• DOI:10.4230/LIPIcs.ISAAC.2019.59
• Corpus ID: 53250178

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}
}

• Serge Gaspers, Joshua Lau
• Published 1 November 2018
• Mathematics, Computer Science
• ArXiv

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 problem remains open in the case where $G$ is a tree. We show that MinIGL on trees is subexponential by giving a $2^{O\left((n \log n)^{5/6}\right)}$ time, polynomial space algorithm, where $n$ is the number of vertices of the input graph.
The distance distribution of a graph $G$ is a sequence $\{a_i\}$ describing the number of vertex pairs distance $i$ apart in $G$: $a_i = |\{ \{u, v\}: d_G(u, v) = i\}|$. Given only the distance distribution, one can easily determine graph parameters such as diameter, Wiener index, and particularly, the IGL. We show that the distance distribution of a tree can be computed in $O(n \log^2 n)$ time by reduction to polynomial multiplication. We also extend the result to graphs of bounded treewidth by showing that the first $P$ values of the distance distribution can be computed in $2^{O(\text{tw}(G))} n^{1 + \varepsilon} \sqrt{P}$ time.