Node-balancing by edge-increments

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.Comment: 10 page

Fully Indecomposable and Nearly Decomposable Graphs

LetA be an n-square non-negative matrix. If A contains no s\times t zero submatrix, where s + t = n, then it is called fully indecomposable. Also, a graph G is said to be fully indecomposable if its adjacency matrix is fully indecomposable. In this paper we provide some necessary and sucient conditions for a graph to be fully indecomposable. Among other results we prove that a regular connected graph is fully indecomposable if and only if it is not bipartite.  

A partial refining of the Erdős-Kelly regulation

The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs & Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple $n$-vertex graph $G$ with maximum vertex degree $\Delta$, the exact minimum number, say $\theta =\theta(G)$, of new vertices in a $\Delta$-regular graph $H$ which includes $G$ as an induced subgraph. The number $\theta(G)$, which we call the cost of regulation of $G$, has been upper-bounded by the order of $G$, the bound being attained for each $n\ge4$, e.g. then the edge-deleted complete graph $K_n-e$ has $\theta=n$. For $n\ge 4$, we present all factors of $K_n$ with $\theta=n$ and next $\theta=n-1$. Therein in case $\theta=n-1$ and $n$ odd only, we show that a specific extra structure, non-matching, is required

Randomisierte Approximation für das Matching- und Knotenüberdeckung Problem in Hypergraphen: Komplexität und Algorithmen

This thesis studies the design and mathematical analysis of randomized approximation algorithms for the hitting set and b-matching problems in hypergraphs. We present a randomized algorithm for the hitting set problem based on linear programming. The analysis of the randomized algorithm rests upon the probabilistic method, more precisely on some concentration inequalities for the sum of independent random variables plus some martingale based inequalities, as the bounded difference inequality, which is a derived from Azuma inequality. In combination with combinatorial arguments we achieve some new results for different instance classes that improve upon the known approximation results for the problem (Krevilevich (1997), Halperin (2001)). We analyze the complexity of the b-matching problem in hypergraphs and obtain two new results. We give a polynomial time reduction from an instance of a suitable problem to an instance of the b-matching problem and prove a non-approximability ratio for the problem in l-uniform hypergraphs. This generalizes the result of Safra et al. (2006) from b=1 to b in O(l/log(l)). Safra et al. showed that the 1-matching problem in l-uniform hypergraphs can not be approximated in polynomial time within a ratio O(l/log(l)), unless P = NP. Moreover, we show that the b-matching problem on l-uniform hypergraphs with bounded vertex degree has no polynomial time approximation scheme PTAS, unless P=NP.Diese Arbeit befasst sich mit dem Entwurf und der mathematischen Analyse von randomisierten Approximationsalgorithmen für das Hitting Set Problem und das b-Matching Problem in Hypergraphen. Zuerst präsentieren wir einen randomisierten Algorithmus für das Hitting Set Problem, der auf linearer Programmierung basiert. Mit diesem Verfahren und einer Analyse, die auf der probabilistischen Methode fußt, erreichen wir für verschiedene Klassen von Instanzen drei neue Approximationsgüten, die die bisher bekannten Ergebnisse (Krevilevich [1997], Halperin [2001]) für das Problem verbessern. Die Analysen beruhen auf Konzentrationsungleichungen für Summen von unabhängigen Zufallsvariablen aber auch Martingal-basierten Ungleichungen, wie die aus der Azuma-Ungleichung abgeleitete Bounded Difference-Inequality, in Kombination mit kombinatorischen Argumenten. Für das b-Matching Problem in Hypergraphen analysieren wir zunächst seine Komplexität und erhalten zwei neue Ergebnisse. Wir geben eine polynomielle Reduktion von einer Instanz eines geeigneten Problems zu einer Instanz des b-Matching-Problems an und zeigen ein Nicht-Approximierbarkeitsresultat für das Problem in uniformen Hypergraphen. Dieses Resultat verallgemeinert das Ergebnis von Safra et al. (2006) von b = 1 auf b in O(l/log(l))). Safra et al. zeigten, dass es für das 1-Matching Problem in uniformen Hypergraphen unter der Annahme P != NP keinen polynomiellen Approximationsalgorithmus mit einer Ratio O(l/log(l)) gibt. Weiterhin beweisen wir, dass es in uniformen Hypergraphen mit beschränktem Knoten-Grad kein PTAS für das Problem gibt, es sei denn P = NP

Why CAT(0) cube complexes should be replaced with median graphs

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.Comment: 9 pages. Comments are welcome

Square Function Estimates and Functional Calculi

In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F, Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f) for certain H-valued functions f on O. The assignment f to Phi_gamma(f) then becomes a vectorial functional calculus, and a "square function estimate" for f simply means the boundedness of Phi_gamma(f). In this view, all results linking square function estimates with the boundedness of a certain (usually the H-infinity) functional calculus simply assert that certain square function estimates imply other square function estimates. In the present paper several results of this type are proved in an abstract setting, based on the principles of subordination, integral representation, and a new boundedness concept for subsets of Hilbert spaces, the so-called ell-1 -frame-boundedness. These abstract results are then applied to the H-infinity calculus for sectorial and strip type operators. For example, it is proved that any strip type operator with bounded scalar H-infinity calculus on a strip over a Banach space with finite cotype has a bounded vectorial H-infinity calculus on every larger strip.Comment: 49

Arbitrarily regularizable graphs

A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative regularizable graphs have been thoroughly investigated in the literature. In this work, we propose and study arbitrarily regularizable graphs. In particular, we investigate necessary and sufficient regularization conditions on the topology of the graph and of the corresponding adjacency matrix. Moreover, we study the computational complexity of the regularization problem and characterize it as a linear programming model

An Application of the SS-Functional Calculus to Fractional Diffusion Processes

In this paper we show how the spectral theory based on the notion of $S$-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $H^\infty$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier laws for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T=e_1\,a(x)\partial_{x_1} + e_2\,b(x)\partial_{x_2} + e_3\,c(x)\partial_{x_3},$$ where $e_\ell$, $e_\ell=1,2,3$ are orthogonal unit vectors, $a$, $b$, $c$ are suitable real valued function that depend on the space variables $x=(x_1,x_2,x_3)$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator $T$ so we can define the non local version $T^\alpha$, for $\alpha\in (0,1)$, of the Fourier law defined by $T$. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our theory based on the $S$-spectrum for vector operators. This paper is devoted to researchers in different research fields such as: fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis