Recent advances on the matroid secretary problem

Michael Dinitz
2013 ACM SIGACT News  
The matroid secretary problem was introduced by Babaioff, Immorlica, and Kleinberg in SODA 2007 as an online problem that was both mathematically interesting and had applications to online auctions. In this column I will introduce and motivate this problem, and give a survey of some of the exciting work that has been done on it over the past 6 years. While we have a much better understanding of matroid secretary now than we did in 2007, the main conjecture is still open: does there exist an
more » ... -competitive algorithm? $&0 6,*$&7 1HZV -XQH YRO QR $&0 6,*$&7 1HZV -XQH YRO QR Definition 1.1. A matroid M = (E, I) is an ordered pair consisting of a finite set E and a collection I of subsets (called the independent sets) of E satisfying the following three conditions: 1. ∅ ∈ I, 2. If I ∈ I and I ⊆ I, then I ∈ I, and 3. If I 1 and I 2 are in I and |I 1 | < |I 2 |, then there is an element e ∈ I 2 \ I 1 such that I 1 ∪{e} ∈ I. The matroid secretary problem, as defined by [BIK07] , is exactly what it sounds like. For a given matroid M = (E, I) and value function w, let OP T (M, w) denote the value of the most valuable set in I (where the value of a set is the sum of the values of its elements). For any algorithm A and ordering σ, let A(M, w, σ) denote the value of the set returned by A (which must be an independent set). We say that A is α-competitive for matroid M if OP T (M, w)/ E [A(M, w, σ)] ≤ α for any value function w : E → R + , where the expectation is taken over the uniformly random choice of σ and the internal randomness of A. Babaioff, Immorlica, and Kleinberg defined this problem, and made the following conjecture: Conjecture 1.2 ([BIK07]). For every matroid M, there is an algorithm that is O(1)-competitive.
doi:10.1145/2491533.2491557 fatcat:yfft66mjpzb2lhyszq5x3cpwyy