Approximation algorithms for spanner problems and Directed Steiner Forest

  • Authors:

  • Piotr Berman;Arnab Bhattacharyya;Konstantin Makarychev;Sofya Raskhodnikova;Grigory Yaroslavtsev

  • Affiliations:

  • Pennsylvania State University, University Park, PA 16802, United States;Princeton University, Princeton, NJ 08540, United States;Microsoft Research, Redmond, WA 98052, United States;Pennsylvania State University, University Park, PA 16802, United States;Pennsylvania State University, University Park, PA 16802, United States

  • Venue:
  • Information and Computation
  • Year:
  • 2013

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Abstract

We present an O(nlogn)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G=(V,E) with nonnegative edge lengths d:E-R^=^0 and a stretchk=1, a subgraph H=(V,E"H) is a k-spanner of G if for every edge (s,t)@?E, the graph H contains a path from s to t of length at most k@?d(s,t). The previous best approximation ratio was O@?(n^2^/^3), due to Dinitz and Krauthgamer (STOC @?11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths from O@?(n) to O(n^1^/^3logn). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS @?10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer@?s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n^1^/^3logn)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n^2^/^3^+^@e) for any constant @e0, which improves the O(n^@e@?min(n^4^/^5,m^2^/^3)) ratio due to Feldman, Kortsarz and Nutov (JCSS@?12).