Englert, Matthias, Matsakis, Nicolaos and Mucha, Marcin (2014) New bounds for online packing LPs. In: Latin American Theoretical INformatics (LATIN) 2014, Montevideo, Uruguay , 31 Mar- 4 Apr 2014. Published in: Lecture Notes in Computer Science series pp. 1-12. doi:10.1007/978-3-642-54423-1_28

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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
Ο(1/√α).
We significantly improve the upper bound and show that any deterministic online algorithm for LPs with δ variables is at most O(δ2 α1/m/δ/α)-competitive. For randomized online algorithms we show an upper bound of O(m2 α 1/m/α) for LPs with mm!ln α variables.
For LPs with sufficiently many variables, these bounds are O(ln2 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 Θ(1/√α)-competitive
online algorithm for packing LPs with two variables. This is tight, since the previously known upper bound of O(1/√α) still holds for 2-dimensional LPs.

Item Type:	Conference Item (Paper)
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions:	Faculty of Science > Computer Science
Library of Congress Subject Headings (LCSH):	Computer science, Computer science -- Mathematics, Linear programming
Journal or Publication Title:	Lecture Notes in Computer Science series
Publisher:	Springer
Official Date:	2014
Dates:	Date Event 2014 UNSPECIFIED
Date of first compliant deposit:	26 December 2015
Page Range:	pp. 1-12
DOI:	10.1007/978-3-642-54423-1_28
Status:	Peer Reviewed
Description:	Accepted conference paper
Funder:	University of Warwick. Centre for Discrete Mathematics and Its Applications, Engineering and Physical Sciences Research Council (EPSRC), National Communications Network (U.S.) (NCN)
Grant number:	EP/D063191/1 (EPSRC) ; N N206 567940 (NCN)
Embodied As:	1
Conference Paper Type:	Paper
Title of Event:	Latin American Theoretical INformatics (LATIN) 2014
Type of Event:	Conference
Location of Event:	Montevideo, Uruguay
Date(s) of Event:	31 Mar- 4 Apr 2014

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