7.04d Travelling salesman lower bound: using minimum spanning tree

7 Rob delivers bread to six shops \(A , B , C , D , E\) and \(F\). Each day, Rob starts at shop \(A\), travels to each of the other shops before returning to shop \(A\). The table shows the distances, in miles, between the shops.

1. 1. Find the length of the tour \(A B C D E F A\).
   2. Find the length of the tour obtained by using the nearest neighbour algorithm starting from \(A\).
2. By deleting \(A\), find a lower bound for the length of a minimum tour.
3. By deleting \(F\), another lower bound of 45 miles is obtained for the length of a minimum tour. The length of a minimum tour is \(T\) miles. Write down the smallest interval for \(T\) which can be obtained from your answers to parts (a) and (b), and the information given in this part.
   (3 marks)

7 The diagram shows the locations of some schools. The number on each edge shows the distance, in miles, between pairs of schools. \includegraphics[max width=\textwidth, alt={}, center]{5a414265-6273-41c5-ad5f-f6316bd774d0-14_1031_1231_428_392} Sam, an adviser, intends to travel from one school to the next until he has visited all of the schools, before returning to his starting school. The shortest distances for Sam to travel between some of the schools are shown in Table 1 opposite.

1. Complete Table 1.
2. 1. On the completed Table 1, use the nearest neighbour algorithm, starting from \(B\), to find an upper bound for the length of Sam's tour.
   2. Write down Sam's actual route if he were to follow the tour corresponding to the answer in part (b)(i).
   3. Using the nearest neighbour algorithm, starting from each of the other vertices in turn, the following upper bounds for the length of Sam's tour were obtained: 77, 77, 77, 76, 77 and 76. Write down the best upper bound. 7
3. 1. On Table 2 below, showing the order in which you select the edges, use Prim's algorithm, starting from \(A\), to find a minimum spanning tree for the schools \(A , B , C\), \(D , F\) and \(G\).
   2. Hence find a lower bound for the length of Sam's minimum tour.
   3. By deleting each of the other vertices in turn, the following lower bounds for the length of a minimum tour were found: \(50,48,52,51,56\) and 64 . Write down the best lower bound.
4. Given that the length of a minimum tour is \(T\) miles, use your answers to parts (b) and (c) to write down the smallest interval within which you know \(T\) must lie.
   (2 marks) \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { F }\)	G
   A	-	2	6	4	16	27
   \(\boldsymbol { B }\)	2	-	8	3	15	26
   \(\boldsymbol { C }\)	6	8	-	10	22	32
   \(\boldsymbol { D }\)	4	3	10	-	12	23
   \(\boldsymbol { F }\)	16	15	22	12	-	20
   G	27	26	32	23	20	-

   \end{table}
   \includegraphics[max width=\textwidth, alt={}]{5a414265-6273-41c5-ad5f-f6316bd774d0-17_2486_1714_221_153}

8 Tony delivers paper to five offices, \(A , B , C , D\) and \(E\). Tony starts his deliveries at office \(E\) and travels to each of the other offices once, before returning to office \(E\). Tony wishes to keep his travelling time to a minimum. The table shows the travelling times, in minutes, between the offices.

4 David, a tourist, wishes to visit five places in Rome: Basilica ( \(B\) ), Coliseum ( \(C\) ), Pantheon ( \(P\) ), Trevi Fountain ( \(T\) ) and Vatican ( \(V\) ). He is to start his tour at one of the places, visit each of the other places, before returning to his starting place. The table shows the times, in minutes, to travel between these places. David wishes to keep his travelling time to a minimum.

1. 1. Find the total travelling time for the tour TPVBCT.
   2. Find the total travelling time for David's tour using the nearest neighbour algorithm starting from \(T\).
   3. Explain why your answer to part (a)(ii) is an upper bound for David's minimum total travelling time.
2. 1. By deleting \(B\), find a lower bound for the total travelling time for the minimum tour.
   2. Explain why your answer to part (b)(i) is a lower bound for David's minimum total travelling time.
3. Sketch a network showing the edges that give the lower bound found in part (b)(i) and comment on its significance.

5 Angelo is visiting six famous places in Palermo: \(A , B , C , D , E\) and \(F\). He intends to travel from one place to the next until he has visited all of the places before returning to his starting place. Due to the traffic system, the time taken to travel between two places may be different dependent on the direction travelled. The table shows the times, in minutes, taken to travel between the six places.

1. Give an example of a Hamiltonian cycle in this context.
2. 1. Show that, if the nearest neighbour algorithm starting from \(F\) is used, the total travelling time for Angelo would be 95 minutes.
   2. Explain why your answer to part (b)(i) is an upper bound for the minimum travelling time for Angelo.
3. Angelo starts from \(F\) and visits \(E\) next. He also visits \(B\) before he visits \(D\). Find an improved upper bound for Angelo's total travelling time.
   
   \includegraphics[max width=\textwidth, alt={}]{44bbec2c-32f4-4d28-9dd6-d89387228454-11_2484_1709_223_153}

8 Mrs Jones is a spy who has to visit six locations, \(P , Q , R , S , T\) and \(U\), to collect information. She starts at location \(Q\), and travels to each of the other locations, before returning to \(Q\). She wishes to keep her travelling time to a minimum. The diagram represents roads connecting different locations. The number on each edge represents the travelling time, in minutes, along that road. \includegraphics[max width=\textwidth, alt={}, center]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-16_524_866_612_587}

1. 1. Explain why the shortest time to travel from \(P\) to \(R\) is 40 minutes.
   2. Complete Table 1, on the opposite page, in which the entries are the shortest travelling times, in minutes, between pairs of locations.
2. 1. Use the nearest neighbour algorithm on Table 1, starting at \(Q\), to find an upper bound for the minimum travelling time for Mrs Jones's tour.
   2. Mrs Jones decides to follow the route given by the nearest neighbour algorithm. Write down her route, showing all the locations through which she passes.
3. By deleting \(Q\) from Table 1, find a lower bound for the travelling time for Mrs Jones's tour. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 1}

   	\(\boldsymbol { P }\)	\(Q\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(T\)	\(\boldsymbol { U }\)
   \(P\)	-	25
   \(Q\)	25	-	20	21	23	11
   \(\boldsymbol { R }\)		20	-
   \(\boldsymbol { S }\)		21		-
   \(T\)		23			-	12
   \(\boldsymbol { U }\)		11			12	-

   \end{table}
   \includegraphics[max width=\textwidth, alt={}]{3b7f04ff-e340-41ad-b50e-a02f94f02e8b-18_2486_1714_221_153}

7 Rupta, a sales representative, has to visit six shops, \(A , B , C , D , E\) and \(F\). Rupta starts at shop \(A\) and travels to each of the other shops once, before returning to shop \(A\). Rupta wishes to keep her travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.

4 Sarah is a mobile hairdresser based at \(A\). Her day's appointments are at five places: \(B , C , D , E\) and \(F\). She can arrange the appointments in any order. She intends to travel from one place to the next until she has visited all of the places, starting and finishing at \(A\). The following table shows the times, in minutes, that it takes to travel between the six places.

1. Sarah decides to visit the places in the order \(A B C D E F A\). Find the travelling time of this tour.
2. Explain why this answer can be considered as being an upper bound for the minimum travelling time of Sarah's tour.
3. Use the nearest neighbour algorithm, starting from \(A\), to find another upper bound for the minimum travelling time of Sarah's tour.
4. By deleting \(A\), find a lower bound for the minimum travelling time of Sarah's tour.
5. Sarah thinks that she can reduce her travelling time to 75 minutes. Explain why she is wrong.

3 The diagram shows a network. The weights on the arcs represent distances in miles. The direct path between any two adjacent vertices is never longer than any indirect path. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-02_490_748_1372_699}

1. By deleting vertex \(U\) and all arcs connected to \(U\), find a lower bound for the length of the shortest cycle that visits every vertex of this network.
2. Find a vertex that can be used as the start vertex for the nearest neighbour method to give a cycle that passes through every vertex of this network. Give your cycle and its length.

6 The arcs in the network represent the tracks in a forest. The weights on the arcs represent distances in km . \includegraphics[max width=\textwidth, alt={}, center]{cec8d4db-4a72-43a3-88f3-ff9df2a11d2c-7_535_1267_395_440} Richard wants to walk along every track in the forest. The total weight of the arcs is \(26.7 + x\).

1. Find, in terms of \(x\), the length of the shortest route that Richard could use to walk along every track, starting at \(A\) and ending at \(G\). Show all of your working.
2. Now suppose that Richard wants to find the length of the shortest route that he could use to walk along every track, starting and ending at \(A\). Show that for \(x \leqslant 1.8\) this route has length \(( 32.4 + 2 x ) \mathrm { km }\), and for \(x \geqslant 1.8\) it has length \(( 34.2 + x ) \mathrm { km }\). Whenever two tracks join there is an information board for visitors to the forest. Shauna wants to check that the information boards have not been vandalised. She wants to find the length of the shortest possible route that starts and ends at \(A\), passing through every vertex at least once. Consider first the case when \(x\) is less than 3.2.
3. (a) Apply Prim's algorithm to the network, starting from vertex \(A\), to find a minimum spanning tree. Draw the minimum spanning tree and state its total weight. Explain why the solution to Shauna's problem must be longer than this.
   (b) Use the nearest neighbour strategy, starting from vertex \(A\), and show that it stalls before it has visited every vertex. Now consider the case when \(x\) is greater than 3.2 but less than 4.6.
4. (a) Draw the minimum spanning tree and state its total weight.
   (b) Use the nearest neighbour strategy, starting from vertex \(A\), to find a route from \(A\) to \(G\) passing through each vertex once. Write down the route obtained and its total weight. Show how a shortcut can give a shorter route from \(A\) to \(G\) passing through each vertex. Hence, explaining your method, find an upper bound for Shauna's problem.

5 Jess and Henry are out shopping. The network represents the main routes between shops in a shopping arcade. The arcs represent pathways and escalators, the vertices represent some of the shops and the weights on the arcs represent distances in metres. \includegraphics[max width=\textwidth, alt={}, center]{ccb12789-cd5f-40dc-9f10-f8bb45399580-6_586_1084_367_495} The total weight of all the arcs is 1200 metres.
The table below shows the shortest distances between vertices; some of these are indirect distances.

1. Use a standard algorithm to find the shortest distance that Jess must travel to cover every arc in the original network, starting and ending at \(M\).
2. Find the shortest distance that Jess must travel if she just wants to cover every arc, but does not mind where she starts and where she finishes. Which two points are her start and finish? Henry suggests that Jess only needs to visit each shop.
3. Apply the nearest neighbour method to the network, starting at \(M\), to write down a closed tour through all the vertices. Calculate the weight of this tour. What does this value tell you about the length of the shortest closed route that passes through every vertex? Henry thinks that Jess does not need to visit shop \(W\). He uses the table of shortest distances to list all the possible connections between \(M , N , P , R , S\) and \(V\) by increasing order of weight. Henry's list is given in your answer book.
4. Use Kruskal's algorithm on Henry's list to find a minimum spanning tree for \(M , N , P , R , S , T\) and \(V\). Draw the tree and calculate its total weight. Jess insists that they must include shop \(W\).
5. Use the weight of the minimum spanning tree for \(M , N , P , R , S , T\) and \(V\), and the table of shortest distances, to find a lower bound for the length of the shortest closed route that passes through all eight vertices.

5 This question uses the same network as question 4.
The network below represents a treasure trail. The arcs represent paths and the weights show distances in units of 100 metres. \includegraphics[max width=\textwidth, alt={}, center]{cdad4fbe-4b94-4c8f-bb42-24d20eeaab4d-5_680_1154_431_459} Gus wants to hunt for the treasure. He assumes that the treasure is located at a vertex, but he does not know which one.

1. (a) Use the nearest neighbour method starting at \(G\) to find an upper bound for the length of the shortest closed route through every vertex.
   (b) Gus follows this route, but starting at \(A\). Write down his route, starting and ending at \(A\).
2. Use Prim's algorithm on the network, starting at \(A\), to find a minimum spanning tree. You should write down the arcs in the order they are included, draw the tree and give its total weight (in units of 100 metres).
3. (a) Vertex \(H\) and all arcs joined to \(H\) are removed from the original network. Write down the weight of the minimum spanning tree for vertices \(A , B , C , D , E , F\) and \(G\) in the resulting reduced network.
   (b) Use this minimum spanning tree for the reduced network to find a lower bound for the length of the shortest closed route through every vertex in the original network.
4. Find a route that passes through every vertex, starting and ending at \(A\), that is longer than the lower bound from part (iii)(b) but shorter than the upper bound from part (i)(a). Give the length of your route, in metres. Assume that Gus travels along paths at a rate of \(x\) minutes for every 100 metres and that he spends \(y\) minutes at each vertex hunting for the treasure. Gus starts by hunting for the treasure at \(A\). He then follows the route from part (iv), starting and finishing at \(A\) and hunting for the treasure at each vertex. Unknown to Gus, the treasure is found before he gets to it, so he has to search at every vertex. Gus can take at most 2 hours from when he starts searching at \(A\) to when he arrives back at \(A\).
5. Use this information to write down a constraint on \(x\) and \(y\). The treasure was at \(H\) and was found 40 minutes after Gus started. This means that Gus takes more than 40 minutes to get to \(H\).
6. Use this information to write down a second constraint on \(x\) and \(y\).

4. The following matrix gives the capacities of the pipes in a system.

1. Represent this information as a digraph.
2. Find the minimum cut, expressing it in the form \(\{ \} \mid \{ \}\), and state its value.
3. Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow.
4. Explain how you know that this flow is maximal.

6 [Figures 4, 5, 6 and 7, printed on the insert, are provided for use in this question.]

1. The network shows a flow from \(S\) to \(T\) along a system of pipes, with the capacity, in litres per minute, indicated on each edge. \includegraphics[max width=\textwidth, alt={}, center]{3ac580ff-f9c8-4e47-b4ca-97d186b0936c-6_350_878_532_591}
   1. Show that the value of the cut shown on the diagram is 97 .
   2. The cut shown on the diagram can be represented as \(\{ S , C \} , \{ A , B , T \}\). Complete the table on Figure 4, giving the value of each of the 8 possible cuts.
   3. State the value of the maximum flow through the network, giving a reason for your answer.
   4. Indicate on Figure 5 a possible flow along each edge corresponding to this maximum flow.
2. Extra pipes, \(B D , C D\) and \(D T\), are added to form a new system, with the capacity, in litres per minute, indicated on each edge of the network below. \includegraphics[max width=\textwidth, alt={}, center]{3ac580ff-f9c8-4e47-b4ca-97d186b0936c-6_483_977_1724_520}
   1. Taking your values from Figure 5 as the initial flow, use the labelling procedure on Figure 6 to find the new maximum flow through the network. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
   2. State the value of the new maximum flow, and, on Figure 7, indicate a possible flow along each edge corresponding to this maximum flow.

6 A retail company has warehouses at \(P , Q\) and \(R\), and goods are to be transported to retail outlets at \(Y\) and \(Z\). There are also retaining depots at \(U , V , W\) and \(X\). The possible routes with the capacities along each edge, in van loads per week, are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-14_673_1193_577_429}

1. On Figure 5 opposite, add a super-source, \(S\), and a super-sink, \(T\), and appropriate edges so as to produce a directed network with a single source and a single sink. Indicate the capacity of each edge that you have added.
2. On Figure 6, write down the maximum flows along the routes SPUYT and SRVWZT.
3. 1. On Figure 7, add the vertices \(S\) and \(T\) and the edges connecting \(S\) and \(T\) to the network. Using the maximum flows along the routes SPUYT and SRVWZT found in part (b) as the initial flow, indicate the potential increases and decreases of the flow on each edge of these routes.
   2. Use flow augmentation to find the maximum flow from \(S\) to \(T\). You should indicate any flow-augmenting routes on Figure 6 and modify the potential increases and decreases of the flow on Figure 7.
4. Find a cut with value equal to the maximum flow. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_629_1100_342_477}
   \end{figure} \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 6}

   Route	Flow
   SPUYT
   SRVWZT

   \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 7} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_634_1109_1838_470}
   \end{figure}

6 The network shows a system of pipes with the lower and upper capacities for each pipe in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-14_807_1472_429_276}

1. Find the value of the cut \(Q\).
2. Figure 2 shows most of the values of a feasible flow of 34 litres per second from \(S\) to \(T\).
   1. Insert the missing values of the flows along \(D E\) and \(F G\) on Figure 2.
   2. Using this feasible flow as the initial flow, indicate potential increases and decreases of the flow along each edge on Figure 3.
   3. Use flow augmentation on Figure 3 to find the maximum flow from \(S\) to \(T\). You should indicate any flow-augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
3. 1. State the value of the maximum flow.
   2. Illustrate your maximum flow on Figure 4.
4. Find a cut with capacity equal to that of the maximum flow. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_1646_1463_280_381}
   \end{figure} Figure 3 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_668_1230_1998_404}
   \end{figure}

8 The network below represents a system of pipes. The capacity of each pipe, in litres per second, is indicated on the corresponding edge. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-22_743_977_404_536}

1. Find the maximum flow along each of the routes \(A B E H , A C F H\) and \(A D G H\) and enter their values in the table on Figure 4 opposite.
2. 1. Taking your answers to part (a) as the initial flow, use the labelling procedure on Figure 4 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
   2. State the value of the maximum flow and, on Figure 5 opposite, illustrate a possible flow along each edge corresponding to this maximum flow.
3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4}

   Route	Flow
   \(A B E H\)
   \(A C F H\)
   \(A D G H\)

   \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-23_746_972_397_845}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-23_739_971_1311_539}
   \end{figure}
   \includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-24_2253_1691_221_153}

6 The network shows a system of pipes with the lower and upper capacities for each pipe in litres per minute. \includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-12_810_1433_429_306}

1. Find the value of the cut \(Q\).
2. Figure 3 opposite shows a partially completed diagram for a feasible flow of 24 litres per minute from \(S\) to \(T\). Indicate, on Figure 3, the flows along the edges \(B T , D E\) and \(E T\).
3. 1. Taking your answer from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 4 opposite.
   2. Use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
   3. Illustrate the maximum flow on Figure 5 opposite.
4. Find a cut with value equal to that of the maximum flow. You may wish to show the cut on the network above. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-13_759_1442_276_299}
   \end{figure} Figure 4 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-13_764_1438_1930_296}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-15_2349_1691_221_153} \includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-16_2489_1719_221_150}

5 The network shows the evacuation routes along corridors in a college, from two teaching areas to the exit, in case of a fire alarm sounding. \includegraphics[max width=\textwidth, alt={}, center]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-14_729_1013_434_497} The two teaching areas are at \(A\) and \(G\) and the exit is at \(X\). The number on each edge represents the maximum number of people that can travel along a particular corridor in one minute.

1. Find the value of the cut shown on the diagram.
2. Find the maximum flow along each of the routes \(A B D X , G F B X\) and \(G H E X\) and enter their values in the table on Figure 3 opposite.
3. 1. Taking your answers to part (b) as the initial flow, use the labelling procedure on Figure 3 to find the maximum flow through the network. You should indicate any flow augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
   2. State the value of the maximum flow, and, on Figure 4, illustrate a possible flow along each edge corresponding to this maximum flow.
4. During one particular fire drill, there is an obstruction allowing no more than 45 people per minute to pass through vertex \(B\). State the maximum number of people that can move through the network per minute during this fire drill. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-15_905_1559_331_292}
   \end{figure} Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-15_689_851_1370_598} Maximum flow is \(\_\_\_\_\) people per minute.

2 The network below represents a system of pipes. The number not circled on each edge represents the capacity of each pipe in litres per second. The number or letter in each circle represents an initial flow in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{5123be51-168e-4487-8cd3-33aee9e3b23f-04_1060_1076_434_466}

1. Write down the capacity of edge \(E F\).
2. State the source vertex.
3. State the sink vertex.
4. Find the values of \(x , y\) and \(z\).
5. Find the value of the initial flow.
6. Find the value of a cut through the edges \(E B , E C , E D , E F\) and \(E G\).

7 Figure 2 shows a network of pipes. Water from two reservoirs, \(R _ { 1 }\) and \(R _ { 2 }\), is used to supply three towns, \(T _ { 1 } , T _ { 2 }\) and \(T _ { 3 }\).
In Figure 2, the capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow.

1. Add a supersource, supersink and appropriate weighted edges to Figure 2. (2 marks)
2. 1. Use the initial flow and the labelling procedure on Figure 3 to find the maximum flow through the network. You should indicate any flow augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
   2. State the value of the maximum flow and, on Figure 4, illustrate a possible flow along each edge corresponding to this maximum flow.
3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-18_1077_1246_1475_395}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_1049_1264_308_386}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_835_1011_1738_171}
   \end{figure}
   \includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_688_524_1448_1302}
   \includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-20_2256_1707_221_153}

6. The network in the figure above, shows the distances in km , along the roads between eight towns, A, B, C, D, E, F, G and H. Keith has a shop in each town and needs to visit each one. He wishes to travel a minimum distance and his route should start and finish at A . By deleting D, a lower bound for the length of the route was found to be 586 km .
By deleting F, a lower bound for the length of the route was found to be 590 km .

1. By deleting C, find another lower bound for the length of the route. State which is the best lower bound of the three, giving a reason for your answer.
2. By inspection complete the table of least distances. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{(8)
   (8)
   (Total 13 marks)} \includegraphics[alt={},max width=\textwidth]{a5d69a77-c196-483c-a550-1a55363555af-3_780_889_1069_1078}
   \end{figure} (4) The table can now be taken to represent a complete network. The nearest neighbour algorithm was used to obtain upper bounds for the length of the route: Starting at D, an upper bound for the length of the route was found to be 838 km .
   Starting at F, an upper bound for the length of the route was found to be 707 km .
3. Starting at C , use the nearest neighbour algorithm to obtain another upper bound for the length of the route. State which is the best upper bound of the

   	A	B	C	D	E	F	G	H
   A	-	84	85	138	173		149	52
   B	84	-	130	77	126	213	222	136
   C	85	130	-	53	88	83	92
   D	138	77	53	-	49			190
   E	173	126	88	49	-	100	180	215
   F		213	83		100	-	163	115
   G	149	222	92		180	163	-	97
   H	52	136		190	215	115	97	-

   three, giving a reason for your answer.
   (4) (Total 13 marks)

6. The table below shows the distances, in km, between six towns \(A , B , C , D , E\) and \(F\).

1. Starting from \(A\), use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
   (4)
2. 1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
   2. Use a short cut to reduce the upper bound to a value less than 680 .
      (4)
3. Starting by deleting \(F\), find a lower bound for the solution of the travelling salesman problem.
   (4)

2. (a) Explain the difference between the classical and practical travelling salesman problems. \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-1_691_1297_1014_383} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at \(A\), visit each restaurant at least once and cover a minimum distance.
(b) Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
(c) Use your answer to part (b) to determine an initial upper bound for the length of the route.
(d) Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.

2. \includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-1_613_1269_1318_392} The network in the diagram shows the distances, in km , of the cables between seven electricity relay stations \(A , B , C , D , E , F\) and \(G\). An inspector needs to visit each relay station. He wishes to travel a minimum distance, and his route must start and finish at the same station. By deleting C, a lower bound for the length of the route is found to be 129 km .

1. Find another lower bound for the length of the route by deleting \(F\). State which is the better lower bound of the two.
2. By inspection, complete the table of least distances. The table can now be taken to represent a complete network.
3. Using the nearest-neighbour algorithm, starting at \(F\), obtain an upper bound to the length of the route. State your route.
   (4) (Total 11 marks)