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In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of…
Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary…
We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a…
We analyze integer linear programs which we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness…
We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical…
Random networks are a powerful tool in the analytical modeling of complex networks as they allow us to write approximate mathematical models for diverse properties and behaviors of networks. One notable shortcoming of these models is that…
Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in…
For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the…
Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their…
In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence…
A strongly connected graph is strongly biconnected if after ignoring the direction of its edges we have an undirected graph with no articulation points. A 3-vertex strongly biconnected graph is a strongly biconnected digraph that has the…
In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…
Random geometric networks consist of 1) a set of nodes embedded randomly in a bounded domain $\mathcal{V} \subseteq \mathbb{R}^d$ and 2) links formed probabilistically according to a function of mutual Euclidean separation. We quantify how…
Motivated by a $2$-dimensional (unsupervised) image segmentation task whereby local regions of pixels are clustered via edge detection methods, a more general probabilistic mathematical framework is devised. Critical thresholds are…
This work presents exact expressions for size distributions of weak/multilayer connected components in two generalisations of the configuration model: networks with directed edges and multiplex networks with arbitrary number of layers. The…
Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that…
Flexible network design deals with building a network that guarantees some connectivity requirements between its vertices, even when some of its elements (like vertices or edges) fail. In particular, the set of edges (resp. vertices) of a…
Thinning is the removal of contour pixels/points of connected components in an image to produce their skeleton with retained connectivity and structural properties. The output requirements of a thinning procedure often vary with…
The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…
A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an approximate…