# Constructive Discrepancy Minimization for Convex Sets

@article{Rothvoss2014ConstructiveDM, title={Constructive Discrepancy Minimization for Convex Sets}, author={Thomas Rothvoss}, journal={2014 IEEE 55th Annual Symposium on Foundations of Computer Science}, year={2014}, pages={140-145} }

A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(√(n)). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett-Meka algorithm finds a half integral point in any "large enough" polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set K with measure at…

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## References

SHOWING 1-10 OF 29 REFERENCES

Constructive Discrepancy Minimization by Walking on the Edges

- Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk which is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

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

- Computer Science, MathematicsSODA '97
- 1997

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.

The ellipsoid method and its consequences in combinatorial optimization

- Mathematics, Computer ScienceComb.
- 1981

The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.

Beck's Three Permutations Conjecture: A Counterexample and Some Consequences

- Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

This work constructs three permutations whose corresponding set system has discrepancy Ω(log n), and proves an interesting implication of this construction in the reverse direction: there are instances of bin packing and corresponding optimal basic feasible solutions for the Gilmore-Gomory LP relaxation such that any packing that contains only patterns from the support of these solutions requires at least opt + Ω (log m) bins.

The discrepancy of permutation families

- Mathematics
- 1997

In this note, we show that the discrepancy of any family of ‘ permutations of [n] = f1; 2;:::;ng is O( p ‘ logn), improving on the O(‘ logn) bound due to Bohus (Random Structures & Algorithms,…

Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem

- Mathematics
- 2013

We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison{Singer problem. The rst is Weaver’s conjecture…

On the Discrepancy of 3 Permutations

- MathematicsRandom Struct. Algorithms
- 1990

It is shown that for any constant number of orderings the discrepancy is O(log n) and the proof also gives an efficient algorithm to determine such a coloring.

The discrepancy method - randomness and complexity

- Computer Science
- 2000

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…

Probability in Banach Spaces: Isoperimetry and Processes

- Mathematics
- 1991

Notation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and…

EXTREMAL PROPERTIES OF ORTHOGONAL PARALLELEPIPEDS AND THEIR APPLICATIONS TO THE GEOMETRY OF BANACH SPACES

- Mathematics
- 1989

It is proved that the distribution function for the maximum of the modulus of a set of jointly Gaussian random variables with given variance and zero mean is minimal if these variables are…