Geometry of Online Packing Linear Programs

Marco Molinaro, R. Ravi

Abstract: We consider packing LP’s with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1 − ε)-competitive algorithms require the right-hand side of the LP to be \(\Omega (\frac{m}{\epsilon^2} \log \frac{n}{\epsilon})\), 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 and known lower bounds for the general case are also independent of n.