Feodor F. Dragan, Anh Tran and Chenyu Yan Algorithmics Lab, Spring 2006, Kent State University

1. Network Flow Spanners Light Flow Spanner Design (LFSD) PROBLEM Given an undirected graph G=(V,E), edge-costsp(e) and edge-capacities c(e),and integers B, t. Does G admit a spanning subgraph H =(V,E’)such that the cost of building H is bounded by B ( ) and for any vertices x, y the (x,y)-maxFlow in G is at most t times the (x,y)-maxFlow in H ( ) ? EXAMPLE 2 denotes the capacity; 3 denotes the price In H, those 7 packages can be sent in two rounds (5+2) In G, we can send 7 packages from source to sink at once OUR RESULTS • The Light Flow-Spanner, Sparse Flow-Spanner, Light-Edge-Connectivity-Spanner and Sparse Edge-Connectivity-Spanner problems are NP-complete. • The Light Tree Flow-Spanner problem is NP-complete. • We give two approximation algorithms for the Light Tree Flow-Spanner problem • Our problems belong to the class of Network Design and Network Survivability problems • They take into account Fault Tolerance, Bandwidth Constraints and Link Failures VARIATIONS OF LFSD Feodor F. Dragan, Anh Tran and Chenyu Yan Algorithmics Lab, Spring 2006, Kent State University RELATED WORK k-Edge-Connected-Spanning-Subgraph problem: • Given a graph Galong with an integer k, one seeks a spanning subgraph of G that is k-edge-connected • MAX SNP-hard [Fernandes’98], (1+2/k)-approx. algorithm [Gabow et. al.’05] • Original edge-connectivities are not taking into account Survivable-network-design problem (SNDP): • Given a graph G=(V, E), a non-negative cost p(e)for every edge e∊E and a non-negative connectivity requirement rijfor every (unordered) pair of vertices i, j. One needs to find a minimum-cost subgraph in which each pair of vertices i, j is joined by at least rijedge-disjoint paths. • NP-hard problem, 2(1+1/2+1/3+…+1/k)-approximation algorithm [Gabow et. al.’98, Goemans et. al.’94] • By settingrij=FG(i, j)/tfor each pair of verticesi, j,our LECSproblem can be reduced to SNDP. EXPERIMENTAL RESULTS Experimental results for both approximation algorithms Experimental results on Internet-like graphs Experimental results on random graphs CONCLUSION TheLight Tree t-Flow Spanner approximation algorithm works better for small stretch factors t while the Light Tree Flow Spanner approximation algorithm works better for larger t. Results were partially presented at LATIN’ 2006 Conference, March 20-24, Valdivia, Chile