Aleksander Madry

Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs

We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1-ε)-approximately maximum s-t flow in time ~O(mn1/3ε-11/3). A dual version of our approach gives the fastest known algorithm for computing a (1+ε)-approximately minimum s-t cut. It takes ~O(m+n4/3ε-16/3) time. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time ~O({m√nε-1), and approximately minimum s-t cuts in time ~O(m+n3/2ε-3).

Faster Generation of Random Spanning Trees

In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative

Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithms

(1+\delta)

A Polylogarithmic-Competitive Algorithm for the k -Server Problem

of uniform in expected time

Computing Maximum Flow with Augmenting Electrical Flows

\TO(m\sqrt{n}\log 1/\delta)

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

. This improves the sparse graph case of the best previously known worst-case bound of

On the Resiliency of Randomized Routing Against Multiple Edge Failures

O(\min \{mn, n^{2.376}\})

The Semi-stochastic Ski-rental Problem

, which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory.

Towards Deep Learning Models Resistant to Adversarial Attacks

We combine the work of Garg and Konemann, and Fleischer with ideas from dynamic graph algorithms to obtain faster (1-ε)-approximation schemes for various versions of the multicommodity flow problem. In particular, if ε is moderately small and the size of every number used in the input instance is polynomially bounded, the running times of our algorithms match -- up to poly-logarithmic factors and some provably optimal terms -- the Ω(mn) flow-decomposition barrier for single-commodity flow.

An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem

We give the first polylogarithmic-competitive randomized algorithm for the k-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log2 k) for any metric space on n points. This improves upon the (2k-1)-competitive algorithm of Koutsoupias and Papadimitriou (J. ACM 1995) whenever n is sub-exponential in k.