Comparing hardness NP-complete problems are the “hardest” in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare …

NP-complete? An independent set in an undirected graph is a set of nodes that have no edges between them.

NP-complete problems are the hardest problems in NP, in the sense that they have a polynomial-time algorithm if and only if P =NP. Many natural problems that seemingly have nothing to do with Turing …

Now we are ready to introduce a formal definition for the NP class of problems.

Formal definitions: decision problems, P and NP. Examples of showing NP-completeness. Bipartite matching can be solved with a max flow algorithm. The max flow problem can be solved by a linear …

• NP-hard problems that are also in NP are NP- complete by definition • If you could reduce an NP problem to an NP-hard problem and then solve it in polynomial time, you could solve all NP problems …

For example, the SET COVER problem is NP-complete because it is a generaliza-tion of VERTEX COVER (and also, incidentally, of 3D MATCHING). See Exercise 8.10 for more examples.