Bypassing Erd\H{o}s' Girth Conjecture: Hybrid Stretch and Sourcewise Spanners

An (α, β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v,H) ≤ α ·dist(u, v,G)+β for every pair (u, v) ∈ V × V , where dist(u, v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1, 0)-spanner of size O(n). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k− 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k-stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G with m edges, there is a (polynomially constructible) k-hybrid spanner with O(k · n) edges. This should be compared against the current best (α, β) spanner construction of [5] that obtains (k, k − 1) stretch with O(k · n) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S × V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more).