An NP-Complete problem can be checked efficiently, but has no known way of being solved in polynomial time.
Then, how is the graph coloring problem (http://en.wikipedia.org/wiki/Graph_coloring_problem) NP-Complete? How does one easily check it?
2026-03-28 03:32:54.1774668774
How is the graph coloring problem NP-Complete?
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For a check, you are given with a particular coloring of the given map. You just go through all the patches, check that the neighbors are of different color, and finally count the total number of colors. This algorithm scales linearly with the number of regions, so it is a polynomial check.
UPDATE: For a general graph (not necessarily planar) this algorithm will be at most quadratic in the number of vertices (colored regions).