Minimizing and Computing the Inverse Geodesic Length on Trees

@article{Gaspers2019MinimizingAC,
  title={Minimizing and Computing the Inverse Geodesic Length on Trees},
  author={Serge Gaspers and Joshua Lau},
  journal={ArXiv},
  year={2019},
  volume={abs/1811.03836}
}
The inverse geodesic length (IGL) of a graph $G=(V,E)$ is the sum of inverse distances between every two vertices: $IGL(G) = \sum_{\{u,v\} \subseteq V} \frac{1}{d_G(u,v)}$. In the MinIGL problem, the input is a graph $G$, an integer $k$, and a target inverse geodesic length $T$, and the question is whether there are $k$ vertices whose deletion decreases the IGL of $G$ to at most $T$. Aziz et al. (2018) proved that MinIGL is $W[1]$-hard for parameter treewidth, but the complexity status of the… 

Figures from this paper

Computing the Inverse Geodesic Length in Planar Graphs and Graphs of Bounded Treewidth
TLDR
The inverse geodesic length of a graph G is shown to be the sum of the inverse of the distances between all pairs of distinct vertices of G, known as the Harary index or the global efficiency of the graph.
Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length
TLDR
It is shown that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program and FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k are provided.

References

SHOWING 1-10 OF 45 REFERENCES
Weakening Covert Networks by Minimizing Inverse Geodesic Length
TLDR
A study of the classical and parameterized complexity of the MINIGL problem, which is NP-complete even if T = 0 and remains both NP- complete and W[1]-hard for parameter k on bipartite and on split graphs.
Multivariate Analysis of Orthogonal Range Searching and Graph Distances
TLDR
The analysis of an algorithm of Cabello and Knauer in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log d n to d + ⌈ log n ⌉ d , as originally observed by Monier.
Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs
TLDR
Truly subquadratic approximation algorithms for most of the versions of Diameter and Radius with optimal approximation guarantees are found, under plausible assumptions, since even a $(3/2-\delta)$-approximation algorithm that runs in time $2^{o(k)n 2-\epsilon}$ would refute the plausible assumptions.
Algorithms for graphs of bounded treewidth via orthogonal range searching
Computing Graph Distances Parameterized by Treewidth and Diameter
We show that the eccentricity of every vertex in an undirected graph on n vertices can be computed in time n exp O(k*log(d)), where k is the treewidth of the graph and d is the diameter. This means
A ck n 5-Approximation Algorithm for Treewidth
TLDR
This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in $k$ and linear in the input size and can be used to speed up many algorithms to work in time.
Critical nodes for distance-based connectivity and related problems in graphs
TLDR
This study considers a class of critical node detection problems that involves minimization of a distance‐based connectivity measure of a given unweighted graph via the removal of a subset of nodes subject to a budgetary constraint and develops an effective exact algorithm that iteratively solves a series of simpler IPs to obtain an optimal solution for the original problem.
A Faster Computation of All the Best Swap Edges of a Tree Spanner
TLDR
Two efficient linear-space solutions for both the weighted and the unweighted case, running in Om2 logαm,n and Omn logn time, respectively are provided, which improve on the time complexity of previous results provided for other related settings of the problem.
Computing All-Pairs Shortest Paths by Leveraging Low Treewidth
TLDR
Two new and efficient algorithms for computing all-pairs shortest paths that make use of directed path consistency along a vertex ordering d to arrive at a run time of O(n w_d^2 + n^2 s_d) on general graphs.
An Experimental Study of the Treewidth of Real-World Graph Data (Extended Version)
TLDR
This article is the first large-scale experimental study of treewidth and tree decompositions of real-world database instances (25 datasets from 8 different domains, with sizes ranging from a few thousand to a few million vertices).
...
...