Extended formulations for the cardinality constrained subtree of a tree problem
Introduction
We consider an undirected tree rooted at node 0 with node set and edge set , a profit associated with each node and a positive integer . We define the profit of a subtree as the sum of the profits of the nodes included in it. The Cardinality Constrained Subtree of a Tree Problem (CSTP) is to find, in T, a subtree rooted at node 0 with maximum profit and such that . When the cardinality constraint becomes redundant and we obtain the (unconstrained) subtree of a tree problem which is studied in [1]. We assume w.l.o.g.: (i) and (ii) the unique path from node 0 to each node has length at most .
Let denote the predecessor of node in and, for all , consider the binary variables indicating whether node is in the solution. The CSTP can be formulated as the following integer linear programming problem. Constraint (1.1) ensures that the root node is selected. This condition is assumed w.l.o.g. to simplify the formulations presented in Section 3. Constraints (1.2) establish that a node is in the solution only if its predecessor is in the solution. Constraint (1.3) states that the number of nodes in the solution is at most . And constraints (1.4) are the variables integrality constraints.
This problem is a particular case of the constrained subtree of a tree problem (see, e.g., [2], [3], [4]), where (1.3) is replaced by the knapsack constraint where . For that reason the CSTP can be solved in polynomial time by dynamic programming (see, e.g., [5], [2], [3]).
The CSTP was previously studied in [1], [5]. Aghezzaf et al. [5] have given a description of co(CSTP) (where co(P) gives the convex hull of the integer solutions of problem P) for and 4. Noticing that the difficulty of finding a complete description of the CSTP polytopes seems to increase with and by showing how to reduce the knapsack problem to a special case of the CSTP, Aghezzaf et al. [5] observe that “this close tie between the knapsack and cardinality constrained problems suggests that the polyhedral structure of conv (co(CSTP)) is unlikely to have a simple form”. This observation motivates the study of the so-called extended descriptions of co(CSTP). More precisely, the study of formulations that use additional sets of variables and such that the projection of the feasible set of the corresponding linear programming relaxation into the space of the variables equals co(CSTP). In this paper we provide one such extended formulation. The underlying idea for deriving the formulation given in this paper is based on the following two observations: (i) our strong conviction that “good” variables to be introduced in such an extended formulation should have information on an “ordering” of the nodes in the subtree solution and (ii) the knowledge that this additional information should be given as an extra index of the original 0/1 variables. The second observation is based on the knowledge taken from other works, where similar re-indexations have been used (see, e.g., [6], [7]), to construct models with a strong linear programming relaxation. With the new set of variables, we will be able to obtain a compact and extended formulation for the CSTP which has variables and constraints and such that the projection of the linear programming feasible set of the formulation is equal to co(CSTP) for all values of . To show this result we give an alternate characterization of the problem (namely that it can be modeled as a constrained shortest path problem) which permits us to provide a different extended formulation (although less compact than the previous one) for the problem and whose linear programming relaxation is easily proven to be tight. Since this new formulation is defined in a space strictly containing the variable set of the ordering formulation, we, then, use projection to show that the set of feasible solutions of the linear programming relaxation of the new formulation can be projected into the set of feasible solutions of the linear programming relaxation of the more compact “ordering” formulation.
The paper is organized as follows. In Section 2, we introduce “ordering” formulations. In Section 3, we relate the CSTP with the cardinality constrained shortest path problem and as a consequence, produce the new reformulation for the CSTP whose linear programming relaxation is easily proven to be tight (i.e., the extreme points are integer valued). In Section 4 we use projection techniques to show that the set of feasible solutions of the linear programming relaxation of the new formulation can be projected into the set of feasible solutions of the linear programming relaxation of the “ordering” formulation.
In the paper we denote by the set of feasible solutions of a given model with optimal value and denote by its Linear Programming relaxation.
Section snippets
A reformulation with ordering variables
In this section we introduce the concept of “ordering” of a node in a subtree solution, introduce new variables which also give information on the ordering of the corresponding node in the subtree and derive a compact formulation with variables and constraints.
We assume that the nodes of the rooted tree are numbered by the Depth First Ordering (DFO), from left to right, once a planar representation of with 0 as a root has been fixed. DFO numbers a node as far as possible along
The CSTP and the cardinality constrained shortest path problem
In this section we show that the CSTP can be modeled as a cardinality constrained shortest path problem in an adequate graph. Then, by using [8] we obtain a formulation whose linear programming relaxation is integer.
To show that the CSTP can be modeled as a cardinality constrained shortest path problem, consider a digraph where is a destination node (dummy node) and is the set of arcs. The set A contains an arc , for all if and only if node can be
The proof of Theorem 2.4
To prove that the extreme points of the linear programming relaxation of SO-CST are integral we will show that the set of feasible solutions of the linear programming relaxation of SO-CST is equal to the projection, via a suitable linear transformation, of the polyhedron SPL into the space defined by the variables.
The variables of the order formulation SO-CST can be related with the variables of the shortest path formulation SPL as follows:
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The Minimal Number of Subtrees with a Given Degree Sequence
2015, Graphs and Combinatorics
- 1
Tel.: +351 234 372551; fax: +351 234 370 066.
- 2
Research partially funded by CEOC from the FCT (Fundação para a Ciência e a Tecnologia) cofinanced by the European Community Fund FEDER/POCI 2010.
- 3
Research partially funded by research project POCTI-ISFL-1-152. Tel.: +351 21 750 0409.