file.pdf (432.64 kB)
Download file

How the Experts Algorithm Can Help Solve LPs Online

Download (432.64 kB)
journal contribution
posted on 01.01.1984, 00:00 by Anupam Gupta, Marco Molinaro

We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention: the main open question is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraint) to get (1 + ε)-approximations online? It is known that the RHS has to be Ω(ε − 2 logm) times the left-hand sides, where m is the number of constraints.

In this paper we show how to achieve this bound for all packing LPs, and also for a wide class of mixed packing/covering LPs. Our algorithms construct dual solutions using a regret-minimizing online learning algorithm in a black-box fashion, and use them to construct primal solutions. The adversarial guarantee that holds for the constructed duals help us to take care of most of the correlations that arise in the algorithm; the remaining correlations are handled via martingale concentration and maximal inequalities. These ideas lead to conceptually simple and modular algorithms, which we hope will be useful in other contexts.

History

Publisher Statement

All Rights Reserved

Date

01/01/1984

Usage metrics

Exports