The Price of Diversity in Assignment Problems

We introduce and analyze an extension to the matching problem on a weighted bipartite graph: Assignment with Type Constraints. The two parts of the graph are partitioned into subsets called types and blocks; we seek a matching with the largest sum of weights under the constraint that there is a pre-specified cap on the number of vertices matched in every type-block pair. Our primary motivation stems from the public housing program of Singapore, accounting for over 70% of its residential real estate. To promote ethnic diversity within its housing projects, Singapore imposes ethnicity quotas: each new housing development comprises blocks of flats and each ethnicity-based group in the population must not own more than a certain percentage of flats in a block. Other domains using similar hard capacity constraints include matching prospective students to schools or medical residents to hospitals. Limiting agents' choices for ensuring diversity in this manner naturally entails some welfare loss. One of our goals is to study the trade-off between diversity and social welfare in such settings. We first show that, while the classic assignment program is polynomial-time computable, adding diversity constraints makes it computationally intractable; however, we identify a $\tfrac{1}{2}$-approximation algorithm, as well as reasonable assumptions on the weights that permit poly-time algorithms. Next, we provide two upper bounds on the price of diversity -- a measure of the loss in welfare incurred by imposing diversity constraints -- as functions of natural problem parameters. We conclude the paper with simulations based on publicly available data from two diversity-constrained allocation problems -- Singapore Public Housing and Chicago School Choice -- which shed light on how the constrained maximization as well as lottery-based variants perform in practice.

A Characterization of Monotone Influence Measures for Data Classification

In this work we focus on the following question: how important was the i-th feature in determining the outcome for a given datapoint? We identify a family of influence measures; functions that, given a datapoint x, assign a value phi_i(x) to every feature i, which roughly corresponds to that i's importance in determining the outcome for x. This family is uniquely derived from a set of axioms: desirable properties that any reasonable influence measure should satisfy. Departing from prior work on influence measures, we assume no knowledge of - or access to - the underlying classifier labelling the dataset. In other words, our influence measures are based on the dataset alone, and do not make any queries to the classifier. While this requirement naturally limits the scope of explanations we provide, we show that it is effective on real datasets.