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There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity $q_z=q_z(epsilon)$ of an estimable parameter $z$ is the size of a random sample of a graph $G$ required to ensure that the value of $z(G)$ may be estimated within an error of $epsilon$ with probability at least 2/3. In this paper, for any fixed monotone graph property $mathcal{P}=mbox{Forb}(mathcal{F})$, we study the sample complexity of estimating a bounded graph parameter $z_{mathcal{P}}$ that, for an input graph $G$, counts the number of spanning subgraphs of $G$ that satisfy $mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.
Given a family of graphs $mathcal{F}$, we prove that the normalized edit distance of any given graph $Gamma$ to being induced $mathcal{F}$-free is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $mathcal{F}$.
We investigate how minor-monotone graph parameters change if we add a few random edges to a connected graph $H$. Surprisingly, after adding a few random edges, its treewidth, treedepth, genus, and the size of a largest complete minor become very larg
An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_vcap T_
We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions
The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they beco