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

Independent Set
● Vertex coloring - A coloring of a graph G is in fact a partitioning of the vertices of G into a small
number of independent sets, since any two vertices of the same color cannot have an edge
between them. In fact, most scheduling applications of independent set are really coloring
problems, since all tasks eventually must be completed.
● Clique - The complement of a graph G = (V,E) is a graph G' = (V,E'), where iff (i,j) is
not in E. In other words, we replace each edge by a nonedge and vica versa. The maximum
independent set in G is exactly the maximum clique in G', so these problems are essentially
identical. Algorithms and implementations in Section can thus be easily used for independent
set.
The simplest reasonable heuristic is to find the lowest-degree vertex, add it to the independent set, and
delete it and all vertices adjacent to it. Repeating this process until the graph is empty gives a maximal
independent set, in that it can't be made larger just by adding vertices. Using randomization or perhaps
some exhaustive search to distinguish among the low-degree vertices might result in somewhat larger
independent sets.
The independent set problem is in some sense dual to the graph matching problem. The former asks for
a large set of vertices with no edge in common, while the latter asks for a large set of edges with no
vertex in common. This suggests trying to rephrase your problem as a matching problem, which can be
computed in polynomial time, while the maximum independent set problem is NP-complete.
The maximum independent set of a tree can be found in linear time by (1) stripping off the leaf nodes,
(2) adding them to the independent set, (3) deleting the newly formed leaves, and then (4) repeating from
the first step on the resulting tree until it is empty.
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. Programs and
data from the challenge are available by anonymous ftp from dimacs.rutgers.edu. Source codes are
available under pub/challenge/graph and test data under pub/djs. See Section .
Combinatorica [Ski90] provides (slow) Mathematica implementations of cliques, independent sets, and
vertex covers. See Section for further information on Combinatorica.
Neural-network heuristics for vertex cover and related problems such as clique and vertex coloring have
been implemented in C and Mathematica by Laura Sanchis and Arun Jagota, and are available in the
algorithm repository http://www.cs.sunysb.edu/ algorith.
Notes: Independent set remains NP-complete for planar cubic graphs [GJ79]. However, it can be solved
efficiently for bipartite graphs [Law76].
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