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 $\lfloor 4D/5\rfloor \leq \hat{D}\leq D$.

💡 Research Summary

The paper addresses the classic problem of approximating the diameter (D) of an unweighted directed or undirected graph with (n) vertices and (m) edges. The authors first improve upon the seminal algorithm of Aingworth, Chekuri, Indyk, and Motwani (1999), which runs in (\tilde O(m\sqrt n + n^2)) time and guarantees an estimate (\hat D) such that (\lfloor 2D/3\rfloor \le \hat D \le D). By employing a randomized sampling technique, they show that the same guarantee can be achieved in expected (\tilde O(m\sqrt n)) time. The key idea is to select (\sqrt n) random source vertices, run a BFS from each, and use the distances discovered to infer a lower bound on the diameter. A careful probabilistic analysis demonstrates that, with high probability, the maximum distance uncovered among these BFS trees is at least (\lfloor 2D/3\rfloor), while the total work remains proportional to (m\sqrt n).

The second major contribution is a conditional lower bound. The authors prove that any algorithm running in (O(m^{2-\varepsilon})) time (for some constant (\varepsilon>0)) that yields an approximation better than ((2/3+\varepsilon)D) would imply a sub‑exponential algorithm for SAT, specifically an (O^*((2-\delta)^n)) algorithm for CNF‑SAT for some (\delta>0). This reduction ties the diameter‑approximation problem to the Strong Exponential Time Hypothesis (SETH) and suggests that improving the approximation factor beyond (2/3) while staying within truly sub‑quadratic time is unlikely unless SETH fails.

Motivated by this hardness result, the paper investigates special cases where a better approximation is feasible. For graphs whose diameter can be expressed as (D = 3h + z) with (h \ge 0) and (z \in {0,1,2}), the authors present an algorithm that runs in (\tilde O(\min{m^{2/3} n^{4/3},, m^{2-1/(2h+3)}})) time and returns an estimate (\hat D) satisfying (2h + z \le \hat D \le D). When (z \neq 0) this yields a factor better than (3/2), which is significant because the hardest instances for diameter approximation are those with small constant diameters. For the specific case of unweighted undirected graphs, they give a deterministic algorithm with running time (\tilde O(m^{2/3} n^{4/3})) that guarantees (\lfloor 4D/5\rfloor \le \hat D \le D), improving the approximation ratio to (4/5) while still staying well below the quadratic barrier.

In summary, the paper makes three intertwined contributions: (1) a practical speed‑up of the classic (2/3)-approximation to expected (\tilde O(m\sqrt n)) time; (2) a conditional lower bound linking any (O(m^{2-\varepsilon})) algorithm with a better-than‑(2/3) factor to the falsity of SETH; and (3) specialized algorithms that achieve stronger approximation ratios for graphs with small or structured diameters, including a (4/5)-approximation for undirected unweighted graphs in (\tilde O(m^{2/3} n^{4/3})) time. These results deepen our understanding of the trade‑off between runtime and approximation quality in diameter computation and delineate the boundaries of what can be achieved under widely believed complexity assumptions.