# Proportional Approval Voting, Harmonic k-median, and Negative Association

## TL;DR

This paper introduces a unified framework connecting multi-facility location problems and multiwinner election rules, showing that harmonic weight structures enable constant-factor approximation algorithms using advanced rounding techniques.

## Contribution

The paper demonstrates that problems with harmonic weight structures, including Proportional Approval Voting, admit constant-factor approximation algorithms despite lacking triangle inequality assumptions.

## Key findings

- Harmonic weights enable constant-factor approximation for related problems.
- Dependent rounding with negative association properties is effective for these problems.
- The approach applies to generalized k-median and multiwinner election models.

## Abstract

We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables.