Secretary Problems with Convex Costs

12/2011; DOI: 10.1007/978-3-642-31594-7_7
Source: arXiv

ABSTRACT 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 function. 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 requires designing 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 any
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.

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