Applications of the Lovász Local Lemma and related methods

V této práci se zabýváme aplikacemi Lovászova lokálního lemmatu a s ním souvisejících metod. Popíšeme postupný vývoj těchto metod a ukážeme konkrétní příklady jejich užití na příkladech z oblasti výzkumu nezávislých transverzál a hypergrafů.ObhájenoIn this thesis we investigate applications of the Lovász local lemma and its related methods. We are going to describe the gradual development of these methods and show the specific examples of its use in the field of research on independent transversals and hypergraphs

Approximability of Sparse Integer Programs

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx:Ax≥b,0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k≥2 and ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx:Ax≤b,0≤x≤d} where A has at most k nonzeroes per column, we give a (2k 2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A ij is small compared to b i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per colum