Fractional arboricity, strength, and principal partitions in graphs and matroids
Discrete Applied Mathematics - Special issue: graphs in electrical engineering, discrete algorithms and complexity
Every matroid is a submatroid of a uniformly dense matroid
Discrete Applied Mathematics
Improved Algorithms and Analysis for Secretary Problems and Generalizations
SIAM Journal on Discrete Mathematics
A multiple-choice secretary algorithm with applications to online auctions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Matroid Theory (Oxford Graduate Texts in Mathematics)
Matroid Theory (Oxford Graduate Texts in Mathematics)
Matroids, secretary problems, and online mechanisms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Online auctions and generalized secretary problems
ACM SIGecom Exchanges
Competitive Weighted Matching in Transversal Matroids
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
A Knapsack Secretary Problem with Applications
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Secretary problems: weights and discounts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Algorithms for Secretary Problems on Graphs and Hypergraphs
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Submodular secretary problem and extensions
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Secretary problems via linear programming
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Hiring a secretary from a poset
Proceedings of the 12th ACM conference on Electronic commerce
On variants of the matroid secretary problem
ESA'11 Proceedings of the 19th European conference on Algorithms
Improved competitive ratio for the matroid secretary problem
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Secretary problems with convex costs
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Geometry of online packing linear programs
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Advances on matroid secretary problems: free order model and laminar case
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Recent advances on the matroid secretary problem
ACM SIGACT News
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In the Matroid Secretary Problem, introduced by Babaioff et al. [5], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it. The objective is to return a maximum weight independent set of the matroid. There are different variants for this problem depending on the information known about the weights beforehand. In the random assignment model, a hidden list of weights is randomly assigned to the matroid ground set, independently from the random order they are revealed to the algorithm. Our main result is the first constant competitive algorithm for this version of the problem, solving an open question of Babaioff et al. Our algorithm achieves a competitive ratio of 2e2/(e − 1). It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a 2e-competitive algorithm for uniformly dense matroids we also develop. We also present constant competitive algorithms in the standard model where the weights are assigned adversarially, for various classes of matroids including cographic, low density, k-column sparse linear matroids and the case when every element is in a small cocircuit. In the same model, we give a new O(log r)-competitive algorithm for matroids of rank r which only uses the relative order of the weights seen and not their actual values, as previously needed.