Secretary Problems with Convex Costs

TL;DR

This paper studies online resource allocation with convex costs in secretary problems, proposing algorithms with competitive ratios close to those without costs, including multi-dimensional extensions.

Contribution

It introduces new online algorithms for secretary problems with convex costs, achieving near-optimal competitive ratios under various constraints and extending to multi-dimensional cases.

Findings

Algorithms with constant-factor competitive ratios for convex cost secretary problems.

Extension to multi-dimensional knapsack secretary problem with O(l) competitive ratio.

Handling of positive and negative objective functions in online decision-making.

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.