Further Instructions: You must prove that the schedule produced by your algorithm has an average completion time that is lower than or equal to the average completion time of any other schedules. Appeals to general principle (e.g. "It is always better to ...") are not acceptable proofs because they simply restate what needs to be proven.
Further Instructions: follow guidelines for proof of optimality given above.
Questions:
Show how you can use this black box to find the vertices of a maximum clique in a graph in polynomial time.
DOM = { (G, k) } | G has a dominating set with k or fewer vertices. }Show that DOM is NP-complete. Hint: reduce from Vertex Cover.
Show how you can use this black box to find a 3-coloring of G in polynomial time. (I.e., you must be able to assign red, green or blue to each vertex of the graph such that no two adjacent vertices have the same color.)
Construct a polynomial-time ≤_{m}-reduction from the Hamiltonian Cycle problem to HAM_PATH.
Note: a graph can have an exponential number of cycles. So, when you show that X ∈ NP, you cannot simply guess E' and check every cycle in G. That would take exponential time.