Hardness and Approximation Algorithms for Balanced Districting Problems

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.