We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem

Tasnádi, Attila

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Optimal partisan districting on planar geographiesAttila Tasna´diMTA-BCE “Lendu¨let” Strategic Interactions Research Group, Department of Mathe-matics, Corvinus University of Budapest, H – 1093 Budapest, Fo˝va´m te´r 8, Hungary,attila.tasnadi@uni-corvinus.huJune 2013Abstract. We show that optimal partisan districting in the plane with geographicalconstraints is an NP-complete problem.Keywords: districting, gerrymandering, NP-complete problems.1 IntroductionIn electoral systems with single-member districts (or even with at least two multi-member districts) redistricting has to be carried out to resolve geographic malappor-tionment caused by migration and different district population growth rates. An in-herent difficulty associated with redistricting is that it may favor a party. The problembecomes even worse if redistricting is manipulated for an electoral advantage, which isreferred to as gerrymandering.In the middle of the previous century it was hoped that the problem of gerryman-dering could be overcome by computer programs using only data on voters geographicdistribution without any statistical information on voters preferences (e.g. Vickrey,1961) and thus determining an ‘unbiased’ districting. The first algorithm finding alldistrictings with (i) equally sized, (ii) connected and (iii) compact districts was givenby Garfinkel and Nemhauser (1970).1 The computational difficulty of the problem wasclear from the very beginning. Nagel (1972) documented in an early survey the com-putational limitations of automated redistricting by considering the available programsof his time. Altman (1997) showed that the problems of achieving any of the threementioned criteria are NP-hard. Moreover, he also demonstrated that maximizing thenumber of competitive districts is also NP-hard. Because of the computational diffi-culty of the problem there is a growing literature on new approaches to finding unbiaseddistrictings (see, for instance, Mehrotra et al. (1998), Bozkaya et al. (2003), Bac¸a˜o etal. (2005), Chou and Li (2006), Ricca and Simeone (2008) or Ricca et al. (2008)). Forrecent surveys we refer to Ricca et al. (2011) and Tasndi (2011).Though finding an equally sized districting is already computationally hard, fromanother point of view it is feared by the public that the continuously increasing compu-tational power makes the problem of carrying out an optimal partisan gerrymanderingpossible. However, the underlying difficulty that hinders us in finding an unbiaseddistricting, does not allow us to determine an optimal partisan redistricting. Indeed,Altman and McDonald (2010) provide recent evidence that current computer programsare far away from finding an optimal gerrymandering.A formal proof establishing that a simplified version of the optimal gerrymander-ing problem is NP-complete was given by Puppe and Tasna´di (2009). Though theytake geographical constraints into account, planarity is not prescribed explicitly. Thecurrent paper overcomes this shortcoming by locating voters in the plane.2 The FrameworkWe assume that parties A and B compete in an electoral system consisting only ofsingle member districts. In addition, voters with known party preferences are locatedin the plane and have to be divided into a given number of almost equally sized districts.The districting problem is defined by the following structure:Definition 1. A districting problem is given by Π = (X,N, (xi)i∈N , v,K,D), where• X is a bounded and strictly connected2 subset of R2,1Earlier Hess et al. (1965) provided an algorithm striving for similar goals; however, their algorithmdid not always obtain optimal solutions.2We call a bounded subset A of R2 strictly connected if its boundary ∂A is a closed Jordan curve.1• the finite set of voters is denoted by N = {1, . . . , n},• the distinct locations of voters are given by x1, . . . , xn ∈ int(X),• the voters’ party preferences are given v : N → {A,B},• the set of district labels is denoted by K = {1, . . . , k}, where bn/kc ≥ 3, and• D denotes the finite set of admissible districts consisting of bounded and strictlyconnected subsets of X and each of them containing the location of bn/kc ordn/ke voters,3 and furthermore,• we shall assume that based on their locations the n voters can be partitioned intok districts {D1, . . . , Dk} ⊆ D.Observe that in defining the districting problem, we assumed that obtaining an al-most equally sized districting is possible, which can be justified by the fact that findingan admissible districting for real-life problems is possible, while finding a districtingsatisfying additional requirements such as partisan optimality is difficult. In particu-lar, the staff hired to produce a districting map could always construct a districtingmap consisting of almost equally sized districts although other properties as partisanoptimality are difficult to prove or confute. Producing a districting with almost equallysized districts, is a tractable problem if there are not to many geographical restrictionssince we can obtain a result by drawing districts from left to right and from top tobottom on a map of a state by keeping the average district size in mind.We shall mention that in reality the basic units of a districting problem from whichdistricts have to be created are census blocks or counties rather than voters. In thiscase voter preferences v : N → {A,B} have to be replaced by a function of typev′ : N ′ → [0, 1], where N ′ stands for the finite set of counties, expressing the fractionof party A voters. However, our results obtained in this paper can be extended to thismore general setting, by allowing the case of almost equally sized counties, for whichdistrict outcomes are determined by the number of winning counties for party A, whichhappens to be the case, for instance, if v′(N ′) = {α, 1− α} for a given α ∈ [0, 1/2).Turning back to our districting problem defined on the level of voters, we have toassign each voter to a district.Definition 2. An f : N → D is a districting for problem Π if there exists a set ofdistricts D1, . . . , Dk ∈ D such that• f(N) = {D1, . . . , Dk},• Di ∩Dj = ∅ if i 6= j and i, j ∈ K,• {xi | i ∈ f−1(Dj)} ⊂ int(Dj) for any j ∈ K.Observe that without loss of generality we do not explicitly require that a districtingcovers the entire country but just the inhibited areas.Definition 3. Two districtings f : N → D and g : N → D with districts D1, . . . , Dkand D′1, . . . , D′k, respectively, are equivalent if there exists a bijection between theseries of sets {xi | i ∈ f−1(D1)}, . . . , {xi | i ∈ f−1(Dk)} and the series of sets {xi | i ∈g−1(D′1)}, . . . , {xi | i ∈ g−1(D′k)} such that the respective sets are identical.3bxc stands for the largest integer not greater than x ∈ R and dxe stands for the smallest integernot less than x ∈ R.2Clearly, by defining equivalent districtings we have defined an equivalence relationabove the set of districtings for problem Π.A districting f and voters’ preferences v determine the number of districts won byparties A and B, which we denote by F (f, v, A) and F (f, v,B), respectively. If the twoparties should receive the same number of votes in a district, its winner is determinedby a predefined tie-breaking rule τ : D → {A,B}.Definition 4. For a given problem Π and tie-breaking rule τ a districting f : N → Dis optimal for party I ∈ {A,B} if F (f, v, I) ≥ F (g, v, I) for any districting g : N → D.Note that due to the above defined equivalence relation the set of districtings hasfinitely many equivalence classes, there exists at least one optimal districting for eachparty.3 Determining an optimal districting is NP-completeWe establish that even the decision problem associated with the optimization problemof determining an optimal partisan districting, i.e. deciding for a given districtingproblem Π whether there exists a districting with at least m winning districts fora party, say party A, is an NP-complete problem; we call this problem WINNINGDISTRICTS. In order to prove this, we shall reduce the INDEPENDENT SET problemon planar cubic4 graphs, a proven NP-complete problem (see Garey and Johnson; 1979,pp. 195), to WINNING DISTRICTS. The INDEPENDENT SET problem asks whethera given graph G has a set of non-neighboring vertices of cardinality not less than m.Theorem 1. WINNING DISTRICTS is NP-complete.Proof. Whether a districting possesses at least m winning districts for party A can beverified easily in polynomial time, and therefore WINNING DISTRICTS is in NP.We establish that INDEPENDENT SET on planar cubic graphs reduces to WIN-NING DISTRICTS. We define the mapping that assigns to an arbitrary planar cubicgraph a districting problem in two steps.Step 1: We start with constructing party A winning districts from a planar cubicgraph G = (V,E). Let each vertex be a party A voter and replace the ‘midpoint’ ofeach edge with a party A voter. In addition, we associate with each vertex v ∈ V afour member district containing the party A voter assigned to vertex v and the threeparty A voters replacing the three edges adjacent to vertex v. Hence, so far we have|V | + |E| = 5|V |/2 party A voters and |V | districts, where each of them is consistingof four party A voters. Step 1 is illustrated in Figure 1.Observe that the given planar cubic graph has an independent set of size m if andonly if we can select m disjoint districts from the districts drawn in Step 1. However,as the right-hand side of Figure 1 shows, based on the districts drawn so far, we cannotpartition the set of voters even if we extend the boundaries of the districts5 in a way thatthe set of contained voters remains the same. Hence, we cannot obtain a districting.This is what Step 2 takes care about.Step 2: Now we associate with each district full of party A voters twelve new partyB voters such that the new voters have to be placed on the ‘same side’ of the districtas illustrated in Figure 2.4A graph is cubic if the degree of each vertex equals 3.5By extending districts appropriately we can assign any uninhabited area of a map to one district.3  @@  @@ r rrrrr     @@@@@     @@@@@r rrrrrr rrrrrr rrrrr r rr rr rr rrrrrr rrrrr@@@@@@      @@@         @@@@@@@@@   @@@@    @      @@@@@@  Figure 1: Associating party A winning districts with a planar cubic graphw w w wffififlg g g gg g g gg g g gFigure 2: Party B votersIn addition, we have to form new districts distinguishing between the two caseswhether a district full of party A voters will be included in our districting. First, if adistrict full of party A voters is not selected by a districting, then the respective party Bvoters should be grouped into party B winning districts as shown in Figure 3. We willrefer to these ‘vertical districts’ as type 1 party B winning districts. However, we havew w w wffififlg g g gg g g gg g g g				Figure 3: Type 1 party B winning districtsto be more careful since each of the three party A voters corresponding to an edge ofour initial planar cubic graph G = (V,E) is even a member of another party A winningdistrict full of party A voters. Therefore, if none of these two districts containing thesame party A voter is contained in a districting, then this specific party A voter canbe only included in one of these two type 1 districts. Hence, to make a districtingpossible we also include three type 2 ‘vertical districts’ associated with each party Avoter corresponding to an edge. Type 2 party B winning districts are illustrated in4Figure 4 in which the second party A voter from the left is a voter corresponding to avertex of our initial planar cubic graph.w w w wffififlg g g gg g g gg g g gFigure 4: Type 2 party B winning districtsSecond, if a district full of party A voters is selected by our districting, then the‘horizontal districts’ illustrated in Figure 5 make a districting possible.w w w wffififlg g g gg g g gg g g g   Figure 5: Type 3 party B winning districtsObserve that the districts introduced in Step 2 make a districting possible and thatthe given planar cubic graph has at least an independent set of size m if and only ifthe associated districting problem has at least m party A winning districts.It remains to be shown that given a planar cubic graph G = (V,E) its associateddistricting problem outlined in Steps 1 and 2 can be determined in polynomial time.Constructing a straight line planar drawing of G with edges of at most five differentslopes (see Keszeg et al. 2008) or a planar embedding in the grid of G (as shown byLiu et al. 1994), which can be obtained in polynomial time, we can easily locate thevoters described in Steps 1 and 2 in the plane such that the respective districts can bedrawn in polynomial time.Remark 1. Considering our reduction, we can observe that the approximability ofWINNING DISTRICTS cannot be better than that of INDEPENDENT SET on pla-nar cubic graphs for which Burns (1989) showed that the approximation ratio of thepolynomial time algorithm given by Choukhmane and Franco (1986) equals 7/8.5References[1] M. Altman, Is automation the answer? The computational complexity of auto-mated redistricting, Rutgers Computer and Law Technology Journal 23 (1997)81–142.[2] M. Altman, M. McDonald, The promise and perils of computers in redistricting,Duke Constitutional Law and Policy Journal 5 (2010) 69–112.[3] F. Bac¸a˜o, V. Lobo, M. Painho, Applying genetic algorithms to zone design, SoftComputing 9 (2005) 341–348.[4] B. Bozkaya, E. Erkut, G. Laporte, A tabu search heuristic and adaptive memoryprocedure for political districting, European Journal of Operational Research 144(2003) 12–26.[5] J.E. 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Optimal partisan districting on planar geographies

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