NDLI: Secretary problems: laminar matroid and interval scheduling

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Secretary problems: laminar matroid and interval scheduling

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Secretary problems: laminar matroid and interval scheduling

Content Provider	ACM Digital Library
Author	Wang, Yajun Im, Sungjin
Abstract	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 $J_{i}$ arrives with interval $I_{i},$ processing time $p_{i}$ and weight $w_{i}.$ If $J_{i}$ is accepted, then it must be scheduled during $I_{i},$ 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.
Starting Page	1265
Ending Page	1274
Page Count	10
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2011-01-23
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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