Approximating Spanners and Directed Steiner Forest: Upper and Lower Bounds

Abstract

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA '16, Bodwin-Williams SODA '16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n 3/5+ε )-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(log n)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation).Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n 3/5+ε )-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O(n 2/3+ε )-approximation due to Berman et al. [ICALP '11] (which holds for general edge costs).

Distributed Spanner Approximation

“…For any constant > 0 and k ≥ 3 there are no polynomial-time algorithms that approximate the k-spanner problem within a factor better than 2 (log 1− n)/k [19], or the directed k-spanner problem within a factor better than 2 (log 1− n) [31]. Similar results are known for additional variants [14,31].…”

“…These approximation ratios are matched by a recent distributed O(k log n)-round algorithm, that uses only polynomial local computations [22]. Approximation algorithms are given also for pairwise spanners and distance preservers [14], for spanners with lowest maximum degree [13,15,22,47], for fault-tolerant spanners [21,23], and more.…”

On the Structure of Unique Shortest Paths in Graphs

“…Shortest path structure is often studied through the lens of distance preservers, in which the goal is to determine the maximum possible number of edges that might be needed in a subgraph that preserves p given pairwise distances in an n-node input graph. Some of the work on distance preservers includes [10,16,9,8,18,14,13]. In particular, the value and limitations of shortest path consistency are directly studied in [16,9,8], and this theme is continued in this work.…”

“…Another example of a well-studied area in theoretical computer science in need of new shortest paths structure is distance preservers [10,16,9,8,14], in which the goal is to determine the maximum possible number of edges that could be needed in a subgraph that preserves p given pairwise distances in an n-node input graph. For e.g.…”

Reachability Preservers: New Extremal Bounds and Approximation Algorithms

“…The most fundamental case of DSN, which captures the essence of its difficulty, is when all the weights are the same, or equivalently, if the graph is unweighted (the UDSN problem). In a recent breakthrough, Chlamtac, Dinitz, Kortsarz, and Laekhanukit [18] achieved a better approximation factor of O(n 3/5+ε ) for UDSN. On the negative side, it is quasi-NP-hard to approximate UDSN to within 2 log 1−ε n for all ε > 0 [25].…”