Approximating the diameter of a graph

In this paper we consider the fundamental problem of approximating the diameter $D$ of directed or undirected graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SIAM J. Comput. 1999] presented an algorithm that computes in $\Ot(m\sqrt n + n^2)$ time an estimate $\hat{D}$ for the diameter of an $n$-node, $m$-edge graph, such that $\lfloor 2/3 D \rfloor \leq \hat{D} \leq D$. In this paper we present an algorithm that produces the same estimate in $\Ot(m\sqrt n)$ expected running time. We then provide strong evidence that a better approximation may be hard to obtain if we insist on an $O(m^{2-\eps})$ running time. In particular, we show that if there is some constant $\eps>0$ so that there is an algorithm for undirected unweighted graphs that runs in $O(m^{2-\eps})$ time and produces an approximation $\hat{D}$ such that $(2/3+\eps) D \leq \hat{D} \leq D$, then SAT for CNF formulas on $n$ variables can be solved in $O^{*}((2-\delta)^{n})$ time for some constant $\delta>0$, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false. Motivated by this somewhat negative result, we study whether it is possible to obtain a better approximation for specific cases. For unweighted directed or undirected graphs, we show that if $D=3h+z$, where $h\geq 0$ and $z\in {0,1,2}$, then it is possible to report in $\tilde{O}(\min{m^{2/3} n^{4/3},m^{2-1/(2h+3)}})$ time an estimate $\hat{D}$ such that $2h+z \leq \hat{D}\leq D$, thus giving a better than 3/2 approximation whenever $z\neq 0$. This is significant for constant values of $D$ which is exactly when the diameter approximation problem is hardest to solve. For the case of unweighted undirected graphs we present an $\tilde{O}(m^{2/3} n^{4/3})$ time algorithm that reports an estimate $\hat{D}$ such that $\lfloor 4D/5\rfloor \leq \hat{D}\leq D$.