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

Vertex Cover
Vertex cover and independent set are very closely related graph problems. Since every edge in E is (by
definition) incident on a vertex in a cover S, there can be no edge for which both endpoints are not in S.
Thus V-S must be an independent set. Further, since minimizing S is the same as maximizing V-S, a
minimum vertex cover defines a maximum independent set, and vice versa. This equivalence means that
if you have a program that solves independent set, you can use it on your vertex cover problem. Having
two ways of looking at it can be helpful if you expect that either the cover or independent set is likely to
contain only a few vertices, for it might pay to search all possible pairs or triples of vertices if you think
that it will pay off.
The simplest heuristic for vertex cover selects the vertex with highest degree, adds it to the cover, deletes
all adjacent edges, and then repeats until the graph is empty. With the right data structures, this can be
done in linear time, and the cover you get ``usually'' should be ``pretty good''. However, for certain input
graphs the resulting cover can be times worse than the optimal cover.
Much better is the following approximation algorithm, discussed in Section , which always finds a
vertex cover whose size is at most twice as large as optimal. Find a maximal matching in the graph, i.e.
a set of edges no two of which share a vertex in common and that cannot be made larger by adding
additional edges. Such a maximal matching can be built incrementally, by picking an arbitrary edge e in
the graph, deleting any edge sharing a vertex with e, and repeating until the graph is out of edges. Taking
both of the vertices for each edge in the matching gives us a vertex cover, since we only delete edges
incident to one of these vertices and eventually delete all the edges. Because any vertex cover must
contain at least one of the two vertices in each matching edge just to cover the matching, this cover must
be at most twice as large as that of the minimum cover.
This heuristic can be tweaked to make it perform somewhat better in practice, without losing the
performance guarantee or costing too much extra time. We can select the matching edges so as to ``kill
off'' as many edges as possible, which should reduce the size of the maximal matching and hence the
number of pairs of vertices in the vertex cover. Also, some vertices selected for our cover may in fact not
be necessary, since all of their incident edges could also have been covered using other selected vertices.
By making a second pass through our cover, we can identify and delete these losers. If we are really
lucky, we might halve the size of our cover using these techniques.
A problem that might seem closely related to vertex cover is edge cover, which seeks the smallest set of
edges such that each vertex is included in one of the edges. In fact, edge cover can be efficiently solved
by finding a maximum cardinality matching in G (see Section ) and then selecting arbitrary edges to
account for the unmatched vertices.
Implementations: Programs for the closely related problems of finding cliques and independent sets
were sought for the Second DIMACS Implementation Challenge, held in October 1993. See Section
for details.
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