A Quadratic Programming Model for Political Districting Problem

Political Districting Problem is one of the critical issues in political elections, it can be expressed as: how to partition a state into reasonable districts for parliament election and presidential election. In this paper, the political districting problem for parliament election is modelled as a doubly weighted graph partition problem and it is formulated into quadratic programming model. Using Lingo software, we can easily find the optimal solution. Our work can be summarized as follows: Firstly, we construct a graph for a given state H, where each node of H represents a county of the state, an edge connecting node u and v indicates that county u and v are adjacent in the map. After that, we extend H to a doubly weighted graph G(V,E,P(v),D(e)), in which each node represents a county of the state and the weight of a node denotes the size of population in that county; each edge (u,v) ∈ E implies that there is a path between node u and v in graph H, and the weight of an edge (u,v) represents the length of the shortest path between u and v in graph H. Then, we convert the districting problem into a doubly weighted graph partition problem and formulate it as a quadratic programming model. Finally, we compile a software using Lingo to solve the quadratic programming problem. By using our software, we partition New York State into 29 election districts which satisfy the following conditions: Each election district has the same size of population; and the topology of each election district is continuous compactness. It is much better than the election district partition of New York in 2002.