The maximum independent set problem is the special case in which all weights are one.

The example that we will use to demonstrate Baker's technique is the maximum weight independent set problem.

One example is the maximum independent set problem:

The maximum independent set problem is NP-hard and it is also hard to approximate.

Clique (see also independent set problem)

Thus, in graphs of bounded treewidth, the maximum independent set problem may be solved in linear time.

Covering problems are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem.

Finding the largest edgeless induced subgraph, or independent set, called the independent set problem (NP-complete).

The problem of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem.

There is an algorithm for maximum weighted independent set problem and an algorithm for the maximum weighted clique problem.

In the maximum independent set problem, the input is an undirected graph, and the output is a maximum independent set in the graph.

For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time.

For example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs.

For the perfect graphs, a number of NP-complete optimization problems (graph coloring problem, maximum clique problem, and maximum independent set problem) are polynomially solvable.

The clique problem and the independent set problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa.

Because comparability graphs are perfect, many problems that are hard on more general classes of graphs, including graph coloring and the independent set problem, can be computed in polynomial time for comparability graphs.

Since maximum matchings may be found in polynomial time, so may the maximum independent sets of line graphs, despite the hardness of the maximum independent set problem for more general families of graphs.

To cite some of them: graph partitioning, multidimensional knapsack, travelling salesman problem, quadratic assignment problem, set cover problem, minimal graph coloring, max independent set problem, bin packing problem, and generalized assignment problem.

The maximum independent set problem may be solved using as a subroutine an algorithm for the maximal independent set listing problem, because the maximum independent set must be included among all the maximal independent sets.

Each interval represents a request for a resource for a specific period of time; the maximum weight independent set problem for the graph represents the problem of finding the best subset of requests that can be satisfied without conflicts.

However, the two problems have different properties when applied to restricted families of graphs; for instance, the clique problem may be solved in polynomial time for planar graphs while the independent set problem remains NP-hard on planar graphs.

These findings were able to explain the lack of progress that had been seen in the research community on the approximability of a number of optimization problems, including 3SAT, the Independent Set problem, and the Travelling Salesman Problem.

Nevertheless the maximum independent set problem can be solved more efficiently than the O(n 2) time that would be given by a naive brute force algorithm that examines every vertex subset and checks whether it is an independent set.

The maximum independent set problem and the maximum clique problem are polynomially equivalent: one can find a maximum independent set in a graph by finding a maximum clique in its complement graph, so many authors do not carefully distinguish between the two problems.

The NP-completeness of the clique problem follows trivially from the NP-completeness of the independent set problem, because there is a clique of size at least 'k' if and only if there is an Independent set (graph theory) of size at least 'k' in the complement graph.