Online Vertex-Weighted Bipartite Matching and Single-bid Budgeted Allocations (1007.1271v1)

Abstract: We study the following vertex-weighted online bipartite matching problem: $G(U, V, E)$ is a bipartite graph. The vertices in $U$ have weights and are known ahead of time, while the vertices in $V$ arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in $U$. When all the weights are equal, this reduces to the classic \emph{online bipartite matching} problem for which Karp, Vazirani and Vazirani gave an optimal $\left(1-\frac{1}{e}\right)$-competitive algorithm in their seminal work~\cite{KVV90}. Our main result is an optimal $\left(1-\frac{1}{e}\right)$-competitive randomized algorithm for general vertex weights. We use \emph{random perturbations} of weights by appropriately chosen multiplicative factors. Our solution constitutes the first known generalization of the algorithm in~\cite{KVV90} in this model and provides new insights into the role of randomization in online allocation problems. It also effectively solves the problem of \emph{online budgeted allocations} \cite{MSVV05} in the case when an agent makes the same bid for any desired item, even if the bid is comparable to his budget - complementing the results of \cite{MSVV05, BJN07} which apply when the bids are much smaller than the budgets.

• The paper presents a (1 - 1/e)-competitive randomized algorithm that combines deterministic greedy strategies with random weight perturbations for vertex-weighted matching.
• The paper generalizes classical online matching techniques to handle budgeted allocation problems in practical settings such as online advertising.
• The paper bridges vertex-weighted matching and budgeted allocation challenges, offering a unified approach that informs scalable, real-world online allocation systems.

Vertex-Weighted Online Bipartite Matching and Budgeted Allocations

The paper by Aggarwal, Goel, Karande, and Mehta expands the foundational work on online bipartite matching by addressing the vertex-weighted variant of this problem. The problem is defined on a bipartite graph $G(U, V, E)$ where $V$ arrives in an online fashion, and each vertex in $U$ has a predetermined weight. The objective is to maximize the sum of the weights of matched vertices in $U$. This generalization is significant because, in practical applications such as online advertising and resource allocation, the value of matching varies for different vertices. The work extends the classic $(1 - \frac{1}{e})$-competitive algorithm of Karp, Vazirani, and Vazirani originally designed for unweighted instances, applying it to cases with arbitrary vertex weights.

The key contribution is the development of a $(1 - \frac{1}{e})$-competitive randomized algorithm that adapts well to the vertex-weighted online matching context. The algorithm utilizes random perturbations of weights. Specifically, it augments deterministic greedy strategies by introducing independent, multiplicative random factors that align expected selection probabilities with the optimal distribution over permutations. This approach, when applied to vertex-weighted problems, not only replicates the success of the KVV algorithm in the unweighted scenario but also establishes a sophisticated link with randomized strategies for online budgeted allocations.

Notably, this paper leverages the natural structure of the vertex-weighted problem to propose solutions to an essential subclass of online budgeted allocations where bids are uniform across desired items. The results assert a $(1 - \frac{1}{e})$ competitive ratio for this subclass, thus addressing an unresolved aspect of the budgeted allocation problem, which is commonly modeled in online advertising as sponsored search auctions. In environments where bids are comparable to budget constraints, the authors show that handling larger bid cases effectively necessitates randomization. This realization deepens our understanding of the role of randomization in complex online allocation problems.

One of the robust implications of the paper is its potential to unify approaches to vertex-weighted matching and budgeted allocations, highlighting that the algorithm's underlying principles can be adapted to tackle each problem's presumed complexities effectively. The algorithm achieves an optimal competitive ratio by carefully combining deterministic and randomized principles, as seen in disparate variations of budgeted allocation problems.

Future directions of this research might include extending this work to even more generalized online matching and allocation problems with complex network structures or additional constraints, such as fairness or diversity. There's also a potential to explore hybrid models that incorporate both random and adversarial vertex arrivals and expand on edge-weight perturbations in broader online allocation contexts.

In conclusion, the paper contributes significantly to the theoretical understanding of online matching problems with weighted vertices and inspires possible extensions into practical, large-scale online allocation systems. As online allocation problems continue to evolve with real-world constraints, the insights gained from these algorithms offer foundational strategies that can be adapted and expanded in various practical applications.