Strong Algorithms for the Ordinal Matroid Secretary Problem

In contrast with the standard and widely studied utility variant, in the ordinal Matroid Secretary Problem (MSP) candidates do not reveal numerical weights but the decision maker can still discern if a candidate is better than another. We consider three competitiveness measures for the ordinal MSP. An algorithm is α ordinal-competitive if for every weight function compatible with the ordinal information, the expected output weight is at least 1/α times that of the optimum; it is α intersection-competitive if its expected output includes at least 1/α fraction of the elements of the optimum, and it is α probability-competitive if every element from the optimum appears with probability 1/α in the output. This is the strongest notion as any α probability-competitive algorithm is also α intersection, ordinal and utility (standard) competitive. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. In fact, we improve upon the guarantees for almost every matroid class considered in the MSP literature: we achieve probability-competitive ratios of e for transversal matroids (matching Kesselheim et al. [29], but under a stronger notion); of 4 for graphic matroids (improving on 2e by Korula and Pál [33]); of 3 √ 3 ≈ 5.19 for laminar matroids (improving on 9.6 by Ma et al. [39]); and of k for a superclass of k column sparse matroids, improving on the ke result by Soto [44]. We also get constant ratios for hypergraphic matroids, for certain gammoids and for graph packing matroids that generalize matching matroids. The forbidden sets technique is inspired by the backward analysis of the classical secretary problem algorithm and by the analysis of the e-competitive algorithm for online weighted bipartite matching by Kesselheim et al. [29]. Additionally, we modify Kleinberg’s 1+O( √ 1/ρ) utility-competitive algorithm for uniform matroids of rank ρ in order to obtain a 1 + O( √ log ρ/ρ) probability-competitive algorithm. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids. We devise an O(1) intersection-competitive algorithm, an O(log ρ) probability-competitive algorithm and an O(log log ρ) ordinal-competitive algorithm for matroids of rank ρ. The last two results are based on the O(log log ρ) utility-competitive algorithm by Feldman et al. [19].