Lower bounds for graph embeddings and combinatorial preconditioners

Given a general graph G, a fundamental problem is to find a spanning tree H that best approximates G by some measure. Often this measure is some combination of the congestion and dilation of an embedding of G into H. One example is the routing time ρ(G, H) ≤ O(congestion + dilation), the number of steps necessary to route pairwise demands G on network links H in the store-and-forward packet routing model. Another is the condition number κf (G, H) ≤ O(congestion·dilation), the square root of which bounds the number of iterations necessary to solve a linear system with coefficient matrix G preconditioned by H using the classical conjugate gradient method. The algorithmic applications of being able to find (efficiently) a good tree approximation H for a graph G are numerous; but what if no good tree exists? In this paper, we seek to identify the class of graphs G which are intrinsically difficult to approximate by a particular measure. It is easily seen that with respect to rout...

• Type: Conference
• Year: 2004
• Where: SPAA
• Authors: Gary L. Miller, Peter C. Richter
• Keywords: Distributed And Parallel Computing, SPAA 2004