arXiv subjects

Feodor F. Dragan

Publications and source records attributed to Feodor F. Dragan.

23 records · Page 2Linked to original sources

arXiv2016

Core congestion is inherent in hyperbolic networks

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset $X$ of vertices of a $δ$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk $D(m,4 δ)$ of radius $4 δ$ centered at $m$ intercepts at least one half of the total flow between all pairs of vertices of $X$, where the flow between two vertices $x,y\in X$ is carried by geodesic (or quasi-geodesic) $(x,y)$-paths. A set $S$ intercepts the flow between two nodes $x$ and $y$ if $S$ intersect every shortest path between $x$ and $y$. Differently from what was conjectured by Jonckheere et al., we show that $m$ is not (and cannot be) the center of mass of $X$ but is a node close to the median of $X$ in the so-called injective hull of $X$. In case of non-uniform traffic between nodes of $X$ (in this case, the unit flow exists only between certain pairs of nodes of $X$ defined by a commodity graph $R$), we prove a primal-dual result showing that for any $ρ>5δ$ the size of a $ρ$-multi-core (i.e., the number of disks of radius $ρ$) intercepting all pairs of $R$ is upper bounded by the maximum number of pairwise $(ρ-3δ)$-apart pairs of $R$.

cs.DS↗

arXiv2015

Minimum Eccentricity Shortest Paths in some Structured Graph Classes

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distance-hereditary graphs (generalizing the previous result for trees) and give a generalised approach which allows to solve the problem in polynomial time for other graph classes. This includes chordal graphs, dually chordal graphs, graphs with bounded tree-length, and graphs with bounded hyperbolicity. Additionally, we give a simple algorithm to compute an additive approximation for graphs with bounded tree-length and graphs with bounded hyperbolicity.

cs.DM↗

arXiv2014

Line-distortion, Bandwidth and Path-length of a graph

We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show: - if a graph $G$ can be embedded into the line with distortion $k$, then $G$ admits a Robertson-Seymour's path-decomposition with bags of diameter at most $k$ in $G$; - for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem; - there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; - AT-free graphs and some intersection families of graphs have path-length at most 2; - for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.

cs.DS↗

arXiv2014

Metric tree-like structures in real-life networks: an empirical study

Based on solid theoretical foundations, we present strong evidences that a number of real-life networks, taken from different domains like Internet measurements, biological data, web graphs, social and collaboration networks, exhibit tree-like structures from a metric point of view. We investigate few graph parameters, namely, the tree-distortion and the tree-stretch, the tree-length and the tree-breadth, the Gromov's hyperbolicity, the cluster-diameter and the cluster-radius in a layering partition of a graph, which capture and quantify this phenomenon of being metrically close to a tree. By bringing all those parameters together, we not only provide efficient means for detecting such metric tree-like structures in large-scale networks but also show how such structures can be used, for example, to efficiently and compactly encode approximate distance and almost shortest path information and to fast and accurately estimate diameters and radii of those networks. Estimating the diameter and the radius of a graph or distances between its arbitrary vertices are fundamental primitives in many data and graph mining algorithms.

cs.SI↗

arXiv2012

Collective Additive Tree Spanners of Bounded Tree-Breadth Graphs with Generalizations and Consequences

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a system of $μ$ collective additive tree $r$-spanners} (resp., {\em multiplicative tree $t$-spanners}) if there is a system $\cT(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $x,y$ of $G$ a spanning tree $T\in \cT(G)$ exists such that $d_T(x,y)\leq d_G(x,y)+r$ (resp., $d_T(x,y)\leq t\cdot d_G(x,y)$). When $μ=1$ one gets the notion of {\em additive tree $r$-spanner} (resp., {\em multiplicative tree $t$-spanner}). It is known that if a graph $G$ has a multiplicative tree $t$-spanner, then $G$ admits a Robertson-Seymour's tree-decomposition with bags of radius at most $\lceil{t/2}\rceil$ in $G$. We use this to demonstrate that there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative tree $t$-spanner, constructs a system of at most $\log_2 n$ collective additive tree $O(t\log n)$-spanners of $G$. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph $G$ admits a multiplicative $t$-spanner with tree-width $k-1$, then $G$ admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most $k$ disks of $G$ of radius at most $\lceil{t/2}\rceil$ each. This is used to demonstrate that, for every fixed $k$, there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative $t$-spanner with tree-width $k-1$, constructs a system of at most $k(1+ \log_2 n)$ collective additive tree $O(t\log n)$-spanners of $G$.

cs.DS↗