Greedy Algorithms for Shortcut Sets and Hopsets

Abstract: We explore the power of greedy algorithms for hopsets and shortcut sets. In particular, we propose simple greedy algorithms that, given an input graph $G$ and a parameter $β$, compute a shortcut set or an exact hopset $H$ of hopbound at most $β$, and we prove the following guarantees about the size $|H|$ of the output: For shortcut sets, we prove the bound $$|H| \le \tilde{O}\left( \frac{n^2}{β^3} + \frac{n^{3/2}}{β^{3/2}} \right).$$ This matches the current state-of-the-art upper bound by Kogan and Parter [SODA '22]. For exact hopsets of $n$-node, $m$-edge weighted graphs, the size of the output hopset is existentially optimal up to subpolynomial factors, under some technical assumptions. Despite their simplicity and conceptual implications, these greedy algorithms are slower than existing sampling-based approaches. Our second set of results focus on faster deterministic algorithms that are based on a certain greedy set cover approximation algorithm on paths in the transitive closure. One consequence is a deterministic algorithm that takes $O(mn^{2/3})$ time to compute a shortcut set of size $\tilde{O}(n)$ and hopbound $O(n^{1/3})$.