Improved Randomized Rounding using Random Walks

TL;DR

This paper introduces a new randomized rounding algorithm for packing integer programs using multidimensional Brownian motion, providing an efficient alternative to classical methods with broad applicability.

Contribution

The paper presents a novel Brownian motion-based rounding algorithm that converges in polynomial time and generalizes to various packing ILPs.

Findings

Converges in polynomial time to a distribution over solutions.

Maintains expected objective value equal to the fractional solution.

Applicable to disjoint path problems and MISR.

Abstract

We describe a novel algorithm for rounding packing integer programs based on multidimensional Brownian motion in $\mathbb{R}^n$. Starting from an optimal fractional feasible solution $\bar{x}$, the procedure converges in polynomial time to a distribution over (possibly infeasible) point set $P \subset {\{0,1 \}}^n$ such that the expected value of any linear objective function over $P$ equals the value at $\bar{x}$. This is an alternate approach to the classical randomized rounding method of Raghavan and Thompson \cite{RT:87}. Our procedure is very general and in conjunction with discrepancy based arguments, yield efficient alternate methods for rounding other optimization problems that can be expressed as packing ILPs including disjoint path problems and MISR.