An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t^{1/2} log n) bound. The result also extends to the more general Koml\'{o}s setting and gives an algorithmic O(log^{1/2} n) bound

On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems where iterated rounding has been useful

Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most $t$ sets. We give an algorithm that finds a coloring with discrepancy $O((t \log n \log s)^{1/2})$ where $s$ is the maximum cardinality of a set. This improves upon the previous constructive bound of $O(t^{1/2} \log n)$ based on algorithmic variants of the partial coloring method, and for small $s$ (e.g.$s=\textrm{poly}(t)$) comes close to the non-constructive $O((t \log n)^{1/2})$ bound due to Banaszczyk. Previously, no algorithmic results better than $O(t^{1/2}\log n)$ were known even for $s = O(t^2)$. Our method is quite robust and we give several refinements and extensions. For example, the coloring we obtain satisfies the stronger size-sensitive property that each set $S$ in the set system incurs an $O((t \log n \log |S|)^{1/2})$ discrepancy. Another variant can be used to essentially match Banaszczyk's bound for a wide class of instances even where $s$ is arbitrarily large. Finally, these results also extend directly to the more general Koml\'{o}s setting

Dependent randomized rounding for clustering and partition systems with knapsack constraints

Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among the cluster-centers. This, and much more general clustering problems, can be formulated with "knapsack" and "partition" constraints. We develop new randomized algorithms targeting such problems, and study two in particular: multi-knapsack median and multi-knapsack center. Our rounding algorithms give new approximation and pseudo-approximation algorithms for these problems. One key technical tool, which may be of independent interest, is a new tail bound analogous to Feige (2006) for sums of random variables with unbounded variances. Such bounds are very useful in inferring properties of large networks using few samples

On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems such as rounding column-sparse LPs, makespan minimization on unrelated machines, degree-bounded spanning trees and multi-budgeted matchings

The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues

A classic result of Banaszczyk (Random Str. & Algor. 1997) states that given any n vectors in Rm with ℓ2-norm at most 1 and any convex body K in Rm of Gaussian measure at least half, there exists a ±1 combination of these vectors that lies in 5K. Banaszczyk’s proof of this result was non-constructive and it was open how to find such a ±1 combination in polynomial time. In this paper, we give an efficient randomized algorithm to find a ±1 combination of the vectors which lies in cK for some fixed constant c > 0. This leads to new efficient algorithms for several problems in discrepancy theory

Approximation-friendly discrepancy rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing

Approximation-friendly discrepancy rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitationswhen compared to approaches such as randomized and iterative rounding.We provide an extension of the discrepancy-based rounding algorithmdue to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing

Approximation-friendly discrepancy rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing

Approximation-friendly discrepancy rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing