New Bounds for Online Packing LPs

Abstract

Solving linear programs online has been an active area of research in recent years and was used with great success to develop new online algorithms for a variety of problems. We study the setting introduced by Ochel et al. as an abstraction of lifetime optimization of wireless sensor networks.

In this setting, the online algorithm is given a packing LP and has to monotonically increase LP variables in order to maximize the objective function. However, at any point in time, the adversary only provides an α-approximation of the remaining slack for each constraint. This is designed to model scenarios in which only estimates of remaining capacities (e.g. of batteries) are known, and they get more and more accurate as the remaining capacities approach 0.

Ochel et al. (ICALP’12) gave a Θ(ln α/α)-competitive online algorithm for this online packing LP problem and showed an upper bound on the competitive ratio of any online algorithm, even randomized, of $O(1/\sqrt{\alpha})$. We significantly improve the upper bound and show that any deterministic online algorithm for LPs with d variables is at most O(d ² α ^1/d/α)-competitive. For randomized online algorithms we show an upper bound of O(m ² α ^1/m/α) for LPs with m ^{m!ln α} variables. For LPs with sufficiently many variables, these bounds are O(ln ² α/α), nearly matching the known lower bound.

On the other hand, we also show that the known lower bound can be significantly improved if the number of variables in the LP is small. Specifically, we give a deterministic $\Theta(1/\sqrt{\alpha})$-competitive online algorithm for packing LPs with two variables. This is tight, since the previously known upper bound of $O(1/\sqrt{\alpha})$ still holds for 2-dimensional LPs.

The first and second author are supported by the Centre for Discrete Mathematics and its Applications (DIMAP), University of Warwick, EPSRC award EP/D063191/1. The third author is supported by NCN grant N N206 567940.