Combinatorial Preconditioners

The Conjugate Gradient (CG) method is an iterative algorithm for solving linear systems of equations Ax = b, where A is symmetric and positive de nite. The convergence of the method depends on the spectrum of A; when its eigenvalues are clustered, the method converges rapidly. In particular, CG converges to within a xed tolerance in O( √ κ), where κ = κ(A) = λmax(A)/λmin(A) is the spectral condition number of A. When the spectrum of A is not clustered, a preconditioner can accelerate convergence. The Preconditioned Conjugate Gradients (PCG) method applies the CG iteration to the linear system (B−1/2AB−1/2)(B1/2x) = B−1/2b using a clever transformation that only requires applications of A and B−1 in every iteration; B also needs to be symmetric positive de nite. The convergence of PCG is determined by the spectrum of (B−1/2AB−1/2), which is the same as the spectrum of B−1A. If a representation of B−1 can be constructed quickly and applied quickly, and if B−1A has a clustered spectrum, the method is very e ective. There are also variants of PCG that require only one of A and B to be positive de nite, and variants that allow them to be singular, under some technical conditions on their null spaces. Combinatorial preconditioning is a technique that relies on graph algorithms to construct e ective preconditioners. The simplest applications of combinatorial preconditioning target a class of matrices that are isomorphic to weighted undirected graph. The coe cient matrix A is viewed as its isomorphic graph GA. A specialized graph algorithm constructs another graph GB such that the isomorphic matrix B is a good preconditioner for A. The graph algorithm aims to achieve two goals: the inverse of B should be easy to apply, and the spectrum of B−1A should be clustered. It turns out that the spectrum of B−1A can be bounded in terms of properties of the graphs GA and GB; in particular, the quality of embeddings of GA in GB (and sometimes vice versa) plays a fundamental role in these spectral bounds. This chapter focuses on explaining the relationship between the spectrum of B−1A and quantitative properties of embeddings of the two graphs. The last section surveys algorithms that construct combinatorial preconditioners. We omit most proofs from this chapter; some are trivial, and the others appear in the paper cited in the statement of the theorem or lemma.