A Simple $O(\log\log(\mathrm{rank}))$-Competitive Algorithm for the Matroid Secretary Problem

Abstract: Only recently progress has been made in obtaining $o(\log(\mathrm{rank}))$-competitive algorithms for the matroid secretary problem. More precisely, Chakraborty and Lachish (2012) presented a $O(\sqrt{\log(\mathrm{rank})})$-competitive procedure, and Lachish (2014) later presented a $O(\log\log(\mathrm{rank}))$-competitive algorithm. Both these algorithms and their analyses are very involved, which is also reflected in the extremely high constants in their competitive ratios. Using different tools, we present a considerably simpler $O(\log\log(\mathrm{rank}))$-competitive algorithm for the matroid secretary problem. Our algorithm can be interpreted as a distribution over a simple type of matroid secretary algorithms which are easy to analyze. Due to the simplicity of our procedure, we are also able to vastly improve on the hidden constant in the competitive ratio.