In this article, we delve into the realm of online matching with combinatorial constraints. Specifically, we focus on the problem of maximizing the revenue generated by an online algorithm that matches items with buyers while adhering to certain constraints. Our goal is to provide a comprehensive understanding of the competitive landscape in this field and establish new bounds for both lower and upper limits.
Lower Bounds
To begin with, we establish a tight lower bound on the competitive ratio of any online algorithm for matroid-constrained buyers. This involves demonstrating that any algorithm that can achieve a competitive ratio of 1/2 or better must necessarily have a probability of proposing items that are not available, which leads to a significant loss in revenue.
Next, we present a reduction from the single-item case in [26] to the multi-item setting, which enables us to establish lower bounds on the average reward achieved by any algorithm. This reduction is crucial in demonstrating that our algorithm can achieve an average reward of at least 1/2, thereby proving our main result.
Approximation Algorithm
Moving on to the approval algorithm for matroid-constrained buyers, we observe that there are two key components to our approach: (i) using linear programming constraints to reduce the problem and (ii) proposing items more frequently in certain instances. The former allows us to establish a tighter bound on the competitive ratio, while the latter enables us to improve upon previous upper bounds.
To address the issue of proposing unavailable items, we develop an extension of the CRS algorithm that operates slightly outside the matroid polytope. This enables us to propose more goods and increase our chances of obtaining a better outcome.
Main Result
Our main result shows that any online algorithm for matroid-constrained buyers must have a competitive ratio of at least 1/2, which implies that the optimal algorithm can achieve a revenue of at least $1/2 \cdot B$, where $B$ is the total value of the items. This establishes a tight lower bound on the competitive ratio and demonstrates that our approximation algorithm for matroid-constrained buyers is nearly optimal.
Conclusion
In conclusion, this article provides a comprehensive analysis of online matching with combinatorial constraints, including a new upper bound and a tight lower bound. Our main result shows that any algorithm must have a competitive ratio of at least 1/2, which highlights the challenges involved in solving this problem. Nonetheless, we are able to develop an approximation algorithm for matroid-constrained buyers that achieves an average reward of at least 1/2, which demonstrates the feasibility of solving this problem with high accuracy. Our findings contribute significantly to the field of online matching and provide insights into the development of more sophisticated algorithms in the future.
