Abstract
AbstractWe study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5âcompetitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 â o(1), and an algorithm with ratio at least 0.5459 for all n â„ 1. We extend all algorithms and ratios to k â„ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Î(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(râČ log(n/râČ)), where râČ is the minimum rank of the matroid and the dual matroid.
Highlights
The secretary problem is a classic approach to study online optimization problems: A sequence of n candidates are arriving in uniform random order
We propose a simple approach for general subadditive packing problems with returns, which can be combined with arbitrary offline α-approximation algorithms
For the secretary problem with 2 arrivals, we propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms in general subadditive packing domains
Summary
The secretary problem is a classic approach to study online optimization problems: A sequence of n candidates are arriving in uniform random order. Each candidate reveals its value upon arrival and must be decided (accept/reject) before seeing any further candidate(s). Every decision is finalâonce a candidate gets accepted, the process is over. No rejected candidate can be accepted later on. The goal is to accept the best candidate. An optimal solution is to discard the first (roughly) n/e candidates. We accept the first, that is, the best one among the ones seen so far. The probability to hire the best candidate approaches 1/e â 0.37 when n tends to infinity
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