Jakub W. Pachocki

Solving SDD linear systems in nearly m log 1/2 n time

We show an algorithm for solving symmetric diagonally dominant (SDD) linear systems with m non-zero entries to a relative error of ε in O(m log1/2 n logc n log(1/ε)) time. Our approach follows the recursive preconditioning framework, which aims to reduce graphs to trees using iterative methods. We improve two key components of this framework: random sampling and tree embeddings. Both of these components are used in a variety of other algorithms, and our approach also extends to the dual problem of computing electrical flows. We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. In the graph setting, this leads to ultra-sparsifiers that have optimal behavior in expectation. The improved running time makes previous low stretch embedding algorithms the running time bottleneck in this framework. In our analysis, we relax the requirement of these embeddings to snowflake spaces. We then obtain a two-pass approach algorithm for constructing optimal embeddings in snowflake spaces that runs in O(m log log n) time. This algorithm is also readily parallelizable.

Scalable Large Near-Clique Detection in Large-Scale Networks via Sampling

Extracting dense subgraphs from large graphs is a key primitive in a variety of graph mining applications, ranging from mining social networks and the Web graph to bioinformatics [41]. In this paper we focus on a family of poly-time solvable formulations, known as the k-clique densest subgraph problem (k-Clique-DSP) [57]. When k=2, the problem becomes the well-known densest subgraph problem (DSP) [22, 31, 33, 39]. Our main contribution is a sampling scheme that gives densest subgraph sparsifier, yielding a randomized algorithm that produces high-quality approximations while providing significant speedups and improved space complexity. We also extend this family of formulations to bipartite graphs by introducing the (p,q)-biclique densest subgraph problem ((p,q)-Biclique-DSP), and devise an exact algorithm that can treat both clique and biclique densities in a unified way. As an example of performance, our sparsifying algorithm extracts the 5-clique densest subgraph --which is a large-near clique on 62 vertices-- from a large collaboration network. Our algorithm achieves 100% accuracy over five runs, while achieving an average speedup factor of over 10,000. Specifically, we reduce the running time from ∼2 107 seconds to an average running time of 0.15 seconds. We also use our methods to study how the k-clique densest subgraphs change as a function of time in time-evolving networks for various small values of k. We observe significant deviations between the experimental findings on real-world networks and stochastic Kronecker graphs, a random graph model that mimics real-world networks in certain aspects. We believe that our work is a significant advance in routines with rigorous theoretical guarantees for scalable extraction of large near-cliques from networks.

Geometric median in nearly linear time

In this paper we provide faster algorithms for solving the geometric median problem: given n points in d compute a point that minimizes the sum of Euclidean distances to the points. This is one of the oldest non-trivial problems in computational geometry yet despite a long history of research the previous fastest running times for computing a (1+є)-approximate geometric median were O(d· n4/3є−8/3) by Chin et. al, Õ(dexpє−4logє−1) by Badoiu et. al, O(nd+poly(d,є−1)) by Feldman and Langberg, and the polynomial running time of O((nd)O(1)log1/є) by Parrilo and Sturmfels and Xue and Ye. In this paper we show how to compute such an approximate geometric median in time O(ndlog3n/є) and O(dє−2). While our O(dє−2) is a fairly straightforward application of stochastic subgradient descent, our O(ndlog3n/є) time algorithm is a novel long step interior point method. We start with a simple O((nd)O(1)log1/є) time interior point method and show how to improve it, ultimately building an algorithm that is quite non-standard from the perspective of interior point literature. Our result is one of few cases of outperforming standard interior point theory. Furthermore, it is the only case we know of where interior point methods yield a nearly linear time algorithm for a canonical optimization problem that traditionally requires superlinear time.

Online Row Sampling

Finding a small spectral approximation for a tall

Routing under balance

n \times d

Efficient counting of square substrings in a tree

matrix

Tight Lower Bounds on Graph Embedding Problems

A

On the String Consensus Problem and the Manhattan Sequence Consensus Problem

is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of

Efficient Counting of Square Substrings in a Tree

A

On the string consensus problem and the Manhattan sequence consensus problem

. Row sampling improves interpretability, saves space when