Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

Complexity Estimates for Two Uncoupling Algorithms

Uncoupling algorithms transform a linear differential system of first order into one or several scalar differential equations. We examine two approaches to uncoupling: the cyclic-vector method (CVM) and the Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the scalar equations produced by CVM, and design a fast variant of CVM whose complexity is quasi-optimal with respect to the output size. We exhibit a strong structural link between CVM and DBZ enabling to show that, in the generic case, DBZ has polynomial complexity and that it produces a single equation, strongly related to the output of CVM. We prove that algorithm CVM is faster than DBZ by almost two orders of magnitude, and provide experimental results that validate the theoretical complexity analyses.Comment: To appear in Proceedings of ISSAC'13 (21/01/2013

Cognitive networks: brains, internet, and civilizations

In this short essay, we discuss some basic features of cognitive activity at several different space-time scales: from neural networks in the brain to civilizations. One motivation for such comparative study is its heuristic value. Attempts to better understand the functioning of "wetware" involved in cognitive activities of central nervous system by comparing it with a computing device have a long tradition. We suggest that comparison with Internet might be more adequate. We briefly touch upon such subjects as encoding, compression, and Saussurean trichotomy langue/langage/parole in various environments.Comment: 16 page

Formation of wave-front pattern accompanied by current-driven electrostatic ion-cyclotron instabilities

Formation of a wave-front pattern accompanied by an electrostatic ion-cyclotron instability driven by electrons drifting along a magnetic field is investigated by two-and-half dimensional particle simulations. A clear spatial wave-front pattern appears as the ion cyclotron wave grows due to the instability. When the electron stream is uniform in the system, an obliquely intersected stripe wave-front pattern is formed. When the stream has a bell-shaped pattern across the magnetic field, a V-shaped stripe wave-front pattern appears. The wave fronts have small angles with the magnetic field lines and propagate from the high-stream region to the low-stream region