Online Matroid Embeddings

TL;DR

The paper introduces online matroid embeddings (OMEs) to map unknown matroids revealed online into fixed host matroids, enabling reductions that relate online MSP variants to MSPs with known structure. The authors prove the existence of order-independent OMEs for binary (in particular, complete binary) matroids via the complete binary host, and show how OMEs yield constant-factor reductions between online-revealed MSP, known-matroid MSP, and prophet MSP with pairwise-independent weights. They extend the framework to approximate embeddings, discuss bounds on distortion, and show both positive results (e.g., laminar matroids) and impossibilities (e.g., graphical into a graphically regular host), illuminating the limits and potential of transferring MSP algorithms across matroid classes. The results offer new insights into the role of matroid structure in MSP, provide a novel route to handle binary matroids under correlated weight distributions, and link matroid theory with online embeddings and distributional reductions of broader algorithmic problems.

Abstract

We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure, is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.

Figures (5)

• Figure 1: Reductions for binary matroids. We use $P \succeq Q$ to indicate that $P$ is harder than $Q$, and we use $\succeq_\epsilon$ and $\succeq_C$ to designate an additive $\varepsilon$ or multiplicative factor $C$ loss in approximation.
• Figure 2: Two graphs that generate the same matroid on their edge set
• Figure 3: Two graphic matroids whose restriction to $\{a,b,c\}$ coincide.
• Figure 4: Two sets of points that induce the same matroid on $\{a,b,c,d,e,f\}$
• Figure 5: Once the dependency of point $g$ with respect the remaining points is specified, this constrains the possible geometric arrangements of points $\{a,b,c,d,e,f\}$.

Theorems & Definitions (88)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Lemma 4.3
• proof