The.Algorithm.Design.Manual.Springer-Verlag.1998

Set Cover
There are several variations of set cover problems to be aware of:
● Are you allowed to cover any element more than once? - The distinction here is between set
covering and set packing, the latter of which is discussed in Section . If we are allowed to
cover elements more than once, as in the logic minimization problem above, we should take
advantage of this freedom, as it usually results in a smaller covering.
● Are your sets derived from the edges or vertices of a graph? - Set cover is a very general problem,
and it includes several useful graph problems as special cases. Suppose instead that you seek the
smallest set of edges in a graph that covers each vertex at least once. The solution is to find a
maximum matching in the graph (see Section ), and then add arbitrary edges to cover the
remaining vertices. Suppose you seek the smallest set of vertices in a graph that covers each edge
at least once. This is the vertex cover problem, discussed in Section .
It is instructive to model vertex cover as an instance of set cover. Let the universal set U be the set
of edges . Construct n subsets, with consisting of the edges incident on vertex .
Although vertex cover is just a set cover problem in disguise, you should take advantage of the
fact that better algorithms exist for vertex cover.
● Do your subsets contain only two elements each? - When all of your subsets have at most two
elements each, you are in luck. This is about the only special case that you can solve efficiently to
optimality, by using the matching technique described above. Unfortunately, as soon as your
subsets get to have three elements each, the problem becomes NP-complete.
● Do you want to cover elements with sets, or sets with elements? - In the hitting set problem, we
seek a small number of items that together represent an entire population. Formally, we are given
a set of subsets of the universal set , and we seek the smallest
subset such that each subset contains at least one element of T. Thus for all
. Suppose we desire a small Congress with at least one representative of each ethnic
group. If each ethnic group is represented as a subset of people, the minimum hitting set is the
smallest possible politically correct Congress. Hitting set is illustrated in Figure .
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