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Laminar Matroid Secretary: Greedy Strikes Back

Zhiyi Huang, Zahra Parsaeian, Zixuan Zhu

TL;DR

This work tackles the Laminar Matroid Secretary Problem by proposing a simple greedy algorithm in a continuous-time arrival model. The algorithm delays decisions until a threshold $t_0$ and then accepts elements arriving at time $t$ only if they belong to $OPT(t)$ and maintain independence, with a key improvement arising from using $OPT(t)$ rather than the previous local optimum. The main result is a $4.75$-competitive guarantee, derived via a probabilistic analysis that decomposes the laminar structure into blocks with capacities $c(B_i)$ and leverages Gamma/exponential arrival-time distributions and a union bound, aided by numerical verification at $t_0=0.7$. This advances the state of the art for laminar matroids, achieving a tighter bound than the prior $3\sqrt{3} \approx 5.196$-competitive result, and operates in the ordinal model, enhancing practical applicability.

Abstract

We show that a simple greedy algorithm is $4.75$ probability-competitive for the Laminar Matroid Secretary Problem, improving the $3\sqrt{3} \approx 5.196$-competitive algorithm based on the forbidden sets technique (Soto, Turkieltaub, and Verdugo, 2018).

Laminar Matroid Secretary: Greedy Strikes Back

TL;DR

This work tackles the Laminar Matroid Secretary Problem by proposing a simple greedy algorithm in a continuous-time arrival model. The algorithm delays decisions until a threshold and then accepts elements arriving at time only if they belong to and maintain independence, with a key improvement arising from using rather than the previous local optimum. The main result is a -competitive guarantee, derived via a probabilistic analysis that decomposes the laminar structure into blocks with capacities and leverages Gamma/exponential arrival-time distributions and a union bound, aided by numerical verification at . This advances the state of the art for laminar matroids, achieving a tighter bound than the prior -competitive result, and operates in the ordinal model, enhancing practical applicability.

Abstract

We show that a simple greedy algorithm is probability-competitive for the Laminar Matroid Secretary Problem, improving the -competitive algorithm based on the forbidden sets technique (Soto, Turkieltaub, and Verdugo, 2018).
Paper Structure (5 sections, 3 theorems, 1 figure, 1 algorithm)

This paper contains 5 sections, 3 theorems, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Greedy Algorithm
  4. Competitive Analysis
  5. Conclusion

Key Result

Lemma 2

For any $S, T \in \mathcal{I}$ such that $|S| = |T|$, for any $e \in S \setminus T$, there is an element $e' \in T \setminus S$ such that $S - e + e', T - e' + e \in \mathcal{I}$.

Figures (1)

  • Figure :

Theorems & Definitions (4)

  • Definition 1
  • Lemma 2: c.f., Schrijver Schrijver, Corollary 39.12a
  • Lemma 3
  • Theorem 4