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arXiv · 1902.10765

Reconfiguration of Connected Graph Partitions

Hugo A. Akitaya·Matthew D. Jones·Matias Korman·Christopher Meierfrankenfeld·Michael J. Munje·Diane L. Souvaine·Michael Thramann·Csaba D. Tóth
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Abstract

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $k\geq 1$, a $k$-district map of $G$ is a partition of $V(G)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is NP-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial length is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that takes a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.

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Hugo A. Akitaya, Matthew D. Jones, Matias Korman, Christopher Meierfrankenfeld, Michael J. Munje, Diane L. Souvaine, Michael Thramann, Csaba D. Tóth. 2019-02-27. Reconfiguration of Connected Graph Partitions. https://doi.org/10.1002/jgt.22856

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