型的图给出最优结构结论。
Random Lipschitz functions on graphs with weak expansion
本文将 Benjamini 等人关于次对数最大度图上随机 $\mathbb Z$-同态值域超常数及 $C_{n,k}$ 图上突变现象的结果，推广至随机 $M$-Lipschitz 函数与随机实值 Lipschitz 函数。
Asymmetric Homomorphism Thresholds for Graphs of Large Odd Girth
本文确定了奇数围长至少为7的图到无三角形图的不对称同态阈值为1/9，并推广到更一般的K_t及任意奇数围长情形，否证了相关猜想，下界基于高维Borsuk图，上界使用正则性方法。
On Domination Exponents for Pairs of Graphs
本文针对图密度分布问题的挑战，提出同态密度支配指数$C(H_1, H_2)$以松弛问题，证明需无限多图族实现该指数，并推导了路径、环等图对的指数值或渐近紧界，推进了图对密度分布的研究。
On the complexity of Sandwich Problems for $M$-partitions
本文研究了$M$-划分问题的复杂性，通过结构分类和高效算法，对矩阵划分和沙丁鱼问题进行了P与NP完全的分类。
The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H …
In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is concerned with Cameron's problem of classifying the homogeneous structures in a language of finitely many linear orders, which we call finite-dimensional permutation structures. Towards this problem, we present a construction that we conjecture produces all such structures. Some evidence for this conjecture is given, including the classification of the homogene…
Upper bounds on the $2$-colorability threshold of random $d$-regular $k$-uniform hypergraphs for $k\geq 3$
本文研究了随机$d$-正则$k$-均匀超图$2$-着色与随机$d$-正则$k$-NAE-SAT模型，针对所有$k\geq 3$证明了显式可满足性上界$d_{\ast}(k)$，该上界与统计物理学预测吻合且在$k$足够大时与已知阈值一致。
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