Constructive Algorithms for Discrepancy Minimization

@article{Bansal2010ConstructiveAF,
  title={Constructive Algorithms for Discrepancy Minimization},
  author={Nikhil Bansal},
  journal={2010 IEEE 51st Annual Symposium on Foundations of Computer Science},
  year={2010},
  pages={3-10}
}
  • N. Bansal
  • Published 11 February 2010
  • Mathematics, Computer Science
  • 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Given a set system $(V,\mathcal{S})$, $V=\{1,\ldots,n\}$ and $\mathcal{S}=\{S_1,\ldots,S_m\}$, the minimum discrepancy problem is to find a 2-coloring $\mathcal{X}:V \right arrow \{-1,+1\}$, such that each set is colored as evenly as possible, i.e. find $\mathcal{X}$ to minimize $\max_{j \in [m]} \left|\sum_{i \in S_j} \mathcal{X}(i)\right|$. In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the… 

Constructive Discrepancy Minimization with Hereditary L2 Guarantees

TLDR
The algorithm avoids solving an SDP and instead relies on computing eigendecompositions of matrices and improves over previous work by Chazelle and Lvov, and by Matousek et al.

A spectral bound on hypergraph discrepancy

TLDR
The discrepancy of a random $t$-regular hypergraph with n vertices and m edges is almost surely $O(\sqrt{t} + \lambda)$ as $n$ grows.

The remote set problem on lattices

  • I. Haviv
  • Mathematics, Computer Science
    computational complexity
  • 2014
TLDR
A polynomial-time deterministic algorithm that on rank n lattice L outputs a set of points, at least one of which is $${\sqrt{\log n / n} \cdot \rho(\mathcal{L})}$$logn/n·ρ(L) -far from $$L, where $$ rho(p) stands for the covering radius of L, i.e., the maximum possible distance of a point in space from L.

Vector Balancing in Lebesgue Spaces

TLDR
It is proved that for any fixed constant $\delta>0$, in a centrally symmetric body $K \subseteq \mathbb{R}^n$ with measure at least $e^{-\delta n}$ one can find such a fractional coloring in polynomial time.

Factorization Norms and Hereditary Discrepancy

TLDR
A new lower bound of $\Omega(\log^{d-1} n)$ for the \emph{$d$-dimensional Tusnady problem}, asking for the combinatorial discrepancy of an $n$-point set in $\mathbb{R}^d$ with respect to axis-parallel boxes is proved.

Balancing Polynomials in the Chebyshev Norm

TLDR
The Rudin-Shapiro sequence is extended, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.

An Explicit VC-Theorem for Low-Degree Polynomials

TLDR
This work shows that for any X ⊆ ℝ n and any Boolean function class \({\cal C}\) that is uniformly approximated by degree k polynomials, an e-approximation S can be be constructed deterministically in time poly(n k ,1/e,|X|) provided that W is the weight of the approximating polynomial.

A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

TLDR
This paper shows that there exists a coloring $\chi$ with discrepancy bound O^{*}(|S|^{1/2 - d_1/(2d)} n^{(d_1 - 1)/(2d)})$, for each $S \in \EuScript{S}$, where $O^{*](\cdot)$ hides a polylogarithmic factor in $n$.

On Integer Programming and Convolution

TLDR
It is shown that improving the algorithm for IPs of any fixed number of constraints is equivalent to improving (min, +)-convolution and matching conditional lower bounds are given.
...

References

SHOWING 1-10 OF 35 REFERENCES

Roth’s estimate of the discrepancy of integer sequences is nearly sharp

TLDR
It is proved that R(N)=N1/4+o(1) thus showing that Roth’s original lower bound was essentially best possible, and the notion ofdiscrepancy of hypergraphs is introduced and derive an upper bound from which the above result follows.

Six standard deviations suffice

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Knl/2, K an absolute constant. This improves the basic probabilistic method with

Tight hardness results for minimizing discrepancy

TLDR
It is NP-hard to distinguish between such set systems with discrepancy zero and those with discrepancy Ω(√<i>N</i), which means that even if the optimal solution has discrepancy zero, the authors cannot hope to efficiently find a coloring with discrepancy O(N), and these results are tight for general set systems.

"Integer-making" theorems

Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems

TLDR
Improved upper bounds on the discrepancy of two well-studied families of sparse matrices: e permutations of [n], and rectangles containing n points in Rk are shown, and a discrepancy bound of O(&logn) is shown for the former, improving on the previous-best O( logn) due to Bohus.

Geometric Discrepancy: An Illustrated Guide

1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2.

Non-constructive proofs in Combinatorics

  • N. Alon
  • Mathematics, Computer Science
  • 2002
TLDR
This work describes some representing non-constructive proofs of this type, demonstrating the applications of Topological, Algebraic and Probabilistic methods in Combinatorics, and discusses the related algorithmic problems.

The discrepancy method - randomness and complexity

This book tells the story of the discrepancy method in a few short independent vignettes. It is a varied tale which includes such topics as communication complexity, pseudo-randomness, rapidly mixing

The Probabilistic Method

TLDR
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.

Combinatorial optimization. Polyhedra and efficiency.

TLDR
This book shows the combinatorial optimization polyhedra and efficiency as your friend in spending the time in reading a book.