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

Abstract: We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing 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 [SWAT 2018]. For $O(n)$-sized spanners, we improve the lower bound on the additive stretch from $\Omega(n<sup>{1/11})$ to $\Omega(n<sup>{2/21})$. For $O(n)$-sized emulators, we improve the lower bound on the additive stretch from $\Omega(n<sup>{1/18})$ to $\Omega(n<sup>{1/16})$. For $O(m)$-sized shortcut sets, we improve the lower bound on the graph diameter from $\Omega(n<sup>{1/11})$ to $\Omega(n<sup>{1/8})$. 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.