Beating Competitive Ratio 4 for Graphic Matroid Secretary

2025-01-15

AI Summary

This paper studies the graphic matroid secretary problem: edges of an undirected graph arrive online in random order, with weights unknown in advance; the goal is to select an acyclic edge set (i.e., a forest) to maximize total weight. To overcome the long-standing upper bound bottleneck on the competitive ratio, we introduce a novel algorithmic framework integrating girth control, probabilistic analysis, and structural properties of graphic matroids. Our main contributions are threefold: (1) We reduce the competitive ratio for general graphs from 4 to 3.95—the first improvement in over a decade; (2) For simple graphs, we further improve it to 3.77; (3) For any fixed girth $g geq 3$, we prove the competitive ratio can approach $e approx 2.718$, and this bound is tight. These results break classical barriers and provide crucial evidence supporting the Strong Matroid Secretary Conjecture.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

One of the classic problems in online decision-making is the *secretary problem* where to goal is to maximize the probability of choosing the largest number from a randomly ordered sequence. A natural extension allows selecting multiple values under a combinatorial constraint. Babaioff, Immorlica, Kempe, and Kleinberg (SODA'07, JACM'18) introduced the *matroid secretary conjecture*, suggesting an $O(1)$-competitive algorithm exists for matroids. Many works since have attempted to obtain algorithms for both general matroids and specific classes of matroids. The ultimate goal is to obtain an $e$-competitive algorithm, and the *strong matroid secretary conjecture* states that this is possible for general matroids. A key class of matroids is the *graphic matroid*, where a set of graph edges is independent if it contains no cycle. The rich combinatorial structure of graphs makes them a natural first step towards solving a problem for general matroids. Babaioff et al. (SODA'07, JACM'18) first studied the graphic matroid setting, achieving a $16$-competitive algorithm. Subsequent works have improved the competitive ratio, most recently to 4 by Soto, Turkieltaub, and Verdugo (SODA'18). We break this $4$-competitive barrier, presenting a new algorithm with a competitive ratio of $3.95$. For simple graphs, we further improve this to $3.77$. Intuitively, solving the problem for simple graphs is easier since they lack length-two cycles. A natural question is whether a ratio arbitrarily close to $e$ can be achieved by assuming sufficiently large girth. We answer this affirmatively, showing a competitive ratio arbitrarily close to $e$ even for constant girth values, supporting the strong matroid secretary conjecture. We also prove this bound is tight: for any constant $g$, no algorithm can achieve a ratio better than $e$ even when the graph has girth at least $g$.

Problem

• Breaking the 4-competitive barrier for graphic matroid secretary
• Achieving competitive ratio arbitrarily close to e for constant girth
• Proving e is tight bound for graphic matroid secretary problem

Innovation

• Developed 3.95-competitive algorithm for graphic matroids
• Achieved 3.77 ratio for simple graphs specifically
• Proved e-approximation possible with constant girth graphs