Obstructions to faster diameter computation: Asteroidal sets

TL;DR

This paper introduces new algorithms for efficiently computing the diameter and eccentricities in specific graph classes, overcoming obstructions posed by general computational hardness results.

Contribution

It provides the first deterministic algorithms with subquadratic runtime for diameter and eccentricity approximation in certain graph classes, improving upon previous complexity bounds.

Findings

Deterministic ${ m O}( ext{alpha}^3 m^{3/2})$-time diameter algorithm for graphs in $Ext_{\alpha}$

Linear-time diameter computation for graphs with bounded clique number

Approximate eccentricities within ${\rm O}( ext{alpha}^2 m)$ time

Abstract

An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $\alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{\alpha}$ can be computed in deterministic ${\cal O}(\alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${\cal O}(\alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${\cal O}(m^{2-\epsilon})$ time for any $\epsilon > 0$. As important special cases of our main result, we derive an ${\cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${\cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.