[Problem 5] Let A be a 1-dimensional integer array with random entries, and a decision problem Q = {A is an integer array with n elements : integer z & A.} Prove that problem Q is in NP.

0/1 knapsack problem, generate algorithmic solutions to this problem, with each of the design techniques, of the following algorithms, in Python language: - greedy algorithms - dynamic programming - backtracking - branching and pruning (brach and bound) We have n objects, each with a weight (w_i) and a benefit (v_i), and a backpack in which we can put objects, with a maximum weight capacity M. The objective is to maximize the benefit of the transported objects, where each object can be taken integer (x_i=1) or nothing (x_i=0).

Problem 2: Finding an element.(i) Let n be an even positive integer. Let A be an array containing n not necessarydifferent integers. Elements in A are not necessary sorted. Describe an algorithmwhich in time O(n) tests whether there exists an element x in A that appears in Aat least n/2 times.(ii) Let S be a set containing n integers. An element x ∈ S is an approximate median ofS, if|{y ∈ S | y < x}| ≥ n/4and |{y ∈ S | y > x}| ≥ n/4Given an array A containing n different integers, not necessary sorted, describe analgorithm which in time O(n) finds all approximate medians of the set {A[i] | i =1, 2, . . . , n}.

b A problem Q is in P and there is a polynomial-time reduction from Q to Q'. What can we say about Q'? Is Q' € P? Is Q' € NP? c Let Q be a problem defined as follows: Input: a set of numbers A = a1, a2, ..., an and a number x Output: 1 if and only if there are two values ai, ak € A such that a; +ak = x. Is Q in NP? Is Q in P? Explain your answer.