Towards Bypassing Lower Bounds for Graph Shortcuts

For a given (possibly directed) graph G, a hopset (a.k.a. shortcut set) is a (small) set of edges whose addition reduces the graph diameter while preserving desired properties from the given graph G, such as, reachability and shortest-path distances. The key objective is in optimizing the tradeoff between the achieved diameter and the size of the shortcut set (possibly also, the distance distortion). Despite the centrality of these objects and their thorough study over the years, there are still significant gaps between the known upper and lower bound results. A common property shared by almost all known shortcut lower bounds is that they hold for the seemingly simpler task of reducing the diameter of the given graph, DG, by a constant additive term, in fact, even by just one! We denote such restricted structures by (DG−1)-diameter hopsets. In this paper we show that this relaxation can be leveraged to narrow the current gaps, and in certain cases to also bypass the known lower bound results, when restricting to sparse graphs (with O(n) edges): Hopsets for Directed Weighted Sparse Graphs. For every n-vertex directed and weighted sparse graph G with DG ≥ n¹/⁴, one can compute an exact (DG − 1)-diameter hopset of linear size. Combining this with known lower bound results for dense graphs, we get a separation between dense and sparse graphs, hence shortcutting sparse graphs is provably easier. For reachability hopsets, we can provide (DG − 1)-diameter hopsets of linear size, for sparse DAGs, already for DG ≥ n¹/⁵. This should be compared with the diameter bound of O^e(n¹/³) [Kogan and Parter, SODA 2022], and the lower bound of DG = n¹/⁶ by [Huang and Pettie, SIAM J. Discret. Math. 2018]. Additive Hopsets for Undirected and Unweighted Graphs. We show a construction of +24 additive (DG − 1)-diameter hopsets with linear number of edges for DG ≥ n¹/¹² for sparse graphs. This bypasses the current lower bound of DG = n¹/⁶ obtained for exact (DG − 1)-diameter hopset by [HP’18]. For general graphs, the bound becomes DG ≥ n¹/⁶ which matches the lower bound of exact (DG − 1) hopsets implied by [HP’18]. We also provide new additive D-diameter hopsets with linear size, for any given diameter D. Altogether, we show that the current lower bounds can be bypassed by restricting to sparse graphs (with O(n) edges). Moreover, the gaps are narrowed significantly for any graph by allowing for a constant additive stretch.