CS6515 EXAM 3 (ACTUAL EX AM) WITH CORRECT 150+ QUESTIONS AND CORRECTLY WELL DEFINED ANSWERS LATEST 2024 ALREADY GRADED A+ LP Duality Formulation - ANSWERS -1. Add 1 variable for each constraint in primal - objective function is to minimize sum product of these constraints 2. Add 1 constraint for each variable in primal - constrained value comes from objective function - values to be constrained come from the constraints in the primal NP-Hard - ANSWERS -We are yet to find a polynomial time solution NP-Complete - ANSWERS -Both in NP and at least as hard as NP -Hard problems NP Reduction - Steps - ANSWERS -1. Demonstrate that problem B is in the class of NP Problems - validate a solution in polynomial time The Class P - ANSWERS -A solution may be found in polynomial time The Class NP - ANSWERS -A solution may be verified in polynomial time 2. Demonstrate that problem B is at least as ha rd as a problem believed to be NP -Complete. This is done via a reduction from a known problem A (A->B) a) Show how an instance of A is converted to B in polynomial time b) Show how a solution to B can be converted to a solution for A, again in polynomial time c) Show that a solution for B exists IF AND ONLY IF a solution to A exists - most prove both parts: if you you have a solution to B, you have a solution to A - If there is no solution to B, then no solution exists to A LP: Why optimum occurs at a v ertex - ANSWERS -Feasible region is convex LP: Optimum achieved at a vertex except: - ANSWERS -1. Infeasible - feasible region is empty 2. Unbounded - optimal value of objective function is arbitrarily large Independent Set -> Vertex Cover - ANSWERS -Lemma: I is an independent set of G(V, E) iff V - I is a vertex cover of G Simply check if there is a vertex cover of size V - b in G(V, E). If there is, output V - S 3SAT -> Independent Set - ANSWERS -Lemma: f is satisfiable iff the resulting set has an independent set of size m (# of clauses in f) in G(V, E). To construct G(V, E), create a node for each literal in each clause and connect them by an edge. Also connect any literal with it's negation. Independent Set -> Clique - ANSWERS -Lemma: G(V, E) has an independent set of size g iff G'(V, E) has a clique of size g. To construct G'(V, E), add all the vertices in G(V, E) to G'(V, E) and add edges to G'(V, E) if there is no edge in G(V, E) SAT - ANSWERS -Input: C is a CNF with any # of variables (n) and clauses (m) Output: assignment of each variable s.t. the CNF is True 3SAT - ANSWERS -Input: C is a CNF whose clauses have at most 3 literals Output: Assignment of eac h variable s.t. the CNF is True Clique - ANSWERS -Input: G is an undirected, unweighted graph, k is the size of the clique Output: A Clique in G with at least k vertices
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