Secretary Problems with Convex Costs

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


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.

7 Reads
  • Source
    • "Recently, increased interest arose in nonlinear versions of the secretary problem, with a focus on the maximization of a non-negative submodular function 2 [5] [6] [14] [23] [32], leading to the submodular secretary problem. Submodular functions have widespread use as valuation functions because they reflect the property of diminishing returns, i.e., the marginal value of an element is the bigger the fewer elements have been selected so far. "
    [Show abstract] [Hide abstract]
    ABSTRACT: During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most questions related to SMSP remained open, including whether SMSP may be substantially more difficult than MSP; and more generally, to what extend MSP and SMSP are related. Our goal is to address these points by presenting general black-box reductions from SMSP to MSP. In particular, we show that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O(1)-competitive algorithm for SMSP over the same matroid class. This implies that the matroid secretary conjecture is equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a particular matroid class, it suffices to consider MSP over the same matroid class. Using our reductions we obtain many first and improved O(1)-competitive algorithms for SMSP over various matroid classes by leveraging known algorithms for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive algorithm for SMSP, thus, matching the currently best asymptotic algorithm for MSP, and substantially improving on the previously best O(log(rank))-competitive algorithm for SMSP.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study online procurement markets where agents arrive in a sequential order and a mechanism must make an irrevocable decision whether or not to procure the service as the agent arrives. Our mechanisms are subject to a budget constraint and are designed for stochastic settings in which the bidders are either identically distributed or, more generally, permuted in random order. Thus, the problems we study contribute to the literature on budget-feasible mechanisms as well as the literature on secretary problems and online learning in auctions. Our main positive results are as follows. We present a constant-competitive posted price mechanism when agents are identically distributed and the buyer has a symmetric submodular utility function. For nonsymmetric submodular utilities, under the random ordering assumption we give a posted price mechanism that is O(log n)-competitive and a truthful mechanism that is O(1)-competitive but uses bidding rather than posted pricing.
    Proceedings of the 14th ACM Conference on Electronic Commerce (EC); 01/2012
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the run-ning time of O(n log n), where n is the number of items. Before, only a 3-approximation algorithm was known. We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.
    Mathematical Foundations of Computer Science; 01/2013
Show more

Similar Publications


7 Reads
Available from