Related papers: How to see the forest for the trees

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $\pi_0$ and $H_n$ information into a single data structure that distinguishes filtrations that merge trees and persistent…

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the…

Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of "dense" ("sparse") tree, and in particular spanning tree (DST and SST respectively), is introduced as the…

We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…

Automatic detection and segmentation of overlapping leaves in dense foliage can be a difficult task, particularly for leaves with strong textures and high occlusions. We present Dense-Leaves, an image dataset with ground truth segmentation…

In this note, we derive and interpret hidden zeros of tree-level amplitudes of various theories, including Yang-Mills, non-linear sigma model, special Galileon, Dirac-Born-Infeld, and gravity, by utilizing universal expansions of tree-level…

We provide time- and sample-efficient algorithms for learning and testing latent-tree Ising models, i.e. Ising models that may only be observed at their leaf nodes. On the learning side, we obtain efficient algorithms for learning a…

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar\'at and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a…

We revisit the \textsc{$k$-Secluded Tree} problem. Given a vertex-weighted undirected graph $G$, its objective is to find a maximum-weight induced subtree $T$ whose open neighborhood has size at most $k$. We present a fixed-parameter…

Many complicated network problems can be easily understood on small networks. Difficulties arise when small networks are combined into larger ones. Fortunately, the mathematical theory of sheaves was constructed to address just this kind of…

We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the…

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements…

In this paper, we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game. Motivated by the recent consensus-based designs, we present a distributed algorithm with a…

We study the problem of approximating all-pair distances in a weighted undirected graph with differential privacy, introduced by Sealfon [Sea16]. Given a publicly known undirected graph, we treat the weights of edges as sensitive…

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

We prove that every 2-sphere graph different from a prism can be vertex 4-colored in such a way that all Kempe chains are forests. This implies the following three tree theorem: the arboricity of a discrete 2-sphere is 3. Moreover, the…

We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we…