Problem
Research questions and friction points this paper is trying to address.
matroid secretary problem
graphic matroids
independence oracle
online algorithms
constant-competitive algorithm
Graphic Matroid Secretary without the Graph
🤖 AI Summary
This work addresses the online graph matroid secretary problem under the constraint that the algorithm has access only to an independence oracle for elements that have arrived so far and no prior knowledge of the underlying graph structure. The authors propose a polynomial-time online algorithm that achieves a constant competitive ratio without requiring any additional structural information about the graph. By integrating structural properties of graph matroids, weight optimization under the random arrival model, and an adaptive online decision strategy, the algorithm guarantees a competitive ratio of at least $1/36$ relative to the maximum-weight independent set. This result constitutes the first constant-competitive solution for unknown graph matroids in the secretary setting, overcoming the reliance on extra prior information inherent in previous approaches.
📝 Abstract
The matroid secretary problem (MSP) is one of the cleanest, and most captivating open problems in online algorithms. The famous MSP conjecture stipulates that there exists a constant-competitive algorithm, yet to date the best known algorithms are $O(\log \log (\text{rank}))$ competitive. It is widely believed that all information that an algorithm for the MSP should use is information that is available through an independence oracle on the already arrived elements. Despite this, there are natural classes of matroids where a constant-competitive algorithm is known if we are given additional upfront information about the matroid; while no such algorithm is known if all the algorithm can use is an independence oracle on the arrived elements. In this work, we tackle the perhaps most appealing such class of matroids, graphic matroids. We develop an algorithm for the MSP that has access to the independence oracle only. Our algorithm runs in polynomial time, and if the underlying matroid is graphic, it produces an independent set whose weight is at least $1/36$ of the maximum-weight independent set. Ours is the first constant-competitive algorithm for MSP on unknown graphic matroids.
Innovation
Methods, ideas, or system contributions that make the work stand out.
matroid secretary problem
graphic matroids
independence oracle
constant-competitive algorithm
online algorithms
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2024-03-15arXiv.orgCitations: 2
