Austin Buchanan This paper studies the problems of partitioning the vertices of a graph G = (V,E) into (or covering with) a minimum number of low-diameter clusters from the lenses of approximation algorithms and integer programming. Here, the low-diameter criterion is formalized by an s-club, which is a subset of vertices whose induced subgraph has diameter at … Read more
The most successful approaches for the TSP use the integer programming model proposed in 1954 by Dantzig, Fulkerson, and Johnson (DFJ). Although this model has exponentially many subtour elimination constraints (SECs), it has been observed that relatively few of them are needed to prove optimality in practice. This leads us to wonder: What is the … Read more
Consider the task of dividing a state into k contiguous political districts whose populations must not differ by more than one person, following current practice for congressional districting in the USA. A widely held belief among districting experts is that this task requires at least k-1 county splits. This statement has appeared in expert testimony, … Read more
This mini survey covers optimization, heuristic, and sampling methods for political districting. CitationAccepted for publication in the Encyclopedia of Optimization (Springer, 3rd edition) edited by Panos M. Pardalos and Oleg A. Prokopyev. ArticleDownload View PDF
In the academic literature and in expert testimony, the Polsby-Popper score is the most popular way to measure the compactness of a political district. Given a district with area \(A\) and perimeter \(P\), its Polsby-Popper score is given by \( (4 \pi A)/P^2\). This score takes values between zero and one, with circular districts achieving … Read more
When partitioning a state into political districts, a common criterion is that political subdivisions like counties should not be split across multiple districts. This criterion is encoded into most state constitutions and is sometimes enforced quite strictly by the courts. However, map drawers, courts, and the public typically do not know what amount of splitting … Read more
Motivated by applications in political districting, we consider the task of partitioning the n vertices of a planar graph into k connected components. We propose an extended formulation that has two desirable properties: (i) it uses just O(n) variables, constraints, and nonzeros, and (ii) it is perfect. To explore its ability to solve real-world problems, … Read more
Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the $s$-club, which is a vertex subset that induces a subgraph of diameter at most $s$. This model has found use in a variety … Read more
When constructing political districting plans, prominent criteria include population balance, contiguity, and compactness. The compactness of a districting plan, which is often judged by the “eyeball test,” has been quantified in many ways, e.g., Length-Width, Polsby-Popper, and Moment-of-Inertia. This paper considers the number of cut edges, which has recently gained traction in the redistricting literature … Read more
The usual integer programming formulation for the maximum clique problem has several undesirable properties, including a weak LP relaxation, a quadratic number of constraints and nonzeros when applied to sparse graphs, and poor guarantees on the number of branch-and-bound nodes needed to solve it. With this as motivation, we propose new mixed integer programs (MIPs) … Read more