1. Given an instance I of POM, the algorithm Greedy-POM erate any Pareto optimal matching in I. can gen-
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- True or False: - Best-first search is optimal in the case where we have a perfect heuristic (i.e., h(?) = h∗(?), the true cost to the closest goal state). - Suppose there is a unique optimal solution. Then, A* search with a perfect heuristic will never expand nodes that are not in the path of the optimal solution.- A* search with a heuristic which is admissible but not consistent is complete.Suppose we have a heuristic h that over-estimates h* by at most epsilon (i.e., for all n, 0<= h(n) <= h*(n)+epsilon). Show that A* search using h will get a goal whose cost is guaranteed to be at most epsilon more than that of the optimal goal.Given the figure below, using the Greedy -Best First search find if it is optimal / admissible ornot? Explain why ? I want it written by hand
- consider the problem of finding a minimum Pareto optimal matching. Let MIN-POM denote the problem deciding, given an instance I of POM and an integer K, whether I admits a Pareto optimal matching of size at most K.firstly prove that MIN-POM is NP-complete via a reduction from MMM, which is the problem of deciding, given a graph G and an integer K, whether G admits a maximal matching of size at most K.Let M be a matching in a given instance of POM. Then M is Pareto optimal if and only if M is maximal, trade-in-free and coalition-free.Given an instance 1 of POM, the relation forms a strict partial order over the set of matchings in I.Given the figure below, using the Greedy-Best First search find ifit is optimal /admissible or not? Explain why?
- Please answer the following question in full detail. Please be specifix about everything: You have learned before that A∗ using graph search is optimal if h(n) is consistent. Does this optimality still hold if h(n) is admissible but inconsistent? Using the graph in Figure 1, let us now show that A∗ using graph search returns the non-optimal solution path (S,B,G) from start node S to goal node G with an admissible but inconsistent h(n). We assume that h(G) = 0. Give nonnegative integer values for h(A) and h(B) such that A∗ using graph search returns the non-optimal solution path (S,B,G) from S to G with an admissible but inconsistent h(n), and tie-breaking is not needed in A∗.he heuristic path algorithm is a best-first search in which the objective function is f(n) = (4 −w)g(n) + wh(n). For what values of w is this algorithm guaranteed to be optimal? What kind of search does this perform when w = 0? When w = 1? When w = 4?The “Half-Manhattan distance” heuristic is equal to the Manhattan distance divided by two. explain how the use of this heuristic would or would not affect the optimality properties of (i). Greedy Best-First Search,(ii). Uniform-Cost Search, and (iii.) A* Search
- Show how to transform the weight function of a weighted matroid problem, wherethe desired optimal solution is a minimum-weight maximal independent subset, tomake it a standard weighted-matroid problem. Argue carefully that your transformation is correct.In every instance (i.e., example) of the TSP, we are given n cities, where each pair of cities is connected by a weighted edge that measures the cost of traveling between those two cities. Our goal is to find the optimal TSP tour, minimizing the total cost of a Hamiltonian cycle in G. Although it is NP-complete to solve the TSP, there is a simple 2-approximation achieved by first generating a minimum-weight spanning tree of G and using this output to determine our TSP tour. Prove that our output is guaranteed to be a 2-approximation, provided the Triangle Inequality holds. In other words, if OPT is the total cost of the optimal solution, and APP is the total cost of our approximate solution, clearly explain why APP≤ 2∗OPT.Insufficient Overlapping Subproblems and the Principle of Optimality (Optimal Substructure) are dynamic programming questions. As an example, consider All-Pairs Shortest Paths.