Randomised Rounding with Applications

Abstract

We develop new techniques for rounding packing integer programs using iterative randomized rounding. It is based on a novel application of multidimensional Brownian motion in $\mathbb{R}^n$. Let $\overset{\sim}{x} \in {[0,1]}^n$ be a fractional feasible solution of a packing constraint $A x \leq 1,\ \ $ $A \in {\{0,1 \}}^{m\times n}$ that maximizes a linear objective function. The independent randomized rounding method of Raghavan-Thompson rounds each variable $x_i$ to 1 with probability $\overset{\sim}{x_i}$ and 0 otherwise. The expected value of the rounded objective function matches the fractional optimum and no constraint is violated by more than $O(\frac{\log m} {\log\log m})$.In contrast, our algorithm iteratively transforms $\overset{\sim}{x}$ to $\hat{x} \in {\{ 0,1\}}^{n}$ using a random walk, such that the expected values of $\hat{x}_i$'s are consistent with the Raghavan-Thompson rounding. In addition, it gives us intermediate values $x'$ which can then be used to bias the rounding towards a superior solution.The reduced dependencies between the constraints of the sparser system can be exploited using {\it Lovasz Local Lemma}. For $m$ randomly chosen packing constraints in $n$ variables, with $k$ variables in each inequality, the constraints are satisfied within $O(\frac{\log (mkp\log m/n) }{\log\log (mkp\log m/n)})$ with high probability where $p$ is the ratio between the maximum and minimum coefficients of the linear objective function. Further, we explore trade-offs between approximation factors and error, and present applications to well-known problems like circuit-switching, maximum independent set of rectangles and hypergraph $b$-matching. Our methods apply to the weighted instances of the problems and are likely to lead to better insights for even dependent rounding.