Online Mixed Packing and Covering

In many problems, the inputs to the problem arrive over time. As each input is received, it must be dealt with irrevocably. Such problems are online problems. An increasingly common method of solving online problems is to solve the corresponding linear program, obtained either directly for the problem or by relaxing the integrality constraints. If required, the fractional solution obtained is then rounded online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints Px ≤ p are given offline and covering constraints Cx ≥ c are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. For general mixed packing and covering linear programs, no prior sublinear competitive algorithms are known. We give the first such — a polylogarithmic-competitive algorithm for solving mixed packing and covering linear programs online. We also show a nearly tight lower bound. Our techniques for the upper bound use an exponential penalty function in conjunction with multiplicative updates. While exponential penalty functions are used previously to solve linear programs offline approximately, offline algorithms know the constraints beforehand and can optimize greedily. In contrast, when constraints arrive online, updates need to be more complex. We apply our techniques to solve two online fixed-charge problems with congestion. These problems are motivated by applications in machine scheduling and facility location. The linear program for these problems is more complicated than mixed packing and covering, and presents unique challenges. We show that our techniques combined with a randomized rounding procedure can be used to obtain polylogarithmic-competitive integral solutions. These problems generalize online set-cover, for which there is a polylogarithmic lower bound. Hence, our results are close to tight. Dartmouth College, 6211 Sudikoff Lab, Hanover NH 03755. {umang, lkf}@cs.dartmouth.edu. This work was supported in part by NSF grants CCF-0728869 and CCF-1016778.