Randomized Rounding in the Presence of a Cardinality Constraint

We consider the problem of generating randomized roundings that satisfy a single cardinality constraint and admit Chernoff-type large deviation bounds for weighted sums of the variables. That this can be done efficiently was proven by Srinivasan [2001], a different approach was later given by the first author [Doerr 2006]. In this work, we (a) present an improved version of the bitwise derandomization given by Doerr, (b) give the first derandomization of Srinivasan's tree-based randomized approach and prove its correctness, and (c) experimentally compare the resulting algorithms. Our experiments show that adding a single cardinality constraint typically reduces the rounding errors and only moderately increases the running times. In general, our derandomization of the tree-based approach is superior to the derandomized bitwise one, while the two randomized versions produce very similar rounding errors. When implementing the derandomized tree-based approach, however, the choice of the tree is important.