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Hardness and Approximation Algorithms for Balanced Districting Problems

Authors Prathamesh Dharangutte , Jie Gao , Shang-En Huang , Fang-Yi Yu

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  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Approximation algorithms
  • algorithmic fairness

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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-δ} for any constant δ > 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(√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) which requires a new design of a separation oracle specific for our balanced districting problem. To turn the fractional solution to a feasible integer solution, we adopt the randomized rounding algorithm by [Chan and Har-Peled, SoCG 2009]. To get a good approximation ratio of the rounding procedure, a crucial element in the analysis is the 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. Further, our algorithm is versatile - the very same algorithm can be analyzed in different ways on various graph classes, which leads to class-dependent approximation ratios. We also provide a FPTAS algorithm for complete graphs and tree graphs, as well as greedy algorithms and approximation ratios when the district cardinality is bounded, the graph has bounded degree or the weights are binary. We refer the readers to the full version of the paper for complete set of results and proofs.

Cite As Get BibTex

Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu. Hardness and Approximation Algorithms for Balanced Districting Problems. In 6th Symposium on Foundations of Responsible Computing (FORC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 329, pp. 4:1-4:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FORC.2025.4

Author Details

Prathamesh Dharangutte
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Jie Gao
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Shang-En Huang
  • Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan
Fang-Yi Yu
  • Department of Computer Science, George Mason University, Fairfax, VA, USA

Funding

  • Dharangutte, Prathamesh: Supported by NSF IIS-2229876, DMS-2220271, DMS-2311064, CCF-2208663, CCF-2118953.
  • Gao, Jie: Supported by NSF IIS-2229876, DMS-2220271, DMS-2311064, CCF-2208663, CCF-2118953.
  • Huang, Shang-En: Author conducted part of this research in Boston College and in visiting Osaka University and is currently supported by NTU Grant 113L7491.

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