Generalized Preconditioning and Network Flow Problems

We consider approximation algorithms for the problem of finding $x$ of minimal norm $\|x\|$ satisfying a linear system $\mathbf{A} x = \mathbf{b}$, where the norm $\|\cdot \|$ is arbitrary and generally non-Euclidean. We show a simple general technique for composing solvers, converting iterative solvers with residual error $\|\mathbf{A} x - \mathbf{b}\| \leq t^{-\Omega(1)}$ into solvers with residual error $\exp(-\Omega(t))$, at the cost of an increase in $\|x\|$, by recursively invoking the solver on the residual problem $\tilde{\mathbf{b}} = \mathbf{b} - \mathbf{A} x$. Convergence of the composed solvers depends strongly on a generalization of the classical condition number to general norms, reducing the task of designing algorithms for many such problems to that of designing a \emph{generalized preconditioner} for $\mathbf{A}$. The new ideas significantly generalize those introduced by the author's earlier work on maximum flow, making them more widely applicable. As an application of the new technique, we present a nearly-linear time approximation algorithm for uncapacitated minimum-cost flow on undirected graphs. Given an undirected graph with $m$ edges labelled with costs, and $n$ vertices labelled with demands, the algorithm takes $\epsilon^{-2}m^{1+o(1)}$-time and outputs a flow routing the demands with total cost at most $(1+\epsilon)$ times larger than minimal, along with a dual solution proving near-optimality. The generalized preconditioner is obtained by embedding the cost metric into $\ell_1$, and then considering a simple hierarchical routing scheme in $\ell_1$ where demands initially supported on a dense lattice are pulled from a sparser lattice by randomly rounding unaligned coordinates to their aligned neighbors. Analysis of the generalized condition number for the preconditioner follows that of the classical multigrid algorithm for lattice Laplacian systems.