Assigning Agents to Increase Network-Based Neighborhood Diversity

TL;DR

The paper studies assigning two demographic agent types to graph vertices to maximize the index of integration IoA, i.e., the number of agents adjacent to at least one neighbor of the other type. It presents a suite of algorithms with provable guarantees: a local-improvement method achieving a 1/2-approximation on general graphs, and a semidefinite-programming approach that yields better-than-1/2 guarantees when the minority fraction α is nontrivial; it also provides a PTAS for planar graphs via Baker’s technique and a polynomial-time DP for treewidth-bounded graphs. Empirical results on synthetic and real networks show the local-improvement method often far outperforms its worst-case bound, approaching optimal IoA in practice. The work connects social integration objectives to rigorous graph-theoretic optimization, with implications for public housing allocation and other placement problems where diversity is desired across network proximity.

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.

Figures (10)

• Figure 1: An example assignment of two type-1 agents (blue) and six type-2 agents (red) on a graph $\mathcal{G}{}$. Vertices with integrated agents are labeled by dashed circles. The index of integration for this assignment (ie., the number of integrated agents) is $6$.
• Figure 2: Two assignments $\mathcal{P}$ and $\mathcal{P}^*$ where type-1 and type-2 vertices are highlighted in blue and red, respectively. In this case, $\Tilde{\mathcal{V}{}}_{2-1}= \{x_3, x_4\}$ and $\Tilde{\mathcal{V}{}}_{1-2} = \{x_1, x_2\}$. We may then transform $\mathcal{P}$ into $\mathcal{P}^*$ by swapping types between the pair $(x_1, x_3)$ and between $(x_2, x_4)$. Note that this example is only to demonstrate how $\Tilde{\mathcal{V}{}}_{2-1}$ and $\Tilde{\mathcal{V}{}}_{1-2}$ are defined, as $\mathcal{P}$ cannot be a saturated assignment returned by the algorithm.
• Figure 3: The empirical approximation ratio $\gamma$ for algorithms. The number of vertices and edges ($n$, $m$) for each subgraph are as follows. City*: $(1607, 50112)$, Arena*: $(1981, 9132)$, Google+*: ($2000$, $5042$).
• Figure 4: The change of the fraction of integrated agents as the fraction of minority agents increases. The networks are Gnp and City shown in Table (\ref{['tab:networks']}).
• Figure 5: The change in the number of integrated agents as Local-Improvement proceeds. The underlying gnp networks have $1,000$ vertices; the average degree varies from $10$ to $30$.

Theorems & Definitions (62)

• Definition 3.1: Index of agent-integration (IoA) agarwal2020swap
• Definition 3.2: IM-IoA
• Lemma 4.2
• proof
• Lemma 4.3: Subcase 2.1
• proof
• Lemma 4.4: Subcase 2.1
• proof
• Lemma 4.5: Subcase 2.1
• proof