We present a nearly-linear time approximation algorithm for uncapacitated minimum-cost flow in undirected graphs, along with a more general framework for approximately solving problems of the form: find x satisfying Ax = b with minimal norm ‖x‖, where the norm is generally non- Euclidean. For most of the extensive applications of the latter problem, the exact constraints are essential, so an x satisfying Ax = b with almost-minimal norm is acceptable, while relaxing Ax = b to Ax ≈ b significantly beyond numerical precision is not. On the other hand, existing nearly-linear time solvers for non-Euclidean norms use dual or penalty methods, yielding the opposite notion where | x| is minimal while ‖b — Ax‖ < ί−Ω(1) after t iterations. We show that by composing solvers of the latter type, we may obtain solvers of the more-useful former type. Convergence of the composed solvers depends strongly on a generalization of the classical condition number to general norms. Following our framework, the task of the algorithm designer for such problems is reduced to that of designing a generalized preconditioner for A. Applying the framework to uncapacitated minimum- cost flow, we present an algorithm that, given an undirected graph with m edges labelled with costs, and n vertices labelled with demands, takes ∊−2m1+o(1)-time and outputs a flow routing the demands with total cost at most (1 + ∊) times larger than minimal, along with a dual solution proving near-optimality. The generalized preconditioner is obtained by embedding the cost metric into ℓ1, and then considering a simple hierarchical routing scheme in £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 corresponding preconditioner follows that of the classical multigrid algorithm for lattice Laplacian systems.

We show the strong-convexity assumption of regularization-based methods for solving bilinear saddle point problems may be relaxed to a weaker notion of area-convexity with respect to an alternating bilinear form. This allows bypassing the infamous '' barrier for strongly convex regularizers that has stalled progress on a number of algorithmic problems.

We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using congestion-approximators: easy-to-compute functions that approximate the congestion required to route single-commodity demands in a graph to within some factor α. Our algorithm maintains an arbitrary flow that may have some residual excess and deficits, while taking steps to minimize a potential function measuring the congestion of the current flow plus an over-estimate of the congestion required to route the residual demand. Since the residual term over-estimates, the descent process gradually moves the contribution to our potential function from the residual term to the congestion term, eventually achieving a flow routing the desired demands with nearly minimal congestion after Õ(α2ε-2log2n) iterations. Our approach is similar in spirit to that used by Spielman and Teng (STOC 2004) for solving Laplacian systems, and we summarize our approach as trying to do for ℓ∞-flows what they do for ℓ∞-flows. Together with a nearly linear time construction of a no(1)-congestion-approximator, we obtain 1 + ε-optimal singlecommodity flows undirected graphs in time m1+o(1)ε-2, yielding the fastest known algorithm for that problem. Our requirements of a congestion-approximator are quite low, suggesting even faster and simpler algorithms for certain classes of graphs. For example, an α-competitive oblivious routing tree meets our definition, even without knowing how to route the tree back in the graph. For graphs of conductance φ, a trivial φ-1-congestionapproximator gives an extremely simple algorithm for finding Õ(mφ-1).

This paper ties the line of work on algorithms that find an O(¿(log n))-approximation to the SPARSEST CUT together with the line of work on algorithms that run in subquadratic time by using only single-commodity flows. We present an algorithm that simultaneously achieves both goals, finding an O(¿(log (n)/¿))-approximation using O(n¿logO(1)n) max-flows. The core of the algorithm is a stronger, algorithmic version of Arora et al.'s structure theorem, where we show that matching-chaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks. By using that specialized algorithm in place of a black-box solver, we are able to solve those instances much more efficiently. We also show the cut-matching game framework can not achieve an approximation any better than ¿(log(n)/log log(n)) without re-routing flow.