Chapter 6 the Traveling Salesman Problem

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Excursions in ModernMathematics

Sixth Edition

Peter Tannenbaum



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Chapter 6The Traveling Salesman Problem

Hamilton Joins theCircuit



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The Traveling Salesman ProblemOutline/learning Objectives

To identify and model Hamilton circuit

and Hamilton path problems. To recognize complete graphs and state

the number of Hamilton circuits that they

have. To identify traveling-salesman problems

and the difficulties faced in solving them.



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The Traveling Salesman ProblemOutline/learning Objectives

To implement brute-force, nearest-

neighbor, repeated nearest-neighbor, andcheapest-link algorithms to findapproximate solutions to traveling ±salesman problems.

To recognize the difference betweenefficient and inefficient algorithms.

To recognize the difference between

optimal and approximate algorithms.



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The Traveling Salesman Problem

6.1 Hamilton Circuits andH

amilton Paths



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Hamilton pathHamilton path

A path that visits each vertex of the graph onceand only once.

Hamilton circuitHamilton circuit

Acircuit that visits each vertex of the graphonce and only once (at the end, of course, the

circuit must return to the starting vertex).



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Hamilton Circuit (Paths vs Euler Circuit Path

Figure (a) shows a graph that has Euler

circuits and has Hamilton circuits. One

such Hamilton circuit is A, F, B, C, G,

D, E, A. Note that once a graph has a

Hamilton circuit, it automatically has aHamilton path-- The Hamilton circuit

can always be truncated into a Hamilton

path by dropping the last vertex of the

circuit.



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Figure (b) shows a graph that has no

Euler circuits but does have Euler paths

(for example C, D, E, B, A, D), has no

Hamilton circuits (sooner or later you

have to go to C, and then you are stuck) but does have Hamilton paths (for

example, A, B, E, D, C ). Aha, a graph

can have a Hamilton path but no

Hamilton Circuit!



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Figure (c) shows a graph that has

neither Euler circuits nor paths (it has

four odd vertices), has Hamilton circuits

(for example A, B, C, D, E, A ± there are

plenty more), and consequently hasHamilton paths (for example, A, B, C,

D, E ).



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Figure (d) shows a graph that has Euler

circuits (the vertices are all even), has

no Hamilton circuits (no matter what,

your are going to have to go through E

more than once!) but has Hamilton paths (for example, A, B, E, D, C ).



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Figure (e) shows a graph that has no

Euler circuits but has Euler paths ( F and

G are the two odd vertices), had neither

Hamilton circuits nor Hamilton paths.



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Figure (f) shows a graph that has

neither Euler circuits nor Euler paths

(too many odd vertices), has neither

Hamilton circuits nor Hamilton paths.



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The lesson in the previous Example is that the existence

of an Euler path or circuit in a graph tells us nothing about

the existence of a Hamilton path or circuit in that graph.

This is important because it implies that Euler¶s circuit

and path theorems from Chapter 5 are useless when it

comes to Hamilton circuits and paths.



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There are, however, nice theorems that identify special

situations where a graph must have a Hamilton circuit.

This best known of these theorems is Dirac¶s theorem:

If a connected graph has N vertices (N > 2) and

all of them have degree bigger or equal to N / 2,

then the graph has a Hamilton circuit.



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6.2 Complete Graphs



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If a graph has a Hamilton circuit, then how

many different Hamilton circuits does a ithave?

A graph with N vertices in which every pair of distinct vertices is joined by an edge is called a

complete graph on N vertices and denoted bythe symbol K N .



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Number of Edges inNumber of Edges inK K

N N

K N has N (N ± 1)/2 edges.

Of all graphs with N vertices and no multipleedges or loops, K N has the most edges.



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Hamilton Circuits in K 4

If we travel the four vertices of K 4 in anarbitrary order, we get aHamilton path. For example, C, A, D, B is aHamilton path.



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D, C, A, B is another Hamilton Path.



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Each of these Hamiltonpaths can be closed intoa Hamilton circuit-- the

path C, A, D, B begetsthe circuit D, A, D, B, C .



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The path D, C, A, B

begets the circuit D, C, A,

B, D.



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It is important toremember that the sameHamilton circuit can bewritten in many ways.



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For example, C, A, D, B,C is the same circuit as A,D, B, C, A ± the only

difference is that in thefirst case we used C asthe reference point in thesecond case we used A.



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Number of Hamilton Circuits inNumber of Hamilton Circuits in K K N N

There are (N ± 1)! Distinct Hamiltoncircuits in K N .



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6.3 TravelingSalesmanProblems



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The ³ traveling salesman´

is a convenient metaphor for many differentimportant real-lifeapplications, all involving

Hamilton circuits incomplete graphs but onlyoccasionally involvingsalespeople.



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Any graph whose edges

have numbers attached tothem is called a weighted

graph, and the numbersare called the weights of

the edges. The graph iscalled a complete

weighted graph.



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The problem we want to

solve is fundamentally thesame± find an optimal

Hamilton circuit (a

Hamilton circuit with least

total weight) for the givenweighted graph.



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6.4 SimpleStrategies for Solving TSPs



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± ± Strategy 1 (ExhaustiveStrategy 1 (Exhaustive

Search)Search)

Make a list of all possibleHamilton circuits. For eachcircuit in the list, calculate

the total weight of thecircuit. From all thecircuits, choose the circuitwith smallest total weight.



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± ± Strategy 2 (Go Cheap)Strategy 2 (Go Cheap)

Start from the home city.From there go to the citythat is the cheapest to getto. From each new city go

to the next city that ischeapest to get to. Whenthere are no more new citiesto go to, go back home.



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6.5 The Brute-Force and NearestNeighbor Algorithms



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The Ex haustive Search strategy can be

formalized into an algorithm generally knownas the brute-force algorithm; the Go Cheap

strategy can be formalized into an algorithmknown as the nearest-neighbor algorithm.

In both cases, the objective of the algorithm isto find an optimal (cheapest, shortest, fastest)Hamilton circuit in a complete weightedgraph.



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Algorithm 1: The BruteAlgorithm 1: The Brute--Force AlgorithmForce Algorithm

Step 1. Make a list of all the possibleHamilton circuits of the graph.



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Step 2. For each Hamilton circuitcalculate its total weight (add the

weights of all the edges in the circuit).



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Step 3. Choose an optimal circuit(there is always more than one optimal

circuit to choose from!).



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Algorithm 2: The NearestAlgorithm 2: The Nearest--Neighbor AlgorithmNeighbor Algorithm

Start. Start at the designated startingvertex. If there is no designated

starting vertex, pick any vertex.



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First step. From the starting vertexgo to its nearest neighbor (the vertex

for which the corresponding edge hasthe smallest weight.



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Middle steps. From each vertex goto its nearest neighbor, choosing onlyamong the vertices that haven¶t been

yet visited. (If there is more than one,choose at random). Keep doing thisuntil all the vertices have been visited.



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The brute-force algorithm is a classic

example of what is formally known asan inefficient algorithm ± analgorithm for which the number of steps needed to carry it out growsdisproportionately with the size of the problem.



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The nearest-neighbor algorithm is an

efficient algorithm. Roughly speaking,an efficient algorithm is an algorithm for which the amount of computational effortrequired to implement the algorithmgrows in some reasonable proportionwith the size of the input to the problem.



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6.6 Approximate Algorithm



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A really good algorithm for solving TSP¶s in generalwould have to be both efficient (like the nearest-neighbor)

and optimal (like the brute-force). Unfortunately, nobody

knows of such an algorithm.

We will use the term approximate algorithm to describeany algorithm that produces solutions that are, most of the

time, reasonably close to the optimal solution.



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6.7 The RepetitiveNearest-Neighbor Algorithm



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Let X be any vertex.Find the nearest-neighbor circuit using X

as the starting vertex andcalculate the total cost of the circuit.

Algorithm 3: The Repetitive Nearest-Neighbor Algorithm



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We compute the nearest-neighbor circuit with A asthe starting vertex, andwe got A, C, E , D, B, A

with a total cost of $773.




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Repeat the process witheach of the other verticesof the graph as thestarting vertex.




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If we use B as thestarting vertex, thenearest-neighbor circuittakes us from B to C,then to A, E, D, and backto B, with a total cost of $722.




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Remember we must startand end the trip at A ±this very same circuitwould take the form A, E ,

D, B, C, A.




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The process is onceagain repeated using C,D, and E as the startingvertices with respectivecosts of $722, $722, and$741.




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Of the nearest-neighbor circuits obtained, keepthe best one. If there is adesignated startingvertex, rewrite the circuitusing that vertex as thereference point.




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6.8 The Cheapest-Link Algorithm



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Step 1. Pick thecheapest link (edge withsmallest weight)available. Among all theedges of the graph, the³cheapest link´ is edge

AC , with a cost of $119.

Algorithm 4: The CheapestAlgorithm 4: The Cheapest--Link AlgorithmLink Algorithm



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Step 2. Pick the nextcheapest link availableand mark it. In this caseedge C E with a cost of $120.




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Step 3, 4, «, N -1Continue picking andmarking the cheapestunmarked link available

that does not(a) close a circuit, or

(b) create three edgescoming out of a single

vertex.




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The next cheapest linkavailable is edge BC($121), but we should notchoose BC± we would

have three edges comingout of vertex C.




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The next cheapest linkavailable is AE ($133),but we can¶t take AE

either-- the vertices A, C,

and E would form a smallcircuit.




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The next cheapest linkavailable is BD ($150).Choosing BD would notviolate either of the two

rules, so we can add it toour budding circuit.




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The next cheapest linkavailable is AD ($152)and it works just fine.




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Step N . Connect the lasttwo vertices to close thered circuit. At this point,we have only one way to

close up the Hamiltoncircuit, edge BE .




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The Hamilton circuit cannow be described usingany vertex as thereference point. For A,

we describe it as A, C, E ,B, D, A with a total costof $741.





Conclusion

How does one find an optimalHow does one find an optimal

Hamilton circuit in a completeHamilton circuit in a completeweighted graph?weighted graph?

The nearestThe nearest--neighbor and cheapestneighbor and cheapest--link algorithms are two fairly simplelink algorithms are two fairly simple

strategies for attacking TSPs.strategies for attacking TSPs. The search for an optimal and efficientThe search for an optimal and efficient

general algorithm.general algorithm.

Chapter 6 the Traveling Salesman Problem

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