Solving Symmetric Diagonally-Dominant Systems by Preconditioning

In this paper we design support-tree preconditioners for n × n matrices with m nonzeros that aresymmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix). This reduces to de-signing such a preconditioner for a Laplacian matrix, A, which can be interpreted as an undirectednonnegatively-weighted graph, G with n vertices and m edges. Preconditioners accelerate the conver-gence of iterative methods for solving linear systems, and our preconditioner allows us to analyze theconvergence of a particular algorithm, due to Gremban and Miller, called support-tree conjugate gra-dient (STCG). An advantage of support-tree preconditioners is that STCG parallelizes well. We showthat STCG equipped with our preconditioner requires O(m log n ·√ dilexp(G)) work and O(m) spaceto solve the system Ax = b, where dilexp(G) is an edge-expansion-based upper bound on the diameterof G.Existing bounds depend only on the size of the matrix (graph), hence our bound is incomparable. Forinstance, if G is a bounded-degree expander graph with uniform edge weights, dilexp(G) = O(log 2 n), andthe work is O(n log n). This is currently the best known bound for Laplacians of expander graphs. Weshow that dilexp(G) is always at most n, hence our bound is at most O(m√n log n) for any Laplacian(or SDD) matrix. For sufficiently dense systems, when m = Ω(n), this bound offers the best knownwork guarantee of any linear-space method.The main technical contributions of this paper include (i) adapting a recent result of Räcke todesigning support-tree preconditioners, (ii) extending a power dissipation approach for bounding supportnumbers of preconditioners, and (iii) applying the methods used in Leighton and Rao’s approximatemax-flow min-cut theorem to the “asymmetric” product flows the arise in Räcke’s construction.