Prove that independent set is NP-compete by proving that 3-COL ≤ poly Independent set. (Hint: A 3-coloring for the graph G3-COL can be thought of as a subset of the pairs u, c where u is a node of G3-COL and c is a color. An independent set of the graph GInd selects a subset of its nodes. Hence, a Optimization Problems 342 way to construct the graph Gind in the instance GInd, NInd = InstanceMap(G3-COL) would be to have a node for each pair u, c. Be careful when defining the edges for the graph GInd = InstanceMap(G3-COL) so that each valid independent set of size n in the constructed graph corresponds to a valid 3-coloring of the original graph. If the constructed graph has unexpected independent sets, you may need to add more edges to it.)

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Polynomial Time

The problems for which the deterministic polynomial-time algorithm exists belong to the complexity class (p) and it is central in the field of computational complexity theory. According to Cobham’s thesis, polynomial time is a synonym for “feasible”, “fast”, and “efficient”.