Contiguity-Based Optimization Models for Political Redistricting Problems

Introduction

Political redistricting is a process of redrawing the boundaries of political districts such that a set of criteria can be satisfied. The most prominent example is the redistricting of congressional districts in the United States where the districts are designed to fulfill the one-person-one-vote doctrine (Morrill 1981). Population equality is often required in this process as a rule that requires that all districts have approximately the same number of voters to overcome malapportionment. Contiguity, as required by most states as well as the supreme court rulings¹, is a rule that states that an individual must be able to travel from any point in a district to any other point in that district without crossing the district’s boundary (Mills 1967; Nagel 1972; Grofman 1985).

Political redistricting has been and continues to be a highly controversial issue in the United States, partly because of a common practice that benefits a certain political group under the disguise of meeting other criteria, especially population equality. This process, known as Gerrymandering, normally lead to strange shapes of districts, which has made many researchers to believe that a computer created plan can be more “objective” (Altman 1997). Although computers have been used in redistricting since the 1960s (Thoreson and Liittschwager 1967), the computational intensity of computer programs has generally made them inferior to human experts (Altman 1995; Altman et al. 2005). The hope of using computers to automatically generate districting plans faded as researchers have realized that computer programs are no more objective than their human developers and users. Another reason is the complexity of the problem that has discouraged the use of computers in this area. The past decade has seen the development of many “perfect” redistricting plans where the population difference between any two districts is either zero or one person. These plans were created with geographical information systems that allowed interactive manipulation of spatial boundaries during the search for desirable plans. This situation presents a significant challenge: Is it possible to develop efficient algorithms or models that can compete with human?

Researchers have approached political redistricting problems as combinatorial optimization problems where the number of feasible solutions exponentially increases with the problem size (Altman 1998). These problems may be difficult to solve because of challenges in formulating their requirements in mathematical forms (Altman 1997; Altman 1998; Eagles et al. 2000), simultaneously satisfying several redistricting criteria (Williams 1995; Wei and Chai 2004), and the size of the solution space (Bação et al. 2005). A solution method for political redistricting problems should be efficient with respect to computational time and effective with respect to its ability to find high quality solutions.