Political districting to minimize cut edges

Vertex covering with capacitated trees

“…Hojny et al [43] compared a single‐commodity flow and a node separator approach in the context of the connected max‐$$$ k $$$ cut problem and found the flow formulation to be a competitive way to model connectivity. Validi and Buchanan [77] achieve the same result in the context of political districting.…”

“…Similar connectivity formulations, apart from the vast literature concerning traveling salesman problems, originate for instance from the generalized minimum spanning tree problem [63,66,67], a connected network design problem [59], or the minimum arborescence problem [26]. Very recently, connectivity in MIPs has also been studied with regard to balanced connected partitions [62] and the connected max-k-cut problem [43], and, close to our setting, with regard to political districting [77,78].…”

“…It should be chosen as small as possible, and $$$ M=\mid V\mid -1 $$$ for each arc $$$ \left(u,v\right) $$$ is an upper bound. Following Validi and Buchanan [77], the upper weight bound allows us to improve it to $$$ M={\max}_{D\subseteq V\setminus \left\{r\right\}}\kern0.2em \left\{|D|\kern0.3em :\kern0.3em w(D)\le {W}_U-{w}_r\right\} $$$, which we can compute with a greedy algorithm. We can further strengthen the formulation by considering arc specific big $$$ M $$$ values.…”

“…Publicly available districting instances are rare. We are aware of political districting instances considered by Validi and Buchanan [77], and the commercial territory design instances by Salazar‐Aguilar et al [72]. Here, we detail the latter work that aims at partitioning a graph, representing a geographic area, into a fixed number of business districts with certain properties.…”

“…Maculan [58] transforms the formulation to the Steiner tree problem. Similar SCF models are used in [20,24,43,52,62,76,77] and [78]. The first two of these papers consider a rooted and budgeted (only with upper bounds) version.…”