Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires $\Omega(n^2)$ time. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance, and adaptive constructions based on barrier functions.Comment: 22 pages. A preliminary version of this paper is to appear in proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015

An Integer Interior Point Method for Min-Cost Flow Using Arc Contractions and Deletions

We present an interior point method for the min-cost flow problem that uses arc contractions and deletions to steer clear from the boundary of the polytope when path-following methods come too close. We obtain a randomized algorithm running in expected $\tilde O( m^{3/2} )$ time that only visits integer lattice points in the vicinity of the central path of the polytope. This enables us to use integer arithmetic like classical combinatorial algorithms typically do. We provide explicit bounds on the size of the numbers that appear during all computations. By presenting an integer arithmetic interior point algorithm we avoid the tediousness of floating point error analysis and achieve a method that is guaranteed to be free of any numerical issues. We thereby eliminate one of the drawbacks of numerical methods in contrast to combinatorial min-cost flow algorithms that still yield the most efficient implementations in practice, despite their inferior worst-case time complexity

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

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time $\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right)$ where $\epsilon$ is the amount of error we are willing to tolerate. Here, $\kappa$ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever $\kappa$ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time $\widetilde{O}(m^{3/2} \log (1/\epsilon))$. In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201

Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

We show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an $n \times n$ Eulerian directed Laplacian with $m$ nonzero entries, we show how to compute an $\epsilon$-approximate solution in time $O(m \log^{O(1)} (n) \log (1/\epsilon))$. Through reductions from [Cohen et al. FOCS'16] , this gives the first nearly-linear time algorithms for computing $\epsilon$-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing $\epsilon$-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time $O((m+n2^{O(\sqrt{\log n \log \log n})})\log^{O(1)}(n \epsilon^{-1}))$. To achieve our results, we provide a structural result that we believe is of independent interest. We show that Laplacians of all strongly connected directed graphs have sparse approximate LU-factorizations. That is, for every such directed Laplacian ${\mathbf{L}}$, there is a lower triangular matrix $\boldsymbol{\mathit{{\mathfrak{L}}}}$ and an upper triangular matrix $\boldsymbol{\mathit{{\mathfrak{U}}}}$, each with at most $\tilde{O}(n)$ nonzero entries, such that their product $\boldsymbol{\mathit{{\mathfrak{L}}}} \boldsymbol{\mathit{{\mathfrak{U}}}}$ spectrally approximates ${\mathbf{L}}$ in an appropriate norm. This claim can be viewed as an analogue of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that, once constructed, they yield nearly-linear time algorithms for solving directed Laplacian systems.Comment: Appeared in FOCS 201