Secretary Problems with Convex Costs

  • Siddharth Barman
  • Seeun Umboh
  • Shuchi Chawla
  • David Malec
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The online algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The “profit” of a set of accepted requests is its total value minus a convex cost function of its total size. This problem falls within the framework of secretary problems. Unlike previous work in that area, one of the main challenges we face is that the objective function can be positive or negative, and we must guard against accepting requests that look good early on but cause the solution to have an arbitrarily large cost as more requests are accepted. This necessitates new techniques. We study this problem under various feasibility constraints and present online algorithms with competitive ratios only a constant factor worse than those known in the absence of costs for the same feasibility constraints. We also consider a multi-dimensional version of the problem that generalizes multi-dimensional knapsack within a secretary framework. In the absence of feasibility constraints, we present an O(ℓ) competitive algorithm where ℓ is the number of dimensions; this matches within constant factors the best known ratio for multi-dimensional knapsack secretary.


Competitive Ratio Online Algorithm Full Version Competitive Algorithm Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Siddharth Barman
    • 1
  • Seeun Umboh
    • 1
  • Shuchi Chawla
    • 1
  • David Malec
    • 1
  1. 1.University of Wisconsin–MadisonUSA

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