Formal Barriers to Simple Algorithms for the Matroid Secretary Problem

Bahrani, Maryam; Beyhaghi, Hedyeh; Singla, Sahil; Weinberg, S. Matthew

doi:10.1007/978-3-030-94676-0_16

Cited by 2 publications

(6 citation statements)

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“…Then we formally define the competitive ratio of an algorithm in a MSP setting. Next we present the result from [4] that any "Greedy" algorithm (formal definition below) for MSP cannot have a constant competitive ratio. Lastly, we provide the Forbidden Set argument from [10] that identifies a sufficient condition for an algorithm for an ordinal MSP setting to have a constant competitive ratio.…”

Section: Framework For Mspmentioning

confidence: 99%

“…Greedy Algorithms. In this part, we present the framework of "Greedy" Algorithms from [4]. This class of algorithms extend the idea of greedy algorithms from Classical or Multiple Choice Secretary Problem into a matroid setting: an element is accepted based on feasibility and optimality.…”

Section: Forbidden Setsmentioning

confidence: 99%

“…An interesting result of [4] was that any instance of Greedy Algorithm cannot achieve a constant utility competitive ratio for the MSP. The paper introduces the following graph, which works as an adversarial input to any Greedy Algorithm.…”

Section: Forbidden Setsmentioning

confidence: 99%

“…Theorem 22 (From [4]). A Greedy Algorithm cannot obtain a constant utility competitive ratio on the Hat Graph…”

Section: Forbidden Setsmentioning

confidence: 99%

“…One notable negative result is presented in [4]. Bahrani et al showed that a certain class of algorithms that are "greedy-like" cannot achieve a constant competitive ratio on general matroids.…”

Section: Introductionmentioning

confidence: 99%

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Extension of Simple Algorithms to the Matroid Secretary Problem

Park¹

2022

Preprint

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Whereas there are simple algorithms that are proven to be optimal for the Classical and the Multiple Choice Secretary Problem, the Matroid Secretary Problem is less thoroughly understood. This paper proposes the generalization of some simple algorithms from the Classical and Multiple Choice versions on the Matroid Secretary Problem. Out of two algorithms that make decisions based on samples, like the Dynkin's algorithm, one is proven to be an instance of Greedy Algorithm [4], while the other is not. A generalized version of the Virtual Algorithm [1] obtains a constant competitive ratio for the Hat Graph, the adversarial example for Greedy Algorithms, but fails to do so when a slight modificiation is introduced to the graph. We show that there is no algorithm with Strong Forbidden Sets [10] of size 1 on all graphic matroids. I pledge my honor that this paper represents my own work in accordance with University regulations. /s Juhyun 'Simon' Park

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Section: Framework For Mspmentioning

confidence: 99%

Section: Forbidden Setsmentioning

confidence: 99%

Section: Forbidden Setsmentioning

confidence: 99%

“…Theorem 22 (From [4]). A Greedy Algorithm cannot obtain a constant utility competitive ratio on the Hat Graph…”

Section: Forbidden Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%