The Norms of Graph Spanners
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A t-spanner of a graph G is a subgraph H in which all distances are preserved up to a multiplicative t factor. A classical result of Althöfer et al. is that for every integer k and every graph G, there is a (2k-1)-spanner of G with at most O(n^1+1/k) edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "few" nodes of "large" degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the ℓ_p-norm of their degree vector, thus simultaneously modeling the number of edges (the ℓ_1-norm) and the maximum degree (the ℓ_∞-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k-1)-spanner has ℓ_p norm of at most (O(n), O(n^(k+p)/(kp))), and that this bound is tight (assuming the Erdős girth conjecture). We also study universal lower bounds, allowing us to give "generic" guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the ℓ_1 and ℓ_∞ norm. Finally, we show that at least in some situations, the ℓ_p norm behaves fundamentally differently from ℓ_1 or ℓ_∞: there are regimes (p=2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.
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