The Geometry of Online Packing Linear Programs

We consider packing linear programs with m rows where all constraint coefficients are in the unit interval. In the online model, we know the total number n of columns that arrive in random order. The goal is to set the decision variables corresponding to the arriving columns irrevocably so as to maximize the expected reward. Previous (1− )-competitive algorithms require that the right-hand sides of the constraints are of magnitude at least Ω( 2 log n ), a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case of online secretary problems. Moreover, known lower bounds for the general case of m rows only demonstrate that the right-hand sides must be as large as Ω( logm 2 ), with no dependence on n to obtain (1− )-competitive algorithms. Our goal is to understand whether the dependence on n is required in the multi-row case, making it fundamentally harder than the single-row version. We show that this is not the case by exhibiting an algorithm which is (1− )-competitive as long as the right-hand sides are Ω( 2 2 log m ). Our techniques refine previous PAC-learning based approaches, which interpret the online decisions as linear classifications of the columns based on dual prices obtained from sampled columns. Our improved bounds are proved by constructing a small set of witnesses for misclassifications, which are then used to obtain improved generalization bounds for the learning algorithm. The key component of our improvement is recognizing why the single-row problem is seemingly easier: if the columns of the LP belong to few one-dimensional subspaces, there is high overlap among the misclassifications and hence the associated learning problem is intrinsically more robust. For general linear programs, the idea is to modify the input to make the columns lie in a few one-dimensional subspaces while not changing the feasible set by much.