Tradeoffs Between Information and Ordinal Approximation for Bipartite Matching

Abstract

We study ordinal approximation algorithms for maximum-weight bipartite matchings. Such algorithms only know the ordinal preferences of the agents/nodes in the graph for their preferred matches, but must compete with fully omniscient algorithms which know the true numerical edge weights (utilities). Ordinal approximation is all about being able to produce good results with only limited information. Because of this, one important question is how much better the algorithms can be as the amount of information increases. To address this question for forming high-utility matchings between agents in $\mathcal {X}$ and $\mathcal {Y}$, we consider three ordinal information types: when we know the preference order of only nodes in $\mathcal {X}$ for nodes in $\mathcal {Y}$, when we know the preferences of both $\mathcal {X}$ and $\mathcal {Y}$, and when we know the total order of the edge weights in the entire graph, although not the weights themselves. We also consider settings where only the top preferences of the agents are known to us, instead of their full preference orderings. We design new ordinal approximation algorithms for each of these settings, and quantify how well such algorithms perform as the amount of information given to them increases.