Abstract
A k-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor k. We focus on the following problem: Given a graph and a value k, can we find a k-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total number of edges. Switching the objective to the degree introduces significant new challenges, and currently the only known approximation bound is an O~(Delta^(3-2*sqrt(2)))-approximation for the special case when k = 2 [Chlamtac, Dinitz, Krauthgamer FOCS 2012] (where Delta is the maximum degree in the input graph). In this paper we give the first non-trivial algorithm and polynomial-factor hardness of approximation for the case of general k. Specifically, we give an LP-based O~(Delta^((1-1/k)^2) )-approximation and prove that it is hard to approximate the optimum to within Delta^Omega(1/k) when the graph is undirected, and to within Delta^Omega(1) when it is directed.
Highlights
A spanner of a graph is a sparse subgraph that approximately preserves distances
Much of this work has focused on the fundamental tradeoffs between stretch, size, and total weight, such as the seminal result of Althöfer et al that every graph admits a (2k − 1)-spanner with at most n1+1/k edges [1] and its many extensions (e. g., to dealing with total weight [9])
Degree objectives are notoriously difficult, and so almost all work on approximation algorithms for spanners has focused on minimizing the number of edges, as opposed to maximum degree
Summary
A spanner of a graph is a sparse subgraph that approximately preserves distances. Formally, a k-spanner of a graph G = (V, E) is a subgraph H of G in which dH(u, v) ≤ k · dG(u, v) for all u, v ∈ V , where dH and dG denote shortest-path distances in H and G, respectively. Graph spanners were originally introduced in the context of distributed computing [25, 26], and since have been extensively studied from both a distributed and a centralized perspective. Much of this work has focused on the fundamental tradeoffs between stretch, size, and total weight, such as the seminal result of Althöfer et al that every graph admits a (2k − 1)-spanner with at most n1+1/k edges [1] and its many extensions Kortsarz and Peleg initiated the study of the maximum degree of a spanner, giving an O(∆1/4)-approximation for LD2S [24] (w√here ∆ is the maximum degree of the input graph). This was only recently improved to O(∆3−2 2+ε ) = O(∆0.17...+ε ) for arbitrarily small constant ε > 0 by Chlamtác, Dinitz, and Krauthgamer [13]. Despite the length of time since minimizing the degree was first considered (over 15 years) and the significant amount of work on other spanner problems, no nontrivial upper or lower bounds were known previous to this work for LDkS when k ≥ 3
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