We study Matching, Clustering, and related problems in a partial information setting, where the agents' true utilities are hidden, and the algorithm only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph of agent utilities, but the algorithm receives as input only a preference ordering for each agent induced by the underlying weights. Against this backdrop, we design algorithms to approximate the true optimum solution with respect to the hidden weights. Perhaps surprisingly, such algorithms are possible for many important problems, as we show using our framework based on greedy and random techniques. Our framework yields a 1.6-approximation algorithm for the maximum weighted matching problem, a 2-approximation for the problem of clustering agents into equal sized partitions, a 4-approximation algorithm for Densest $k$-subgraph, and a 1.88-approximation algorithm for Max TSP as long as the hidden weights constitute a metric. Our results are the first non-trivial ordinal approximation algorithms for such problems, and indicate that in many situations, we can design robust algorithms even when we are agnostic to the precise agent utilities.
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