[Paper Review] Proportional Approval Voting, Harmonic k-median, and Negative Association

Jarosław Byrka, Piotr Skowron+1 more|arXiv (Cornell University)|Apr 7, 2017

Game Theory and Voting Systems27 references12 citations

TL;DR

This paper introduces a unified framework for OWA k-median problems with weighted client costs, showing that harmonic weights (1, 1/2, 1/3, ..., 1/k) enable a constant-factor approximation without requiring the triangle inequality—a surprising result in facility location. The authors devise a dependent rounding algorithm based on Negative Association to achieve a constant approximation ratio, establishing the first such result for a k-median variant without triangle inequality assumptions.

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.

Motivation & Objective

• To unify the study of multiwinner approval voting and multi-facility k-median problems under a common OWA k-median framework.
• To analyze the approximability of OWA k-median problems based on structural properties of weight vectors.
• To show that harmonic weights (1, 1/2, ..., 1/k) yield particularly favorable approximation properties, enabling constant-factor approximation.
• To develop a novel approximation algorithm for the harmonic k-median problem that does not assume triangle inequality on costs.
• To establish strong inapproximability results for other weight sequences, such as p-geometric weights, under standard complexity assumptions.

Proposed method

• Formulate the OWA k-median problem using non-increasing weight vectors to aggregate client costs across selected facilities.
• Use a natural LP relaxation of the OWA k-median problem as the starting point for rounding.
• Apply dependent rounding (Srinivasan, FOCS'01) to the LP solution to produce integral facility sets.
• Leverage the stronger Negative Association (NA) property of the rounding process to apply concentration bounds for conditional random variables.
• Prove that the rounding procedure yields a constant-factor approximation due to the harmonic structure of weights.
• Establish inapproximability results via reductions from exact set cover (X3C) and set cover problems, showing hardness for general and p-geometric weights.

Experimental results

Research questions

• RQ1 Can the OWA k-median problem with harmonic weights be approximated within a constant factor without assuming the triangle inequality on client-facility costs?
• RQ2 What structural properties of weight vectors enable constant-factor approximation in OWA k-median problems?
• RQ3 Is there a polynomial-time approximation algorithm for the harmonic k-median problem when costs do not satisfy the triangle inequality?
• RQ4 What is the approximability threshold for OWA k-median with p-geometric weights (w_i = p^{i-1})?
• RQ5 Can the Proportional Approval Voting (PAV) rule be approximated in polynomial time via this framework?

Key findings

• The harmonic k-median problem admits a constant-factor approximation algorithm even without assuming the triangle inequality on costs, which is the first such result for a k-median variant.
• The key to the approximation lies in the harmonic weight structure, which enables the use of Negative Association in dependent rounding to bound error probabilities.
• A polynomial-time (1+ε)-approximation algorithm exists for the harmonic k-median problem under the given framework.
• For p-geometric weights with p < 1/e, no constant-factor approximation algorithm exists unless P = NP.
• The OWA k-median problem with weights that only consider a λ-fraction of facilities (λ < 1) is inapproximable within any positive factor unless P = NP.
• The Proportional Approval Voting (PAV) rule, which corresponds to 0-1 costs and harmonic weights, is captured by the harmonic k-median framework and inherits the same approximation guarantees.