[Paper Review] Online vertex-weighted bipartite matching and single-bid budgeted allocations

Gagan Aggarwal, Gagan Goel+2 more|arXiv (Cornell University)|Jan 23, 2011

Optimization and Search ProblemsComputer Science13 references137 citations

TL;DR

This paper presents a randomized (1−1/e)-competitive algorithm for the online vertex-weighted bipartite matching problem, where vertex weights in set U are known in advance and vertices in V arrive online. The algorithm uses multiplicative random perturbations of weights to generalize the Karp-Vazirani-Vazirani algorithm, achieving optimality for general weights and solving the single-bid budgeted allocation problem when bids are comparable to budgets.

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 online bipartite matching problem for which Karp, Vazirani and Vazirani gave an optimal (1−1/e)-competitive algorithm in their seminal work [10].Our main result is an optimal (1−1/e)-competitive randomized algorithm for general vertex weights. We use random perturbations of weights by appropriately chosen multiplicative factors. Our solution constitutes the first known generalization of the algorithm in [10] in this model and provides new insights into the role of randomization in online allocation problems. It also effectively solves the problem of online budgeted allocations [14] 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 [14, 3] which apply when the bids are much smaller than the budgets.

Motivation & Objective

• To develop an optimal online algorithm for vertex-weighted bipartite matching when vertex weights in U are known in advance and vertices in V arrive online.
• To generalize the Karp-Vazirani-Vazirani (1−1/e)-competitive algorithm from equal weights to arbitrary vertex weights.
• To provide new insights into the role of randomization in online allocation problems with general weights.
• To solve the single-bid budgeted allocation problem in the regime where bids are comparable to budgets, complementing prior work on small bids.

Proposed method

• Introduce multiplicative random perturbations to the vertex weights in U to break symmetry and guide online matching decisions.
• Use a randomized algorithm that assigns each arriving vertex in V to the highest-weighted available neighbor in U, based on perturbed weights.
• Apply concentration bounds to show that the perturbed weights maintain (1−1/e)-competitiveness in expectation.
• Prove optimality by showing no online algorithm can achieve better than (1−1/e)-competitiveness for this problem.
• Leverage the structure of the online bipartite matching problem to extend results from equal-weight to weighted settings.

Experimental results

Research questions

• RQ1 Can the (1−1/e)-competitive ratio be achieved in online vertex-weighted bipartite matching with arbitrary vertex weights?
• RQ2 How can randomization be effectively used to generalize the Karp-Vazirani-Vazirani algorithm to weighted settings?
• RQ3 What is the performance of online allocation algorithms when bids are comparable to budgets, rather than small relative to them?
• RQ4 Does the use of multiplicative weight perturbations preserve competitiveness in the weighted online bipartite matching model?

Key findings

• The proposed algorithm achieves a (1−1/e)-competitive ratio for online vertex-weighted bipartite matching with general vertex weights.
• The algorithm is optimal, as no online algorithm can achieve a better competitive ratio for this problem.
• Random perturbations using multiplicative factors effectively generalize the classic algorithm to the weighted case.
• The method resolves the single-bid budgeted allocation problem when bids are comparable to budgets, extending prior results that required small bids.