Roundings Respecting Hard Constraints

A problem arising in integer linear programming is to transform a solution of a linear system to an integer one which is “close”. The customary model to investigate such problems is, given a matrix A and a [0,1] valued vector x, to find a binary vector y such that ||A(x−y)||∞ (the violation of the constraints) is small. Randomized rounding and the algorithm of Beck and Fiala are ways to compute such solutions y, whereas linear discrepancy is a lower bound measure. In many applications one is looking for roundings that, in addition to being close to the original solution, satisfy some constraints without violation. The objective of this paper is to investigate such problems in a unified way. To this aim, we extend the notion of linear discrepancy, the theorem of Beck and Fiala and the method of randomized rounding to this setting. Whereas some of our examples show that additional hard constraints may seriously increase the linear discrepancy, the latter two sets of results demonstrate that a reasonably broad notion of hard constraints may be added to the rounding problem without worsening the obtained solution significantly. Of particular interest might be our results on randomized rounding. We provide a simpler way to randomly round fixed weight vectors (cf. Srinivasan, FOCS 2001). It has the additional advantage that it can be derandomized with standard methods.