On the computational tractability of a geographic clustering problem arising in redistricting

Redistricting is the problem of dividing a state into a number $k$ of regions, called districts. Voters in each district elect a representative. The primary criteria are: each district is connected, district populations are equal (or nearly equal), and districts are "compact". There are multiple competing definitions of compactness, usually minimizing some quantity. One measure that has been recently promoted by Duchin and others is number of cut edges. In redistricting, one is given atomic regions out of which each district must be built. The populations of the atomic regions are given. Consider the graph with one vertex per atomic region (with weight equal to the region's population) and an edge between atomic regions that share a boundary. A districting plan is a partition of vertices into $k$ parts, each connnected, of nearly equal weight. The districts are considered compact to the extent that the plan minimizes the number of edges crossing between different parts. Consider two problems: find the most compact districting plan, and sample districting plans under a compactness constraint uniformly at random. Both problems are NP-hard so we restrict the input graph to have branchwidth at most $w$. (A planar graph's branchwidth is bounded by its diameter.) If both $k$ and $w$ are bounded by constants, the problems are solvable in polynomial time. Assume vertices have weight~1. One would like algorithms whose running times are of the form $O(f(k,w) n^c)$ for some constant $c$ independent of $k$ and $w$, in which case the problems are said to be fixed-parameter tractable with respect to $k$ and $w$). We show that, under a complexity-theoretic assumption, no such algorithms exist. However, we do give algorithms with running time $O(c^wn^{k+1})$. Thus if the diameter of the graph is moderately small and the number of districts is very small, our algorithm is useable

Balancing Graph Voronoi Diagrams

Abstract—Many facility location problems are concerned with minimizing operation and transportation costs by par-titioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s logn. Keywords-graph Voronoi diagram; balancing; facility loca-tion; territorial design I

Partitioning a multi-weighted graph to connected subgraphs of almost uniform size

Assume that each vertex of a graph G is assigned a constant number q of nonnegative integer weights, and that q pairs of nonnegative integers li and ui, 1 ≤ i ≤ q, are given. One wishes to partition G into connected components by deleting edges from G so that the total i-th weights of all vertices in each component is at least li and at most ui for each index i, 1 ≤ i ≤ q. The problem of finding such a "uniform" partition is NP-hard for series-parallel graphs, and is strongly NP-hard for general graphs even for q = 1. In this paper we show that the problem and many variants can be solved in pseudo-polynomial time for series-parallel graphs. Our algorithms for series-parallel graphs can be extended for partial k-trees, that is, graphs with bounded tree-width