7.04c Travelling salesman upper bound: nearest neighbour method

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)

1. \begin{figure}[h] \end{figure} Figure 1 shows a network of roads connecting six villages \(A , B , C , D , E\) and \(F\). The lengths of the roads are given in km .

1. Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. The table can now be taken to represent a complete network.
2. Use the nearest-neighbour algorithm, starting at \(A\), on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at \(A\) and visits every village exactly once.
   (3)
3. Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
   (1)
4. By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.
   (2)

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.

1. 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.
2. 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.
3. Use your answer to part (b) to determine an initial upper bound for the length of the route.
4. 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)

1. The network above shows the distances, in miles, between seven gift shops, \(A , B\), \(C , D , E , F\) and \(G\).

The area manager needs to visit each shop. She will start and finish at shop A and wishes to minimise the total distance travelled.

1. By inspection, complete the two copies of the table of least distances below. \includegraphics[max width=\textwidth, alt={}, center]{0e86cb18-2c6e-49f1-b235-aa15eb83e260-1_666_1136_141_863}

   	A	\(B\)	C	\(D\)	\(E\)	\(F\)	\(G\)
   A	-		15	36		53	23
   B		-	17	38	49	80	49
   C	15	17	-	21		62	32
   D	36	38	21	-	11	42
   E		49		11	-	31	61
   \(F\)	53	80	62	42	31	-	30
   \(G\)	23	49	32		61	30	-

   (4)

   	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
   \(A\)	-		15	36		53	23
   \(B\)		-	17	38	49	80	49
   \(C\)	15	17	-	21		62	32
   \(D\)	36	38	21	-	11	42
   \(E\)		49		11	-	31	61
   \(F\)	53	80	62	42	31	-	30
   \(G\)	23	49	32		61	30	-

2. Starting at A, and making your method clear, find an upper bound for the route length, using the nearest neighbour algorithm.
   (3)
3. By deleting A, and all of its arcs, find a lower bound for the route length.
   (4) (Total 11 marks)
   

7. \includegraphics[max width=\textwidth, alt={}, center]{151644c7-edef-448e-ac2a-b374d79f264c-3_738_1052_466_516} The network in the diagram above shows the distances, in km, between eight weather data collection points. Starting and finishing at A, Alice needs to visit each collection point at least once, in a minimum distance.

1. Obtain a minimum spanning tree for the network using Kruskal's algorithm, stating the order in which you select the arcs.
2. Use your answer to part (a) to determine an initial upper bound for the length of the route.
3. Starting from your initial upper bound use short cuts to find an upper bound, which is below 630 km . State the corresponding route.
4. Use the nearest neighbour algorithm starting at B to find a second upper bound for the length of the route.
5. By deleting C, and all of its arcs, find a lower bound for the length of the route.
6. Use your results to write down the smallest interval which you are confident contains the optimal length of the route.
   (2) (Total 16 marks)

2.

1. Explain the difference between the classical and the practical travelling salesperson problems. The table below shows the distances, in km, between six data collection points, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\), and F .

   	A	B	C	D	E	F
   A	-	77	34	56	67	21
   B	77	-	58	58	36	74
   C	34	58	-	73	70	42
   D	56	58	73	-	68	38
   E	67	36	70	68	-	71
   F	21	74	42	38	71	-

   Rachel must visit each collection point. She will start and finish at A and wishes to minimise the total distance travelled.
2. Starting at A, use the nearest neighbour algorithm to obtain an upper bound. Make your method clear. Starting at B, a second upper bound of 293 km was found.
3. State the better upper bound of these two, giving a reason for your answer. By deleting A, a lower bound was found to be 245 km .
4. By deleting B, find a second lower bound. Make your method clear.
5. State the better lower bound of these two, giving a reason for your answer.
6. Taking your answers to (c) and (e), use inequalities to write down an interval that must contain the length of Rachel's optimal route.
   

1. The table below shows the least costs, in pounds, of travelling between six cities, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .

Vicky must visit each city at least once. She will start and finish at A and wishes to minimise the total cost.

1. Use Prim's algorithm, starting at A , to find a minimum spanning tree for this network.
   (2)
2. Use your answer to part (a) to help you calculate an initial upper bound for the length of Vicky's route.
   (1)
3. Show that there are two nearest neighbour routes that start from A . You must make your routes and their lengths clear.
4. State the best upper bound from your answers to (b) and (c).
5. Starting by deleting A , and all of its arcs, find a lower bound for the route length.
   

1. \begin{figure}[h] \end{figure} The network in Figure 1 shows the distances, in km, between six towns, A, B, C, D, E and F. Mabintou needs to visit each town. She will start and finish at A and wishes to minimise the total distance travelled.

1. By inspection, complete the two copies of the table of least distances in your answer book.
2. Starting at A, use the nearest neighbour algorithm to find an upper bound for the length of Mabintou's route. Write down the route which gives this upper bound.
3. Starting by deleting A, and all of its arcs, find a lower bound for the route length.

2. The table shows the least distances, in km, between six towns, A, B, C, D, E and F. Toby must visit each town at least once. He will start and finish at A and wishes to minimise the total distance.

1. Use the nearest neighbour algorithm, starting at A , to find an upper bound for the length of Toby's route.
2. Starting by deleting A, and all of its arcs, find a lower bound for the route length.

2. The table shows the least distances, in km, between six towns, A, B, C, D, E and F. Liz must visit each town at least once. She will start and finish at A and wishes to minimise the total distance she will travel.

1. Starting with the minimum spanning tree given in your answer book, use the shortcut method to find an upper bound below 810 km for Liz's route. You must state the shortcut(s) you use and the length of your upper bound.
   (2)
2. Use the nearest neighbour algorithm, starting at A , to find another upper bound for the length of Liz's route.
3. Starting by deleting F , and all of its arcs, find a lower bound for the length of Liz's route.
4. Use your results to write down the smallest interval which you are confident contains the optimal length of the route.

1. The table shows the least distances, in km, between five hiding places, A, B, C, D and E.
Agent Goodie has to leave a secret message in each of the hiding places. He will start and finish at A , and wishes to minimise the total distance travelled.

1. Use Prim's algorithm to find a minimum spanning tree for this network. Make your order of arc selection clear.
2. Use your answer to part (a) to determine an initial upper bound for the length of Agent Goodie's route.
3. Show that there are two nearest neighbour routes which start from A . State these routes and their lengths.
4. State the better upper bound from your answers to (b) and (c).
5. Starting by deleting B, and all of its arcs, find a lower bound for the length of Agent Goodie's route.
6. Consider your answers to (d) and (e) and hence state an optimal route.

2.

1. Explain the difference between the classical and the practical travelling salesperson problem.

   	A	B	C	D	E	F
   A	-	65	48	15	30	40
   B	65	-	50	51	35	26
   C	48	50	-	37	20	34
   D	15	51	37	-	17	25
   E	30	35	20	17	-	14
   F	40	26	34	25	14	-

   The table above shows the least distances, in km, between six towns, A, B, C, D, E and F. Keith needs to visit each town, starting and finishing at A , and wishes to minimise the total distance he will travel.
2. Starting at A, use the nearest neighbour algorithm to obtain an upper bound. You must state your route and its length.
3. Starting by deleting A, and all of its arcs, find a lower bound for the route length.
4. Use your results to write down the smallest interval which you are confident contains the optimal length of the route.
   

2. The table shows the least times, in seconds, that it takes a robot to travel between six points in an automated warehouse. These six points are an entrance, A , and five storage bins, \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F . The robot will start at A , visit each bin, and return to A . The total time taken for the robot's route is to be minimised.

1. Show that there are two nearest neighbour routes that start from A . You must make the routes and their lengths clear.
2. Starting by deleting F , and all of its arcs, find a lower bound for the time taken for the robot's route.
3. Use your results to write down the smallest interval which you are confident contains the optimal time for the robot's route.

3. The network represented by the table shows the least distances, in km, between seven theatres, A, \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\) and G . Jasmine needs to visit each theatre at least once starting and finishing at A. She wishes to minimise the total distance she travels. The least distance between A and B, is \(x \mathrm {~km}\), where \(21 < x < 27\)

1. Using Prim's algorithm, starting at A , obtain a minimum spanning tree for the network. You should list the arcs in the order in which you consider them.
2. Use your answer to (a) to determine an initial upper bound for the length of Jasmine's route.
3. Use the nearest neighbour algorithm, starting at A , to find a second upper bound for the length of the route. The nearest neighbour algorithm starting at F gives a route of \(\mathrm { F } - \mathrm { E } - \mathrm { B } - \mathrm { A } - \mathrm { G } - \mathrm { D } - \mathrm { C } - \mathrm { F }\).
4. State which of these two nearest neighbour routes gives the better upper bound. Give a reason for your answer. Starting by deleting A, and all of its arcs, a lower bound of 159 km for the length of the route is found.
5. Find \(x\), making your method clear.
6. Write down the smallest interval that you can be confident contains the optimal length of Jasmine's route. Give your answer as an inequality.