Write The Rank-Maximal Matching Algorithm. Let G = (AUP, EU... UE). Initialize G₁ = G₁, and M to any maximum matching in G. For i = 1 tor-1 solve \ the  steps, and output.

Let n = 2 and n = 2 (²). Draw Hs². This is from an optimization and convex analysis class

Two computer scientists-turned-farmers are planning what and how to plant during the upcoming season. There are four possible courses of action they can take, and they seek advice. In each case the outcome is either NP-complete or it can be calculated in polynomial time. You are tasked to assess the existence of a polynomial time algorithms and explain how it is assessed. If you find one, show it. If you did not find one, explain why one does not exist. There is a total of n tons of seeds to plant, with k number of types of crops, labeled for simplicity as c1, c2, c3, ..., ck. The quantities of each crop are given by 2i, where i ˃ 0 is an integer. The two farmers want to each plant an equal mix of crops.

There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Limit 1 (Tom) x x x 2 2 (Mark) x   x 1 3 (Fred)   x x 0