📝 SummarXiv

Abstract

We initiate the study of two-sided online resource allocation with costly cancellations. Our focus is on edge-weighted online bipartite matching (and several of its extensions), where nodes arrive online and request offline resources. In contrast to the classic literature, any fraction of an offline resource that was preallocated to an earlier online node can be reclaimed, resulting in the loss of the previously allocated edge-weight plus an additional penalty equal to a non-negative constant factor $f$ times the edge-weight. Parameterizing the problem by the buyback factor $f$, our main result is the development of optimal competitive algorithms for \emph{all possible values} of $f$ through a novel primal-dual family of algorithms in the fractional (or equivalently, large capacity) setting, and establishing their optimality by deriving matching lower bounds. Interestingly, our results reveal a phase transition: for the small buyback regime ($f < \frac{e-2}{2}$), the optimal competitive ratio is $\frac{e}{e-(1+f)}$, and for the large buyback regime ($f \geq \frac{e-2}{2}$), the competitive ratio is $-W_{-1}\left(\frac{-1}{e(1+f)}\right)$, where $W_{-1}$ is the non-principal branch of the Lambert $W$ function. We also study variants of this model, such as matching with deterministic integral allocations. We again show a phase transition: for the small buyback regime ($f < \frac{1}{3}$), the optimal competitive ratio is $\frac{2}{1-f}$, while for the large buyback regime ($f \geq \frac{1}{3}$), the competitive ratio is $1 + 2f + 2\sqrt{f(1+f)}$. We further consider various extensions, including to configuration allocations and submodular welfare maximization, as well as negative values of $f$, modeling a secondary supply channels or overflow capacities available at discounted rates. Our unifying primal-dual framework achieves the exact optimal competitive ratio across all these variants