# 算法代写｜COMP3027 & COMP3927 Algorithm Design Assignment 3

Problem 1. (20 points)
To begin, let us consider the Ford-Fulkerson algorithm. Consider below the flow

network G,c in Figure 1 and the current s−t flow f in Figure 2. Your task is to run one iteration of the Ford-Fulkerson algorithm on the flow network.

1. a)  Draw the residual graph associated with the flow f. Specify the residual capacity of every edge. You are allowed to draw this by hand and insert a photo.
2. b)  Specify an s − t path used to update the flow.
3. c)  After the update, is the new flow through the network a max flow? Explain why.
4. d)  Specify a Min-Cut associated with graph G.

Problem 2. (80 points) (COMP3027 only)

You and your friends have decided to go on a great go-karting adventure. You’ve heard that there is to be a symposium for algorithms and computing theory (SACT for short) in the bush (with music!), and since your good friend André owns k go-karts, you decide that the most fun way to get there is via an epic four wheel journey. There is only one issue – the go-karts can only go so far before they need to be refueled, meaning you need to design a trip that allows for picking up fuel every 10km. Even worse, you have realised that all these stops only have enough fuel for one go-kart, meaning that each individual go-kart will need to refuel at different stops. Formally, the SACT Go-Kart Problem is the following; given a positive integer k, n (2-dimensional) Cartesian points describing fuel stations and both a starting point S and end point T in the Cartesian plane, determine if k friends have valid paths from S to T. Note that a valid path involves stopping at a fuel station every 10km or less.

Figure 3: Map including go-karts start location, fuel stations, and the bush sympo- sium. A go-kart cannot travel 10km without visiting a fuel station.