Online lower bounds via duality

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Abstract

In this paper, we exploit linear programming duality in the online setting, where input arrives on the y, from the unique perspective of designing lower bounds (i.e., hardness results) on the competitive ratio. In particular, we provide a systematic method (as opposed to ad hoc case analysis that is typically done) for obtaining online deterministic and randomized lower bounds on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight hardness results for three diverse online problems: The Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds. Our approach is in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primal- dual scheme. We design a general recipe with the opposite aim of obtaining negative results via dual- ity. The general idea behind our approach is to construct a parameterized family of primal linear pro- grams based on a candidate collection of input sequences for proving the lower bound, where the objective function corresponds to optimizing the competitive ratio. Solving the parameterized family of primal linear programs optimally would yield a valid lower bound, but is a challenging task and limits the tools that can be applied, since analysis must be done precisely and exact optimality needs to be proved. To this end, we consider the corresponding parameterized family of dual linear programs and provide feasible solutions, where the objective function yields a lower bound on the competitive ratio. This opens up additional doors for analysis, including some of the techniques we employ (e.g., continuous analysis, differential equations, etc.), as we need not be so care- ful about exact optimality. We are confident that our methods can be successfully applied to produce many more lower bounds for a wide array of online problems.

Original languageEnglish
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
EditorsPhilip N. Klein
Pages1038-1050
Number of pages13
ISBN (Electronic)9781611974782
DOIs
StatePublished - 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: 16 Jan 201719 Jan 2017

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume0

Conference

Conference28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Country/TerritorySpain
CityBarcelona
Period16/01/1719/01/17

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics

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