Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

Slimness of graphs

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is called $\delta$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the distance from $u$ to $P(x,z) \cup P(y,z)$ is at most $\delta$, i.e. each path is contained in the union of the $\delta$-neighborhoods of two others. A graph $G$ is called $\delta$-slim, if all geodesic triangles in $G$ are $\delta$-slim. The smallest value $\delta$ for which $G$ is $\delta$-slim is called the slimness of $G$. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter $\Delta(G)$ of a layering partition of $G$, (2) graphs with tree-length $\lambda$, (3) graphs with tree-breadth $\rho$, (4) $k$-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1

On Computing the Average Distance for Some Chordal-Like Graphs

The Wiener index of a graph G is the sum of all its distances. Up to renormalization, it is also the average distance in G. The problem of computing this parameter has different applications in chemistry and networks. We here study when it can be done in truly subquadratic time (in the size n+m of the input) on n-vertex m-edge graphs. Our main result is a complete answer to this question, assuming the Strong Exponential-Time Hypothesis (SETH), for all the hereditary subclasses of chordal graphs. Interestingly, the exact same result also holds for the diameter problem. The case of non-hereditary chordal subclasses happens to be more challenging. For the chordal Helly graphs we propose an intricate O?(m^{3/2})-time algorithm for computing the Wiener index, where m denotes the number of edges. We complete our results with the first known linear-time algorithm for this problem on the dually chordal graphs. The former algorithm also computes the median set

A New Optimality Measure for Distance Dominating Sets

  We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset.   The problem is a different view of the d-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance d".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the d-domatic number problem. We show that the problem is NP-complete for any fixed k greater than two even when the input graph is restricted to split graphs, 2-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering k as a constant, and for strongly chordal graphs, for any k. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time O(2. 73n). Moreover, we prove that the problem cannot be approximated within ratio smaller than 2 even for split graphs, 2-connected graphs, and planar bipartite graphs of degree four. We propose a greedy 3-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work