Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product

We obtain improved lower bounds for additive spanners, additive emulators, and diameterreducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of edges that when added to a directed graph, decreases its diameter. The previous best known lower bounds for these three structures are given by Huang and Pettie [HP18]. For O(n)-sized spanners, we improve the lower bound on the additive stretch from Ω(n 1/11 ) to Ω(n 2/21 ). For O(n)-sized emulators, we improve the lower bound on the additive stretch from Ω(n 1/18 ) to Ω(n 1/16 ). For O(m)-sized shortcut sets, we improve the lower bound on the graph diameter from Ω(n 1/11 ) to Ω(n 1/8 ).

Our key technical contribution, which is the basis of all of our bounds, is an improvement of a graph product known as an alternation product.