Annals of Operations Research
Randomized algorithms
A unified approach to approximating resource allocation and scheduling
Journal of the ACM (JACM)
Adaptive limited-supply online auctions
EC '04 Proceedings of the 5th ACM conference on Electronic commerce
A multiple-choice secretary algorithm with applications to online auctions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Matroids, secretary problems, and online mechanisms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Online budgeted matching in random input models with applications to Adwords
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
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
An online mechanism for ad slot reservations with cancellations
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Selling ad campaigns: online algorithms with cancellations
Proceedings of the 10th ACM conference on Electronic commerce
The adwords problem: online keyword matching with budgeted bidders under random permutations
Proceedings of the 10th ACM conference on Electronic commerce
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
Improved competitive ratios for submodular secretary problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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
Interviewing secretaries in parallel
Proceedings of the 13th ACM Conference on Electronic Commerce
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
Stochastic combinatorial optimization via poisson approximation
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Recent advances on the matroid secretary problem
ACM SIGACT News
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The classical secretary problem studies the problem of hiring the best secretary from among the secretaries who arrive in random order by making immediate and irrevocable decisions. After the interesting connection to online mechanism design was found [19, 20], the random order input assumption has been studied for a variety of problems. Babaioff et al. [4] formalized a general version of the secretary problem, namely the matroid secretary problem. In the problem, a secretary corresponds to an element in the universe U. The goal is to select the maximum weight independent set. They conjectured that the matroid secretary problem, for any matroid, allows a constant competitive algorithm. The conjecture remains open. Some constant approximation algorithms are currently known for some special cases of matroids. Another interesting type of secretary problem was studied where elements have non-uniform sizes, as is the case in the knapsack secretary problem [3, 6]. In this paper, we consider two interesting secretary problems. One is when the matroid is a laminar matroid, which generalizes uniform/partition/truncated partition matroids. For the laminar matroid secretary problem, using a novel replacement rule which we call "kick next," we give the first constant-competitive algorithm. The other is the interval scheduling secretary problem, which generalizes the knapsack secretary problem. In this problem, each job Ji arrives with interval Ii, processing time pi and weight wi. If Ji is accepted, then it must be scheduled during Ii, not necessarily continuously. The goal is to accept the jobs of the maximum total weight which are schedulable. We give a simple O(log D)-competitive algorithm and a nearly matching lower bound on the competitive ratio of any randomized algorithm, where D is the maximum interval length of any job.