week-14 - The Set Cover Problem In the U.S. navy, the SEALS...

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1 The Set Cover Problem In the U.S. navy, the SEALS are each specially trained in a wide variety of skills so that small teams can handle a multitude of missions. If there are k different skills needed for a mission, and n SEAL members that can be assigned to the team, find the smallest team that will cover all of the required skills. Andersen knows hand-to-hand , first aid , and camouflage Butler knows hand-to-hand and snares Cunningham knows hand-to-hand Douglas knows hand-to-hand , sniping , diplomacy , and snares Eckers knows first-aid , sniping , and diplomacy Minimum Set Cover Problem: Given a set S of subsets { S 1 , S 2 , …, S m } out of a universal set U = { u 1 , u 2 , …, u n } and an integer k , is it possible to choose only k subsets of S such that the union of these subsets is U . Theorem: Minimum Set Cover is NP-complete. Proof: MSC is in NP - given a subset of sets, we can count them, and show that all elements of U are included. What problem should we choose to reduce this time? Hamiltonian Cycle Problem : Given a graph G, does it contain a cycle that includes all of the vertices in G? Theorem: Hamiltonian Cycle is NP-complete. Proof: Hamiltonian cycle is in NP - given an ordering on the vertices, we can show that and edge connecting each consecutive pair, and then the final vertex connecting back to the first We now have some graph problems to work with, but how can they really help us with this problem?
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2 Reducing to Hamiltonian Cycle In Out Vertex Cover Hamiltonian Cycle Reduction Return Conversion Warning: This transformation will be tricky…. The Reduction For every edge in the Minimum Vertex Cover problem, we must reduce it to a “contraption” in the Hamiltonian Cycle Problem: v u u u v v Observations…. u u v v u u v v u u v v There are only three possible ways that a cycle can include all of the vertices in this contraption.
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3 u u u u Joining Contraptions u v v w w u x x All components that represent edges connected to u are strung together into a chain. If there are
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week-14 - The Set Cover Problem In the U.S. navy, the SEALS...

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