Randomised Rounding with Applications

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 R. Let ∼ x ∈ [0, 1] be a fractional feasible solution of a packing constraint Ax ≤ 1, A ∈ {0, 1} that maximizes a linear objective function. The independent randomized rounding method of Raghavan-Thompson rounds each variable xi to 1 with probability ∼ xi and 0 otherwise. The expected value of the rounded objective function matches the fractional optimum and no constraint is violated by more than O( logm log logm ). In contrast, our algorithm iteratively transforms ∼ x to x̂ ∈ {0, 1} using a random walk, such that the expected values of x̂i’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 Lovasz Local Lemma. Using the Moser-Tardos’ constructive version, x converges to x̂ in polynomial time to a distribution over the unit hypercube Hn = {0, 1} such that the expected value of any linear objective function over Hn equals the value at ∼x. Form randomly chosen packing constraints in n variables, with k variables in each inequality, the constraints are satisfied within O( log(mkp logm/n) log log(mkp logm/n)) with high probability where p is the ratio between the maximum and minimum coefficients of the linear objective function. For example, when m, k = √ n and p = polylog(n), this yields O(log logn/ log log logn) error for polylogarithmic weighted objective functions that significantly improves the O( logm log logm ) error incurred by the classical randomized rounding method of Raghavan and Thompson [RT87]. 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. Email:[email protected] Email:[email protected]