Ramsey properties of randomly perturbed graphs: cliques and cycles

Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$ or a blue copy of $H_2$. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability $(K_3,K_t)$-Ramsey (for $t\ge 3$). They also raised the question of generalising this result to pairs of graphs other than $(K_3,K_t)$. We make significant progress on this question, giving a precise solution in the case when $H_1=K_s$ and $H_2=K_t$ where $s,t \ge 5$. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the $(K_3,K_t)$-Ramsey question. Moreover, we give bounds for the corresponding $(K_4,K_t)$-Ramsey question; together with a construction of Powierski this resolves the $(K_4,K_4)$-Ramsey problem. We also give a precise solution to the analogous question in the case when both $H_1=C_s$ and $H_2=C_t$ are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability $(C_s,K_t)$-Ramsey (for odd $s\ge 5$ and $t\ge 4$).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil Powierski, stated results for cliques in graphs of low positive density separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to appear in CP

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

A graph $G$ is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from $G$ to any graph that is isomorphic to $G$. We say that $G=(V,E)$ is robustly self-ordered if the size of the symmetric difference between $E$ and the edge-set of the graph obtained by permuting $V$ using any permutation $\pi:V\to V$ is proportional to the number of non-fixed-points of $\pi$. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.Comment: Slightly modified and revised version of a CCC 2021 paper that also appeared on ECCC 27: 149 (2020

Beyond Conjugacy for Chain Event Graph Model Selection

Chain event graphs are a family of probabilistic graphical models that generalise Bayesian networks and have been successfully applied to a wide range of domains. Unlike Bayesian networks, these models can encode context-specific conditional independencies as well as asymmetric developments within the evolution of a process. More recently, new model classes belonging to the chain event graph family have been developed for modelling time-to-event data to study the temporal dynamics of a process. However, existing model selection algorithms for chain event graphs and its variants rely on all parameters having conjugate priors. This is unrealistic for many real-world applications. In this paper, we propose a mixture modelling approach to model selection in chain event graphs that does not rely on conjugacy. Moreover, we also show that this methodology is more amenable to being robustly scaled than the existing model selection algorithms used for this family. We demonstrate our techniques on simulated datasets

Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs

Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares (SOS) proof of nonisomorphism requires degree $\Omega(n)$. In other words, we show an $\Omega(n)$-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs $G$ and $H$ which are not even $(1-10^{-14})$-isomorphic. (Here we say that two $n$-vertex, $m$-edge graphs $G$ and $H$ are $\alpha$-isomorphic if there is a bijection between their vertices which preserves at least $\alpha m$ edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem is hard. I.e.\ for every $\epsilon > 0$, there is no efficient algorithm which can distinguish graph pairs which are $(1-\epsilon)$-isomorphic from pairs which are not even $(1-\epsilon_0)$-isomorphic for some universal constant $\epsilon_0$. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest

Discrete Multi-modal Hashing with Canonical Views for Robust Mobile Landmark Search

Mobile landmark search (MLS) recently receives increasing attention for its great practical values. However, it still remains unsolved due to two important challenges. One is high bandwidth consumption of query transmission, and the other is the huge visual variations of query images sent from mobile devices. In this paper, we propose a novel hashing scheme, named as canonical view based discrete multi-modal hashing (CV-DMH), to handle these problems via a novel three-stage learning procedure. First, a submodular function is designed to measure visual representativeness and redundancy of a view set. With it, canonical views, which capture key visual appearances of landmark with limited redundancy, are efficiently discovered with an iterative mining strategy. Second, multi-modal sparse coding is applied to transform visual features from multiple modalities into an intermediate representation. It can robustly and adaptively characterize visual contents of varied landmark images with certain canonical views. Finally, compact binary codes are learned on intermediate representation within a tailored discrete binary embedding model which preserves visual relations of images measured with canonical views and removes the involved noises. In this part, we develop a new augmented Lagrangian multiplier (ALM) based optimization method to directly solve the discrete binary codes. We can not only explicitly deal with the discrete constraint, but also consider the bit-uncorrelated constraint and balance constraint together. Experiments on real world landmark datasets demonstrate the superior performance of CV-DMH over several state-of-the-art methods