
A Competitive Analysis of Online Knapsack Problems with Unit Density
We study an online knapsack problem where the items arrive sequentially ...
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Online Maxmin Fair Allocation
We study an online version of the maxmin fair allocation problem for in...
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Competitive analysis of the topK ranking problem
Motivated by applications in recommender systems, web search, social cho...
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Online Simple Knapsack with Reservation Costs
In the Online Simple Knapsack Problem we are given a knapsack of unit si...
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Online Knapsack Problem under Expected Capacity Constraint
Online knapsack problem is considered, where items arrive in a sequentia...
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Competitive DataStructure Dynamization
Datastructure dynamization is a general approach for making static data...
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Maximizing Online Utilization with Commitment
We investigate online scheduling with commitment for parallel identical ...
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New Results for the kSecretary Problem
Suppose that n items arrive online in random order and the goal is to select k of them such that the expected sum of the selected items is maximized. The decision for any item is irrevocable and must be made on arrival without knowing future items. This problem is known as the ksecretary problem, which includes the classical secretary problem with the special case k=1. It is wellknown that the latter problem can be solved by a simple algorithm of competitive ratio 1/e which is optimal for n →∞. Existing algorithms beating the threshold of 1/e either rely on involved selection policies already for k=2, or assume that k is large. In this paper we present results for the ksecretary problem, considering the interesting and relevant case that k is small. We focus on simple selection algorithms, accompanied by combinatorial analyses. As a main contribution we propose a natural deterministic algorithm designed to have competitive ratios strictly greater than 1/e for small k ≥ 2. This algorithm is hardly more complex than the elegant strategy for the classical secretary problem, optimal for k=1, and works for all k ≥ 1. We derive its competitive ratios for k ≤ 100, ranging from 0.41 for k=2 to 0.75 for k=100. Moreover, we consider an algorithm proposed earlier in the literature, for which no rigorous analysis is known. We show that its competitive ratio is 0.4168 for k=2, implying that the previous analysis was not tight. Our analysis reveals a surprising combinatorial property of this algorithm, which might be helpful to find a tight analysis for all k.
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