On Variants of the Matroid Secretary Problem

TL;DR

This paper explores various versions of the matroid secretary problem, providing algorithms with constant-factor and logarithmic approximations, and resolves open questions about the problem's complexity and classical secretary problem variants.

Contribution

It introduces a constant-factor competitive algorithm for the random assignment model and establishes approximation bounds for unknown matroids, solving an open problem.

Findings

Constant-factor algorithm for the random assignment model

O(log r log n)-approximation for unknown matroids

Impossibility of better than O(log n / log log n) approximation

Abstract

We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the "random assignment" model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto \cite{Soto11}). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an $O(\log r \log n)$-approximation, and prove that a better than $O(\log n / \log \log n)$ approximation is impossible. This resolves an open question posed by Babaioff et al. \cite{BIK07}. As a natural special case, we also consider the classical secretary problem where the number of candidates $n$ is unknown in advance. If $n$ is chosen by an adversary from $\{1,...,N\}$, we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least $1/(H_{N-1}+1)$ and prove that a probability better than $1/H_N$ cannot be achieved (where $H_N$ is the $N$-th harmonic number).