Rutgers Discrete Mathematics Seminar

Title: Improved Approximation for the Directed Spanner Problem

Speaker: Arnab Bhattacharyya, Princeton University

Date: Tuesday, September 13, 2011 2:00pm

Location: Hill Center, Room 425, Rutgers University, Busch Campus, Piscataway, NJ


We present an O(sqrt(n) log n)-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 real edge lengths and a stretch parameter k>= 1, a subgraph H = (V,E') is a k-spanner of G if for every edge (s,t) in E, the graph H contains a path from s to t of length at most k times the length of the shortest path from s to t in G. 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~(sqrt{n}) to O(n^{1/3} log n). 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^{1/3} log n)-approximation for the k-spanner problem on undirected graphs with unit lengths. An easy O(sqrt{n})-approximation algorithm for this problem has been the best known for decades.

Joint work with Piotr Berman, Konstantin Makarychev, Sofya Raskhodnikova and Grigory Yaroslavtsev.