The districting problem: applications and solving methods

1. The districting problem: applications and solving methods Viviane Gascon Département des sciences de la gestion Université du Québec à Trois-Rivières

2. Introduction The districting problem consists in partitioning a geographical region into districts in order to plan some operations while considering different criteria or constraints.

3. Main criteria A district is contiguous if it is possible to travel from any point in the district to any other in the district without having to go through any other district • Contiguity • Compactness • Balance or equity • Respect of natural boundaries • Socio-economic homogeneity

4. Main criteria • Contiguity • Compactness • Balance or equity • Respect of natural boundaries • Socio-economic homogeneity Compactness is a criterion used to prevent the formation of odd-shaped districts that is districts should be circular or square in shape rather than elongated

5. Main criteria • Contiguity • Compactness • Balance or equity • Respect of natural boundaries • Socio-economic homogeneity Balanced in workload or in population in the districts

6. Main criteria • Contiguity • Compactness • Balance or equity • Respect of natural boundaries • Socio-economic homogeneity Rivers, railroads, mountains, administrative boundaries, etc.

7. Main criteria Having a better representation of residents who share common concerns or views (can be based on income revenues, minorities, etc.) • Contiguity • Compactness • Balance or equity • Respect of natural boundaries • Socio-economic homogeneity

8. Applications • Political districting (Hess and Weaver (1965), Garfinkel and Nemhauser (1970), Mehotra, Johnson and Nemhauser (1998), Bozkaya, Erkut and Laporte (2002)) • School districting (Ferland and Guénette (1990)) • Districting for health services (Gascon, Gorvan and Michelon (2010))

9. Political districting The political districting problem consists in partitioning an area into electoral constituencies (districts), each one being assigned a number of representatives. • one representative is assigned to each district; • each population unit is assigned to one district; • the number of districts is usually known (M districts); • all districts must have approximately the same number of voters for better equity

10. Political districting : Hess et al. (1965) • Among the first mathematical programming approach of the political districting problem. • The problem is modeled as an assignmentproblem with additional constraints where each population unit must be assigned to a district center.

11. Political districting : Hess et al. (1965) Mathematical model • Parameters: I : set of population units J : set of potential district centers M : number of district centers pi : population of the ith population unit a : minimum population allowed for a district b : maximum population allowed for a district a and b can be considered as deviations from the average population of all population units which is given by

12. Political districting : Hess et al. (1965) cij , thecost of assigning population unit i to district center j is the Euclidean distance between the district center i and the district center j. dij: distance between the centers of population units i and j. Minimizing the Euclidean distance between population units favours contiguous districts but do not guarantee them.

13. District j Population unit i Center of population unit i Population unit j Center of district j Center of population unit j

14. Political districting : Hess et al. (1965) Mathematical model • Variable: cij = dij2 pj is used in the objective function of the mathematical model by Hess et al.

15. Political districting : Hess et al. (1965) Mathematical model Min Subject to (1) (2) (3)

16. Political districting : Hess et al. (1965) Mathematical model Min Subject to Constraint (1) ensures that each population unit i is assigned to exactly one district (1) (2) (3)

17. Political districting : Hess et al. (1965) Mathematical model Min Subject to (1) Constraint (2) ensures that M districts are chosen. (2) (3)

18. Political districting : Hess et al. (1965) Mathematical model Min Subject to (1) (2) Constraint (3) ensures population equity among districts (3)

19. Political districting : Hess et al. (1965) Solving method : heuristic • Define district centers • Assign population equally to the district centers at minimum costs (with a transportation algorithm) • Adjust assignment so that each population unit is entirely within one district • Compute centroids and use them as improved district centers • Repeat from step 2 until solution converges • Try with other initial district centers

20. Political districting : Hess et al. (1965) Limits of the solving method • No guaranty of convergence • Non contiguous solutions must be rejected • If many solutions, choose the most compact one and one having a good population equity by always verifying that there is no deviation form the minimum and maximum allowable population

21. Political districting : Garfinkel and Nemhauser (1970) Garfinkel and Nemhauser (1970) considers predefined districts to be specified and among which the final districts are chosen.

22. Political districting : Garfinkel and Nemhauser (1970) Mathematical model • Parameters: I : set of population units J : set of potential districts M : number of district pi : population of the ith population unit P(j) : population of district j where

23. Political districting : Garfinkel and Nemhauser (1970) Mathematical model • Parameters: deviation of population of district j from the average population,

24. Political districting : Garfinkel and Nemhauser (1970) Mathematical model • Variable

25. Political districting : Garfinkel and Nemhauser (1970) Mathematical programming problem Minimise st Constraint (1) ensures that each population unit i is assigned to exactly one district (1) (P1) (2)

26. Political districting : Garfinkel and Nemhauser (1970) Minimise st (1) Constraint (2) ensures that that M districts are chosen. (P1) (2)

27. Political districting : Garfinkel and Nemhauser (1970) The problem implies that potential districts must be defined. • Contiguity : Let B = {bik}, a symmetric matrix where If a district is an undirected graph whose vertices are the units of the district, an arc exists between vertices i and k if and only if bik=1. A district is contiguous if and only if the graph is connected (a path exists between every pair of vertices). A district is feasible only if it is contiguous.

28. Connected graph of district j Population unit i Population unit j Center of district j

29. Political districting : Garfinkel and Nemhauser (1970) • A district is feasible only if is the maximum allowable percentage deviation of the population of a district from the average district population.

30. Political districting : Garfinkel and Nemhauser (1970) • Compactness : d(i,k) = distance between units i and k. e(i,k) = exclusion distance between units i and k. District j is feasible only if d(i,k) > e(i,k) implies that aij . akj = 0 (i and j can not be in the same district if the distance between them is higher than e(i,k)) i e(i,k) d(i,k) k

31. Political districting : Garfinkel and Nemhauser (1970) • Compactness : dj = distance between the units of j for district j which are farthest apart. dj = (dj measures the “range” of the district) A(j) = area of district j is a dimensionless measure of the shape compactness of district j District j is feasible only if

32. Political districting : Garfinkel and Nemhauser (1970) Solving method : two phase method • Phase I: Find feasible districts Start at an arbitrary unit and adjoin contiguous units until the combined population becomes feasible. If the district is compact, keep it. If combined population exceeds the upper limit, backtrack on the enumeration tree. It is verified if the district has some enclaves. District with an enclave

33. Political districting : Garfinkel and Nemhauser (1970) Solving method : two phase method • Phase II: Solve the mathematical programming problem (search tree algorithm) (see paper for more details)

34. Political districting : Mehotra, Johnson and Nemhauser (1998) The problem considered by Mehotra et al. (1998) is similar to the problem in Garfinkel and Nemhauser (1970). But their model considers more potential districts. They consider a graph partitioning problem where • A node is associated to every population unit (its weight is equal to the corresponding population) • An edge connects two nodes when the corresponding population units are neighbours • A solution is a connected graph (for contiguity) for which the sum of the node weights is within a population interval (for population equity).

35. Political districting : Mehotra, Johnson and Nemhauser (1998) Same model as Garfinkel and Nemhauser (1970) except for cj which is the cost of district j. The question is : how should cjbe defined ? (1) (P2) (2)

36. Political districting : Mehotra, Johnson and Nemhauser (1998) The cost of district j, cj, measures its non compactness. V: set of population units E: edges connecting units if they share common borders G(V,E): graph G’(V’,E’): connected subgraph defining a district and satisfying population limits Non compactness of G’ will be measured by how far units in the district are from a central unit.

37. Political districting : Mehotra, Johnson and Nemhauser (1998) sij: number of edges in a shortest path from i to j in G. Center of G’ : node Cost of a district with u as the center of the district is given by A district is more compact when the cost is smaller. i sui= 2 u j suj= 2 k suk= 2

38. Political districting : Mehotra, Johnson and Nemhauser (1998) Solving method : column generation method • Start with a subset of feasible districts, J’ • Solve the linear relaxation of (P2) restricted to J’ where This linear relaxation of (P2) is LP-P2(J’). • The optimal solution of the linear relaxation of (P2) is feasible to LP-P2(J). A dual value pi is obtained for each constraint in LP-P2(J). • Determine if the optimal solution of LP-P2(J’) is optimal for LP-P2(J). This is done by solving a subproblem SP.

39. Political districting : Mehotra, Johnson and Nemhauser (1998) Solving method : column generation method Parameters for SP : pi: population of unit i pmin, pmax: lower and upper bounds on the population of a district is the average population of a district

40. Political districting : Mehotra, Johnson and Nemhauser (1998) SP problem

41. Political districting : Mehotra, Johnson and Nemhauser (1998) Contiguity constraints To ensure contiguity of districts, districts are required to be subtrees of a shortest path tree rooted at u(district center). Constraints allowing district j to be selected only if at least one of the nodes that is adjacent to it and closer to u is also selected, are added, that is If then we add the contiguity constraint ensuring that node j is selected only if all nodes along some shortest path from u to j are also selected.

42. Political districting : Mehotra, Johnson and Nemhauser (1998) If the optimal objective value of SP is negative then a district with minimum value is added to the set J’ and LP-P2(J’) is solved again. Otherwise, the current solution to LP-P2(J’) is also optimal to LP-P2(J). In this case, if the solution is integral, then a solution to P2 is found. If it is not integral, a branching rule is applied, based on a depth-first-search strategy, to find another solution.

43. Political districting : Bozkaya, Erkut and Laporte (2003) The political districting problem solved by Bozkaya et al. (2003) considers the contiguity constraint as a hard constraint and all other criteria as soft constraints through a weighted objective function. Other criteria : • population equality • compactness • socio-economic homogeneity • similar districts to the existing districts • integrity of communities

44. Political districting : Bozkaya, Erkut and Laporte (2003) Population equality: J : set of all districts in solution x (feasible or not) Pj(x): population of district j in solution x is the average population of the district The population of a district is required to be in the interval Population equality function : It evaluates the maximum deviation of the population in the district from the maximum and the minimum allowed

45. Political districting : Bozkaya, Erkut and Laporte (2003) Compactness: two measures R : perimeter of the whole territory, used for scaling Rj(x) : perimeter of district j in solution x Compactness measure 1 : Compactness measure 2 :

46. Political districting : Bozkaya, Erkut and Laporte (2003) Socio-economic homogeneity : minimize the sum of the standard deviation of income Sj(x): standard deviation of income in district j : average income Socio-economic homogeneity function:

47. Political districting : Bozkaya, Erkut and Laporte (2003) Similar districts to the existing districts: Oj(x) : largest overlay of district j with a district contained in a solution x A: entire area Similarity objective function: Old and new districts Overlaying sectors

48. Political districting : Bozkaya, Erkut and Laporte (2003) Integrity of communities: Gj(x) : largest population of a given community in district j of solution x Integrity of communities objective function : minimize

49. Political districting : Bozkaya, Erkut and Laporte (2003) Solving method : Tabu search Objective function

50. Political districting : Bozkaya, Erkut and Laporte (2003) Solving method : Tabu search Initial solution : select a seed unit for a district and add to it adjacent units until the district population attains or when no adjacent units are available. If the number of districts created is larger than M, reduce it by merging the least populated unit with the least populated neighbour. If the number of districts created is less than M, gradually increase it by iteratively splitting the most populated district into two while preserving contiguity.