Nearly Linear-Time Approximation Schemes for Mixed Packing/Covering and Facility-Location Linear Programs

We describe nearly linear-time approximation algorithms for explicitly given mixed packing/covering and facility-location linear programs. The algorithms compute $(1+\epsilon)$-approximate solutions in time $O(N \log(N)/\epsilon^2)$, where $N$ is the number of non-zeros in the constraint matrix. We also describe parallel variants taking time $O(\text{polylog}/\epsilon^4)$ and requiring only near-linear total work, $O(N \text{polylog} /\epsilon^2)$. These are the first approximation schemes for these problems that have near-linear-time sequential implementations or near-linear-work polylog-time parallel implementations.