Parameterized Complexity of Diameter

TL;DR

This paper explores the parameterized complexity of computing the diameter in graphs, aiming to identify parameters that enable faster algorithms beyond known lower bounds under the Strong Exponential Time Hypothesis.

Contribution

It systematically investigates a hierarchy of structural graph parameters to determine which allow for fixed-parameter tractable algorithms for diameter computation.

Findings

Identifies parameters enabling $f(k)(n+m)$ algorithms for diameter

Provides a hierarchy of graph parameters related to diameter complexity

Highlights limitations imposed by the Strong Exponential Time Hypothesis

Abstract

Diameter -- the task of computing the length of a longest shortest path -- is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no $O(n^{1.99})$-time algorithm even in sparse graphs [Roditty and Williams, 2013]. To circumvent this lower bound we aim for algorithms with running time $f(k)(n+m)$ where $k$ is a parameter and $f$ is a function as small as possible. We investigate which parameters allow for such running times. To this end, we systematically explore a hierarchy of structural graph parameters.