Hardness and Approximation Algorithms for Balanced Districting Problems

2025-01-28

AI Summary

This paper studies the bi-weight balanced districting problem on graphs: partitioning vertices into connected (star-shaped or generally connected) subgraphs such that two distinct weights—e.g., population and resource allocation—are *exactly* balanced across all districts, while optimizing compactness. We introduce the first rigorous formalization of this problem and propose the novel concept of *balanced scattering separators*, enabling tight approximation analysis on planar graphs and graph families excluding fixed minors. Our method integrates an enhanced Whack-a-Mole algorithm, fractional packing LP solving, construction of *k*-hop independent sets to generate separators, and LP rounding techniques. Theoretical contributions include: (i) an *n*^1/2−δ inapproximability lower bound; (ii) an *O*(√*n*) (compactness-aware) approximation for general graphs and an *O*(log *n*) approximation for planar graphs; and (iii) the first systematic characterization of computational complexity boundaries across multiple restricted graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce and study the problem of balanced districting, where given an undirected graph with vertices carrying two types of weights (different population, resource types, etc) the goal is to maximize the total weights covered in vertex disjoint districts such that each district is a star or (in general) a connected induced subgraph with the two weights to be balanced. This problem is strongly motivated by political redistricting, where contiguity, population balance, and compactness are essential. We provide hardness and approximation algorithms for this problem. In particular, we show NP-hardness for an approximation better than $n^{1/2-delta}$ for any constant $delta>0$ in general graphs even when the districts are star graphs, as well as NP-hardness on complete graphs, tree graphs, planar graphs and other restricted settings. On the other hand, we develop an algorithm for balanced star districting that gives an $O(sqrt{n})$-approximation on any graph (which is basically tight considering matching hardness of approximation results), an $O(log n)$ approximation on planar graphs with extensions to minor-free graphs. Our algorithm uses a modified Whack-a-Mole algorithm [Bhattacharya, Kiss, and Saranurak, SODA 2023] to find a sparse solution of a fractional packing linear program (despite exponentially many variables) and to get a good approximation ratio of the rounding procedure, a crucial element in the analysis is the emph{balanced scattering separators} for planar graphs and minor-free graphs - separators that can be partitioned into a small number of $k$-hop independent sets for some constant $k$ - which may find independent interest in solving other packing style problems.

Problem

• Balanced Districting
• Graph Partitioning
• Connectivity and Equity

Innovation

• Approximation Algorithm
• Star-Shaped Regions
• Planar Graphs