Measuring the Compactness of Political Districting Plans

The United States Supreme Court has long recognized compactness as an important principle in assessing the constitutionality of political districting plans. We propose a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable -- which we coin the relative proximity index. We prove that any compactness measure which satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, requiring us to solve for each state's maximal compactness; an NP-hard problem. Using two properties of maximally compact districts, we prove they are power diagrams and develop an algorithm based on these insights. The correlation between our index and the commonly-used measures of dispersion and perimeter is -.22 and -.06, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when their average vote increases is significantly greater under maximally compact districting plans, relative to the existing plans.

A Graph Partitioning Model of Congressional Redistricting

Redrawing congressional districts in the United States is a constitutionally required, yet politically controversial, task undertaken after each decennial census. Federal law requires contiguous, `relatively compact\u27 congressional districts that maintain `approximately equal\u27 population. Controversy is introduced when individual states redraw their districts, or redistrict, using partisan committees. States such as Ohio continue to redistrict with a committee appointed according to the current proportion of legislators\u27 political parties to the whole. When political parties have majority power in redistricting committees, they can draw districts in a way that gives their party the best chance to keep its majority representation, a process called gerrymandering. Mathematical redistricting models seek an unbiased computational approach to the problem. Rather than trust partisan committees, mathematical modeling approaches rely upon well-defined methods in computational geometry, graph theory, game theory, and other fields. Here, we discuss two such approaches. The first, given as a background for comparison, constructs Voronoi diagrams to redistrict states into convex polygons, which are generally considered `compact\u27. We give greater emphasis to a new model that discretizes a state\u27s population and partitions it into regions of approximately equal population. This model, our main focus, relies upon graph partitioning to achieve the desired result and uses census population data as the sole parameter in redistricting

Measuring the Compactness of Political Districting Plans

We develop a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable, which we coin the relative proximity index. Any compactness measure that satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, which requires us to solve for each state’s maximal compactness—a problem that is nondeterministic polynomial-time hard (NP hard). The correlations between our index and the commonly used measures of dispersion and perimeter are −.37 and −.29, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when its average vote increases is significantly greater under maximally compact districting plans relative to the existing plans.Economic

Districting Problems - New Geometrically Motivated Approaches

This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

Enumerating Graph Partitions Without Too Small Connected Components Using Zero-suppressed Binary and Ternary Decision Diagrams

Partitioning a graph into balanced components is important for several applications. For multi-objective problems, it is useful not only to find one solution but also to enumerate all the solutions with good values of objectives. However, there are a vast number of graph partitions in a graph, and thus it is difficult to enumerate desired graph partitions efficiently. In this paper, an algorithm to enumerate all the graph partitions such that all the weights of the connected components are at least a specified value is proposed. To deal with a large search space, we use zero-suppressed binary decision diagrams (ZDDs) to represent sets of graph partitions and we design a new algorithm based on frontier-based search, which is a framework to directly construct a ZDD. Our algorithm utilizes not only ZDDs but also ternary decision diagrams (TDDs) and realizes an operation which seems difficult to be designed only by ZDDs. Experimental results show that the proposed algorithm runs up to tens of times faster than an existing state-of-the-art algorithm