Assigning Agents to Increase Network-Based Neighborhood Diversity

TL;DR

This paper addresses the challenge of assigning agents to network vertices to maximize neighborhood diversity, proposing approximation algorithms with theoretical guarantees and evaluating their performance on various networks.

Contribution

It introduces new approximation algorithms for maximizing diversity in agent assignments, including a local-improvement method, a semidefinite programming approach, and a PTAS for planar graphs.

Findings

The local-improvement algorithm achieves a 1/2 approximation factor.

The semidefinite programming approach surpasses 1/2 approximation in certain cases.

The PTAS efficiently solves the problem on planar graphs.

Abstract

Motivated by real-world applications such as the allocation of public housing, we examine the problem of assigning a group of agents to vertices (e.g., spatial locations) of a network so that the diversity level is maximized. Specifically, agents are of two types (characterized by features), and we measure diversity by the number of agents who have at least one neighbor of a different type. This problem is known to be NP-hard, and we focus on developing approximation algorithms with provable performance guarantees. We first present a local-improvement algorithm for general graphs that provides an approximation factor of 1/2. For the special case where the sizes of agent subgroups are similar, we present a randomized approach based on semidefinite programming that yields an approximation factor better than 1/2. Further, we show that the problem can be solved efficiently when the underlying graph is treewidth-bounded and obtain a polynomial time approximation scheme (PTAS) for the problem on planar graphs. Lastly, we conduct experiments to evaluate the per-performance of the proposed algorithms on synthetic and real-world networks.