Shashwat Garg

An Algorithm for Komlós 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(t1/2 log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log1/2 n) bound.

Improved Local Search for Geometric Hitting Set

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3,2)-local search and give an (8+\epsilon)-approximation algorithm with expected running time ˜O(n^{2.34}); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n^{15}) -- that too just for unit disks. The techniques and ideas generalize to (4,3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3,2) local search gives an 8-approximation and no better \footnote{This is assuming the use of the standard framework. Improvement of the approximation factor by using additional properties specific to the problem may be possible.}. Similarly (4,3)-local search gives a 5-approximation for all these problems.

Algorithmic discrepancy beyond partial coloring

The partial coloring method is one of the most powerful and widely used method in combinatorial discrepancy problems. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number of times, and the errors can add up in an adversarial way. We give a new and general algorithmic framework that overcomes the limitations of the partial coloring method and can be applied in a black-box manner to various problems. Using this framework, we give new improved bounds and algorithms for several classic problems in discrepancy. In particular, for Tusnady&s problem, we give an improved O(log2 n) bound for discrepancy of axis-parallel rectangles and more generally an Od(logdn) bound for d-dimensional boxes in ℝd. Previously, even non-constructively, the best bounds were O(log2.5 n) and Od(logd+0.5n) respectively. Similarly, for the Steinitz problem we give the first algorithm that matches the best known non-constructive bounds due to Banaszczyk in the 𝓁∞ case, and improves the previous algorithmic bounds substantially in the 𝓁2 case. Our framework is based upon a substantial generalization of the techniques developed recently in the context of the Komlós discrepancy problem.

Tighter estimates for Ε-nets for disks

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in D . In 1994, Bronnimann and Goodrich 5 made an important connection of this problem to the size of fundamental combinatorial structures called ?-nets, showing that small-sized ?-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan 2 showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives O ( 1 ) -factor approximation algorithms in O ? ( n ) time for hitting sets for disks in the plane.This constant factor depends on the sizes of ?-nets for disks; unfortunately, the current state-of-the-art bounds are large - at least 24 / ? and most likely larger than 40 / ? . Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2 / ? , which follows from the Pach-Woeginger construction 32 for halfplanes in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem.The main goal of this paper is to improve the upper-bound to 13.4 / ? for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of ?-nets for a variety of data-sets are lower, around 9 / ? .

The Gram-Schmidt walk: a cure for the Banaszczyk blues

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of 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 which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk’s proof of this result is non-constructive and an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In this paper, we resolve this question and give an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant. This leads to new efficient algorithms for several problems in discrepancy theory.

Faster space-efficient algorithms for subset sum and k-sum

We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O*(20.86n) time, where the O*(·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(nk-0.5(n)) time and O(logn) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(logn) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.

Quasi-PTAS for scheduling with precedences using LP hierarchies

A central problem in scheduling is to schedule

Faster space-efficient algorithms for Subset Sum, k-Sum and related problems

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Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem.

unit size jobs with precedence constraints on

Predicting relevant documents for enterprise communication contexts

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