Online Capacitated General Matching with Knapsack

We study a new online matching problem termed Online Capacitated General Matching with Knapsack (OCGMK), which generalizes the Online General Matching (OGM) problem. In the original OGM, vertices arrive sequentially and need to be paired with other vertices to maximize the total reward of pairing. Our study is the first to consider capacitated vertices in OGM: we allow each vertex to be assigned to multiple vertices up to a capacity limit. We also consider a previously unexamined knapsack constraint in OGM: assigning a pair of vertices has a cost, but the total cost is budgeted. To solve the OCGMK problem, we propose the Online Capacity-Knapsack Assignment (OCKA) algorithm, which constructs capacity-friendly sets and knapsack-friendly sets to simultaneously and effectively address both constraints. OCKA achieves a competitive ratio of α = γ 2β , where γ = 1/(3+e -2 ) and β is the ratio between the overall cost of all edges and the cost budget. When the knapsack constraint is not imposed but the capacitated vertices remain, the competitive ratio of OCKA is α ′ = 1/2, recovering the previous best result for single-capacity OGM. We implement tracedriven experiments to evaluate the practical performance of OCKA on a real-world dating dataset, demonstrating the superior performance of OCKA in online dating applications.