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# Minimizing and Computing the Inverse Geodesic Length on Trees

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LIPIcs.ISAAC.2019.59.pdf
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## Acknowledgements

We thank David Harvey and Ray Li for fruitful discussions and feedback.

## Cite As

Serge Gaspers and Joshua Lau. Minimizing and Computing the Inverse Geodesic Length on Trees. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 59:1-59:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.59

## Abstract

For any fixed measure H that maps graphs to real numbers, the MinH problem is defined as follows: given a graph G, an integer k, and a target tau, is there a set S of k vertices that can be deleted, so that H(G - S) is at most tau? In this paper, we consider the MinH problem on trees. We call H balanced on trees if, whenever G is a tree, there is an optimal choice of S such that the components of G - S have sizes bounded by a polynomial in n / k. We show that MinH on trees is Fixed-Parameter Tractable (FPT) for parameter n / k, and furthermore, can be solved in subexponential time, and polynomial space, whenever H is additive, balanced on trees, and computable in polynomial time. A particular measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the efficiency and connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: IGL(G) = sum_{{u,v} subseteq V} 1/d_G(u,v). While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in 2^{o(k + n + m)} time, even on bipartite graphs with n vertices and m edges, the complexity status of the problem remains open in the case where G is a tree. We show that IGL is balanced on trees, to give a 2^O((n log n)^(5/6)) time, polynomial space algorithm. The distance distribution of G is the 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 with small treewidth by showing that the first p values of the distance distribution can be computed in 2^(O(tw(G))) n^(1 + epsilon) sqrt(p) time, and the entire distance distribution can be computed in 2^(O(tw(G))) n^{1 + epsilon} time, when the diameter of G is O(n^epsilon') for every epsilon' > 0.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Trees
• Mathematics of computing → Graph algorithms
##### Keywords
• Trees
• Treewidth
• Fixed-Parameter Tractability
• Inverse Geodesic Length
• Vertex deletion
• Polynomial multiplication
• Distance distribution

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