17 research outputs found
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Rainbow arborescence in random digraphs
<p>We consider the Erd˝os-R´enyi random directed graph process, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let D(n, m) be a graph with m edges obtained after m steps of this process. Each edge ei (i = 1, 2, . . . , m) of D(n, m) independently chooses a colour, taken uniformly at random from a given set of n(1 + O(log log n/ log n)) = n(1 + o(1)) colours. We stop the process prematurely at time M when the following two events hold: D(n, M) has at most one vertex that has in-degree zero and there are at least n − 1 distinct colours introduced (M = n(n − 1) if at the time when all edges are present there are still less than n − 1 colours introduced; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether D(n, M) has a rainbow arborescence (that is, a directed, rooted tree on n vertices in which all edges point away from the root and all the edges are different colours). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is “yes”.</p
Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry