1 Introduction
The graphs considered here are simple, connected and undirected. If is a graph, then denotes its vertex set and denotes its edge set. Later, for some formulations, we shall refer to directed graphs (or simply, digraphs) , with vertex set and arc set . For an integer , as usual, the symbol denotes the set . Throughout this text, denotes a positive integer number.
Let be a connected graph. A partition of is a collection of nonempty subsets of such that , and for all , . We refer to each set as a class of the partition. In this context, we assume that , otherwise does not admit a partition. We say that a partition of is connected if , the subgraph of induced by , is connected for each . Let be a function that assigns weights to the vertices of . For every subset , we define . To simplify notation, we write when is a graph. In the balanced connected partition problem (), we are given a vertexweighted connected graph, and we seek a connected partition such that the weight of a lightest class of this partition is maximized. A more formal definition of this problem is given below, as well as an example of an instance for (see Figure 1). For some fixed positive integer , each input of is given by a pair . We denote by the weight of a lightest set in an optimal connected partition of ; but write simply when is clear from the context. Furthermore, we simply denote by the instance in which is an assignment of a constant value to all vertices (which we may assume to be ).
Problem 1.
Balanced Connected Partition ()
Instance: a connected graph and a vertexweight function .
Find: a connected partition of .
Goal: maximize .
There are several problems in police patrolling, image processing, data base, operating systems, cluster analysis, education and robotics that can be modeled as a balanced connected partition problem [2, 14, 15, 16, 17, 22]. These different realworld applications indicate the importance of designing algorithms for , and reporting on the computational experiments with their implementations. Not less important are the theoretical studies of the rich and diverse mathematical formulations and the polyhedral investigations leads to.
Let us denote by the restricted version of in which all vertices have unit weight. One may easily check that on 2connected graphs can be solved in polynomial time. This problem also admits polynomialtime algorithms on graphs such that each block has at most two articulation vertices [1, 5, 10, 11]. Dyer and Frieze [8] showed that, for every , is hard on bipartite graphs. Wu [21] proved that is hard even on interval graphs, for all .
For the (weighted version), Chlebíková [5] showed that this problem is hard to approximate within an absolute error guarantee of , for all . In that same paper, Chlebíková designed a approximation algorithm for that problem. For and on connected and connected graphs, respectively, there exist approximation algorithms proposed by Chataigner et al. [4].
Wu [21] showed the first pseudopolynomial algorithm for restricted to interval graphs. Based on this algorithm and using a scaling technique, Wu obtained a approximation with running time , where is the number of vertices of the input graph.
Borndörfer et al. [3] designed a approximation for , where is the maximum degree of an arbitrary spanning tree of the given graph. This is the first known approximation algorithm on general graphs.
Mixed integer linear programming (MILP) formulations were proposed for by Matic [18] and for by Zhou et al. [22]
. Additionally, Matic presented an heuristic algorithm based on a variable neighborhood search (VNS) technique for
, and Zhou et al. devised a genetic algorithm for
. In both works, the authors showed results of computational experiments to illustrate the quality of the solutions constructed by the proposed heuristics and their running times compared to the exact algorithms based on the MILP formulations. No polyhedral study of their formulations was reported.This paper is organized as follows. In Section 2, we present a natural cut formulation for and also two stronger valid inequalities for this formulation. A further polyhedral study of this formulation, when all vertices have unit weight, is presented in Section 3. Two of the inequalities in the formulation are shown to define facets, and one of them is characterized when it is facetdefining. In Section 4, we present a flow and a multicommodity flow based formulations for . In Section 5, we report on the computational experiments with our formulations and also with those presented by Matic [18] and Zhou et al. [22]. We summarize our theoretical and practical contributions for in Section 6.
2 Cut formulation
In this section, the following concept will be useful. Let and be two nonadjacent vertices in a graph . We say that a set is a separator if and belong to different components of . We define as the collection of all minimal separators in .
Let be an input for . We propose the following natural integer linear programming formulation for . For that, for every vertex and , we define a binary variable representing that belongs to the th class if and only if . In this formulation, to get hold of a class with the smallest weight, we impose an ordering of the classes, according to their weights, so that the first class becomes the one whose weight we want to minimize.
s.t.  (1)  
(2)  
(3)  
(4) 
Inequalities (1) imply a nondecreasing weight ordering of the classes. Inequalities (2) impose that every vertex is assigned to at most one class. Inequalities (3) guarantee that every class induces a connected subgraph. The objective function maximizes the weight of the first class. Thus, in an optimal solution no class will be empty, and therefore it will always correspond to a connected partition of .
We observe that the separation problem associated with inequalities (3) can be solved in polynomial time by reducing it to the minimum cut problem. Thus, the linear relaxation of can be solved in polynomial time because of the equivalence of separation and optimization (see [12]).
Since the feasible solutions of the formulation above may have empty classes, to refer to these solutions we introduce the following concept. A connected subpartition of is a connected partition of a subgraph (not necessarily proper) of . Henceforth, we assume that if is a connected subpartition of , then for all . For such a subpartition , we denote by
the binary vector such that its nonnull entries are precisely
for all and (that is, denotes the incidence vector of ).We next show that the previous formulation correctly models . For that, let be the polytope associated with that formulation, that is,
In the next proposition we show that is the convex hull of the incidence vectors of connected subpartitions of .
Proposition 1.
Let be an input for . Then, the following holds.
Proof.
Consider first an extreme point . For each , we define the set of vertices . It follows from inequalities (1) and (2) that is a subpartition of such that for all .
To prove that is a connected subpartition, we suppose to the contrary that there exists such that is not connected. Hence, there exist vertices and belonging to two different components of . Moreover, there is a minimal set of vertices that separates and and such that . This implies that , a contradiction to the fact that satisfies inequalities (3). Therefore, is a connected subpartition of .
To show the converse, consider now a connected subpartition of . By the definition of , it is clear that this vector satisfies inequalities (1) and (2). Take a fixed . For every pair , of nonadjacent vertices in , and every separator in , it holds that , because is connected. Therefore, satisfies inequalities (3). Consequently, belongs to . ∎
In the remainder of this section we present two further classes of valid inequalities for that strenghten the formulation . We start showing a class that dominates the class of inequalities (3).
Proposition 2.
Let be an input for . Let and be two nonadjacent vertices of , and let be a minimal separator. Let , where is a minimumweight path between and in that contains . For every , the following inequality is valid for :
(5) 
Proof.
Inspired by the inequalities devised by de Aragão and Uchoa [6] for a connected assignment problem, we propose the following class of inequalities for .
Proposition 3.
Let be an input for , and be an integer. Let be a subset of , the set of neighbors of , and a subset of containing distinct pair of vertices , , . Moreover, let be an injective function, and let denote the image of , that is, . If there is no collection of vertexdisjoint paths in , then the following inequality is valid for :
(6) 
Proof.
Suppose, to the contrary, that there exists an extreme point of that violates inequality (6). Let and . From inequalities (2), we have that . Since violates (6), it follows that . Thus (because satisfies inequalities (2)). Hence, every vertex in belongs to a class that is different from those indexed by . This implies that every class indexed by contains precisely one of the distinct pairs . Therefore, there exists a collection of vertexdisjoint paths in , a contradiction. ∎
3 Polyhedral results for
In this section we focus on , the special case of in which all vertices have unit weight. In this case, instead of , we simply write , the polytope defined as the convex hull of .
Note that, if the input graph has no matching of size , then has no feasible connected subpartition such that for all , and thus , and it is easy to find an optimal solution. Thus, we assume from now on that has a matching of size (a property that can be checked in polynomial time [9]), and that , where .
For each and we shall construct a binary vector that belongs to . Let us denote by the unit vector such that its single nonnull entry is indexed by and . Now consider any set , , and a bijective function . Since , such a set exists. Fix a pair , where and are as previously defined. Let be the vector . Note that belongs to , it is the incidence vector of a subpartition, say of , in which belongs to the class , and each vertex of belongs to one of the classes , all of which are singletons.
To be rigorous, we should write as different choices of and give rise to different vectors, but we simply write with the understanding that it refers to some and bijection .
Proposition 4.
is fulldimensional, that is, .
Proof.
Let be the set of vectors previously defined, that is, . Assume that . We suppose, with no loss of generality, that the indices of a vector in are ordered as .
Let be the matrix whose columns are precisely the vectors in (in the following order): . One may easily check that is a lower triangular square matrix of dimension . Note that the columns of are precisely the vectors in . Hence the vectors in are linearly independent. Since , we conclude that , that is, is fulldimensional. ∎
In the forthcoming proofs, we have to refer to some connected subpartitions of , defined (not uniquely) in terms of distinct vertices , of , and specific classes , , where . For that, we define a short notation to represent the incidence vectors of these connected subpartitions. Given such , , and , , choose two set of vertices and in , both of cardinality , and bijections and such that

and ;

and .
Let and be vectors in such that their nonnull entries are precisely: for every , and for every . The vectors and clearly belong to . Moreover, note that and for all .
Proposition 5.
For every and , the inequality induces a facet of .
Proof.
Similarly to the proof of Proposition 4, let . Additionally, we define . Note that . Since the null vector and all vectors in are all affinely independent, and they all belong to the face , we conclude that the inequality induces a facet of . ∎
In what follows, considering that the polytope is fulldimensional, to prove that a face is a facet of , we show that if a nontrivial face of contains , then there exists a real positive constant such that and .
Proposition 6.
For every , the inequality induces a facet of .
Proof.
Fix a vertex . Let , where corresponds to . Let be a nontrivial face of such that . We shall prove that and for every and .
Since is nontrivial and connected, it is easy to see that has a set of nested connected subgraphs such that consists solely of the vertex , each for , and . (It suffices to consider a spanning tree in , and starting from , define the subsequent subgraphs by adding at each step an appropriate edge and vertex from this spanning tree.)
Consider the set of vectors , where for every . Since for all , it follows that . As a consequence, for all . Additionally, since .
Let . Suppose that and for every and . Now define the set of vectors . Note that , since belongs to exactly one class of the partition corresponding to each vector in . By the induction hypothesis, we obtain for each . Moreover, observe that belongs to . It follows from the induction hypothesis that .
Therefore, we conclude that . Since (otherwise would be a trivial face), it follows that is a multiple scalar of , and therefore is a facet of . ∎
Let and be two nonadjacent vertices of and let be a minimal separator in . We denote by and the components of which contain and , respectively. Since is minimal, it follows that every vertex in has at least one neighbor in and one in .
In this context of minimal separator , for every , the following two concepts (and notation) will be important for the next result.
We denote by a minimum size connected subgraph of containing , with the following properties: If , then is contained in ; if , then is contained in . Otherwise, contains , and exactly one vertex of , that is, . Clearly, such a subgraph always exists. Moreover, if , then the subgraph is connected (that is, is not a cutvertex of ).
For any integer , we say that admits a robust subpartition if, for each , there is a connected subpartition of such that for all .
Theorem 7.
Let and be nonadjacent vertices in , let be a minimal separator, and let . Then admits a robust subpartition if and only if the following inequality induces a facet of :
4 Flow formulations
We present in this section a mixed integer linear programming formulation for based on flows in a digraph. For that, given an input for , we construct a digraph as follows: First, we add to a set of new vertices. Then, we replace every edge of with two arcs with the same endpoints and opposite directions. Finally, we add an arc from each vertex in to each vertex of (see Figure 2). More formally, the vertex set of is and its arc set is
Now, the idea behind the formulation is the following: find in a maximum flow from the sources in such that every vertex in receives flow only from a single vertex of and consumes of the received flow. As we shall see, for every , the flow sent from source corresponds to the total weight of the vertices in the th class of the desired partition.
To model this concept, with each arc , we associate a nonnegative real variable that represents the amount of flow passing through , and a binary variable that equals one if and only if arc is used to transport a positive flow. The corresponding formulation, shown below, is denoted .
s.t.  (7)  
(8)  
(9)  
(10)  
(11)  
(12)  
(13) 
Inequalities (10) impose that from every source at most one arc leaving it transports a positive flow to a single vertex in . Inequalities (11) require that every nonsource vertex receives a positive flow from at most one vertex of . By inequalities (9), a positive flow can only pass through arcs that are chosen (arcs for which ). Inequalities (8) guarantee that each vertex consumes of the flow that it receives. Finally, inequalities (7) impose that the amount of flow sent by the sources are in a nondecreasing order. This explains the objective function.
Since each nonsource vertex receives flow from at most one vertex, the flows sent by any two distinct sources do not pass through a same vertex. That is, if a source sends an amount of flow, say , this amount is distributed to a subset of vertices, say (with total weight ); and all subsets are mutually disjoint. Moreover, is exactly the sum of the weights of the vertices that receive flow from , and is a connected subgraph of . (See Figure 2.) It follows from these remarks that formulation correctly models .
The proposed formulation has a total of variables (half of them binary), and only constraints, where and . The possible drawbacks of this formulation are the large amount of symmetric solutions and the dependency of inequalities (9) on the weights assigned to the vertices. To overcome such disadvantages, we propose in the next section another model based on flows that considers a total order of the vertices to avoid symmetries and uncouple the weights assigned to the vertices from the flow circulating in the digraph.
4.1 A second flow formulation
Our second formulation for , denoted by , is also based on a digraph that is constructed from as follows. It has vertex set and arc set
Moreover, it assumes that there is a total ordering defined on the vertices of . For simplicity, for a vertex and integer , we use the short notation instead of .
s.t.  (14)  
(15)  
(16)  
(17)  
(18)  
(19)  
(20)  
(21)  
(22) 
To show that the above formulation indeed models , let us consider the following polytope:
Let be a connected subpartition of such that for all . Then, for each integer , there exists in an outarborescence rooted at such that and for all . Now, let be the function defined as follows: if is a leaf of , and , otherwise. It follows from this definition that .
We now define vectors and such that, for every arc and , we have
We are now ready to prove the claimed statement on ,
Proposition 8.
The polytope is precisely the polytope
Proof.
Let be an extreme point of ; and for every , let . It follows from inequalities (16) that, for every vertex , at most one of the arcs entering it is chosen. Observe that inequalities (18) force that a flow of type can only pass through an arc of type if this arc is chosen. Hence, every vertex receives at most one type of flow from its inneighbors. Furthermore, inequalities (19) and (20) guarantee that the flow that enters a vertex and leaves it are of the same type, and that each vertex consumes exactly one unit of such flow.
Inequalities (15) imply that all flow of a given type passes through at most one arc that has tail at the source . Therefore, we have that is a connected partition of .
To prove the converse, let be a connected partition of . We assume without loss of generality that for all . Let be a vector such that and . For each , every vertex in has indegree at most one, and is the smallest vertex in with respect to the order . Thus, inequalities (16) and (17) hold for . From the definition of , the entry of indexed by and equals one, for all . Consequently,
Comments
There are no comments yet.