IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 45
that no two of which are adjacent. The above problem fits this interpretation since the set O can
be thought of as the edges of a graph and D as the nodes of the graph.
The set packing problem is formulated as follows: there is a universal set J = {1, ..., n} and subsets
S i ⊂ J, i = 1, ..., m. The task is to find a collection of subsets C = {S i } i∈{1,...,m} , such that
S i1 ∩ S i2 = ∅, S i1 , S i2 ∈ C. That is, find a maximum number of subsets such that no two of these
subsets have an item in common.
The optimization version of the problem seeks the largest number of subsets such that the aforementioned
condition is true, and there is also weighted version of the problem. Notice that our
decision variables in the optimization version of the problem are the subsets S i which we wish to
include.
To see that independent set is a set packing problem consider a graph G = (V, E) for which we
want to determine an independent set. The decision variables in the optimization version of the
independent set problem are the vertices which we wish to include in the independent set. Let us
consider a set packing problem in which the universal set consists of all the edges of the graph G,
i.e. the universal set is the set E. For each vertex i ∈ V of the graph, let us form a subset S i
which consists of all edges that are adjacent to node i, i.e. let S i = {[i, j]|[i, j] ∈ E} for all i ∈ V .
As we mentioned before, the decision variables in the set packing problem are the subsets which
we want to include, and since each set corresponds to a vertex of the graph of the independent set
problem, each choice of subset S i in the set packing problem corresponds to a choice of vertex i for
the independent set problem. We are seeking to choose the maximum number of subsets S i such
that no two overlap, i.e. the maximum number of vertices such that no two share any edges, which
is the independent set problem.
For the formulation of the set packing optimization problem we define a binary variable x i which
indicates if set S i is selected. For each subset S i we also define J i , the collection of subsets S j , j ≠ i,
that S i shares an element with. That is, if S i is chosen then none of the other subsets S j ∈ J i ,
j ≠ i, can be selected. We get the following problem:
max ∑ m
i=1 x i
s.t.
∑
x i ≤ 1
i∈J i
x i ∈ {0, 1}
i = 1, ..., m
Set packing appears in various contexts. Examples include putting items in storage (packing a
truck), manufacturing (number of car doors in a metal sheet, rolls of paper that can be cut from a
giant roll), the fire station problem and the forestry problem.
20.2 Branch and Cut
As we mentioned in the introduction of this section, cutting planes are useful in getting better
bounds to an integer problem. This is especially useful for branch and bound algorithms since
more nodes of the branch and bound tree can be fathomed, and so that the fathoming can be done
earlier in the process. Recall that we can eliminate node j of the tree if the LP bound for node
j is no more than the value of the incumbent. Hence, the lower the bound of a node that we can
produce the better, and cutting planes are useful in lowering these bounds.
For cutting planes, as for any other known technique for solving general IPs, we can come up with
an instance of a problem which takes an exponential amount of time to solve. The tradeoff of using
cutting planes is that although we can get better bounds at each node of the branch and bound
tree, we end up spending more time in each node of the tree.