Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 6The Traveling Salesman Problem

• Hamilton Joins the Circuit

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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, and cheapest-link
  algorithms to find approximate solutions to
  traveling salesman problems.
• To recognize the difference between efficient and
  inefficient algorithms.
• To recognize the difference between optimal and
  approximate algorithms.

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The Traveling Salesman Problem

• 6.1 Hamilton Circuits and Hamilton Paths

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The Traveling Salesman Problem

• Hamilton path
• A path that visits each vertex of the graph once
  and only once.
• Hamilton circuit
• A circuit that visits each vertex of the graph
  once and only once (at the end, of course, the
  circuit must return to the starting vertex).

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The Traveling Salesman Problem
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 a Hamilton path-- The Hamilton
circuit can always be truncated into a Hamilton
path by dropping the last vertex of the circuit.
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The Traveling Salesman Problem
Hamilton Circuit (Paths vs Euler Circuit Path
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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The Traveling Salesman Problem
Hamilton Circuit (Paths vs Euler Circuit Path
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 has
Hamilton paths (for example, A, B, C, D, E).
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The Traveling Salesman Problem
Hamilton Circuit (Paths vs Euler Circuit Path
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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The Traveling Salesman Problem
Hamilton Circuit (Paths vs Euler Circuit Path
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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The Traveling Salesman Problem
Hamilton Circuit (Paths vs Euler Circuit Path
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 Traveling Salesman Problem
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 Eulers circuit
and path theorems from Chapter 5 are useless when
it comes to Hamilton circuits and paths.
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The Traveling Salesman Problem
There are, however, nice theorems that identify
special situations where a graph must have a
Hamilton circuit. This best known of these
theorems is Diracs theorem If a connected graph
has N vertices (N gt 2) and all of them have
degree bigger or equal to N / 2, then the graph
has a Hamilton circuit.
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The Traveling Salesman Problem

• 6.2 Complete Graphs

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• If a graph has a Hamilton circuit, then how many
  different Hamilton circuits does a it have?
• 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 by the
  symbol KN.

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• Number of Edges in KN
• KN has N(N 1)/2 edges.
• Of all graphs with N vertices and no multiple
  edges or loops, KN has the most edges.

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• Hamilton Circuits in K4
• If we travel the four vertices of K4 in an
  arbitrary order, we get a Hamilton path. For
  example, C, A, D, B is a Hamilton path.

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The Traveling Salesman Problem

• Hamilton Circuits in K4
• D, C, A, B is another Hamilton Path.

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• Hamilton Circuits in K4
• Each of these Hamilton paths can be closed into
  a Hamilton circuit-- the path C, A, D, B begets
  the circuit D, A, D, B, C.

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The Traveling Salesman Problem

• Hamilton Circuits in K4
• The path D, C, A, B begets the circuit D, C, A,
  B, D.

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• Hamilton Circuits in K4
• It is important to remember that the same
  Hamilton circuit can be written in many ways.

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The Traveling Salesman Problem

• Hamilton Circuits in K4
• For example, C, A, D, B, C is the same circuit
  as A, D, B, C, A the only difference is that in
  the first case we used C as the reference point
  in the second case we used A.

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• Number of Hamilton Circuits in KN
• There are (N 1)! Distinct Hamilton circuits in
  KN.

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• 6.3 Traveling Salesman Problems

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• The traveling salesman is a convenient
  metaphor for many different important real-life
  applications, all involving Hamilton circuits in
  complete graphs but only occasionally involving
  salespeople.

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The Traveling Salesman Problem

• Any graph whose edges have numbers attached to
  them is called a weighted graph, and the numbers
  are called the weights of the edges. The graph
  is called a complete weighted graph.

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The Traveling Salesman Problem

• The problem we want to solve is fundamentally
  the same find an optimal Hamilton circuit (a
  Hamilton circuit with least total weight) for the
  given weighted graph.

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• 6.4 Simple Strategies for Solving TSPs

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• Strategy 1 (Exhaustive Search)
• Make a list of all possible Hamilton circuits.
  For each circuit in the list, calculate the total
  weight of the circuit. From all the circuits,
  choose the circuit with smallest total weight.

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• Strategy 2 (Go Cheap)
• Start from the home city. From there go to the
  city that is the cheapest to get to. From each
  new city go to the next city that is cheapest to
  get to. When there are no more new cities to go
  to, go back home.

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• 6.5 The Brute-Force and Nearest Neighbor
  Algorithms

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• The Exhaustive Search strategy can be formalized
  into an algorithm generally known as the
  brute-force algorithm the Go Cheap strategy can
  be formalized into an algorithm known as the
  nearest-neighbor algorithm.
• In both cases, the objective of the algorithm is
  to find an optimal (cheapest, shortest, fastest)
  Hamilton circuit in a complete weighted graph.

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• Algorithm 1 The Brute-Force Algorithm
• Step 1. Make a list of all the possible
  Hamilton circuits of the graph.

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• Algorithm 1 The Brute-Force Algorithm
• Step 2. For each Hamilton circuit calculate
  its total weight (add the weights of all the
  edges in the circuit).

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• Algorithm 1 The Brute-Force Algorithm
• Step 3. Choose an optimal circuit (there is
  always more than one optimal circuit to choose
  from!).

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• Algorithm 2 The Nearest-Neighbor Algorithm
• Start. Start at the designated starting
  vertex. If there is no designated starting
  vertex, pick any vertex.

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• Algorithm 2 The Nearest-Neighbor Algorithm
• First step. From the starting vertex go to its
  nearest neighbor (the vertex for which the
  corresponding edge has the smallest weight.

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• Algorithm 2 The Nearest-Neighbor Algorithm
• Middle steps. From each vertex go to its
  nearest neighbor, choosing only among the
  vertices that havent been yet visited. (If
  there is more than one, choose at random). Keep
  doing this until all the vertices have been
  visited.

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• The brute-force algorithm is a classic example
  of what is formally known as an inefficient
  algorithm an algorithm for which the number of
  steps needed to carry it out grows
  disproportionately 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 effort required to implement the
  algorithm grows in some reasonable proportion
  with 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 TSPs in
general would 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 describe any algorithm
that produces solutions that are, most of the
time, reasonably close to the optimal solution.
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• 6.7 The Repetitive Nearest-Neighbor Algorithm

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Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• Let X be any vertex. Find the nearest-neighbor
  circuit using X as the starting vertex and
  calculate the total cost of the circuit.

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Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• We compute the nearest-neighbor circuit with A as
  the starting vertex, and we got A, C, E, D, B, A
  with a total cost of 773.

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Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• Repeat the process with each of the other
  vertices of the graph as the starting vertex.

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Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• If we use B as the starting vertex, the
  nearest-neighbor circuit takes us from B to C,
  then to A, E, D, and back to B, with a total cost
  of 722.

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The Traveling Salesman Problem
Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• Remember we must start and end the trip at A
  this very same circuit would take the form A, E,
  D, B, C, A.

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The Traveling Salesman Problem
Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• The process is once again repeated using C, D,
  and E as the starting vertices with respective
  costs of 722, 722, and 741.

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Algorithm 3 The Repetitive Nearest-Neighbor
Algorithm

• Of the nearest-neighbor circuits obtained, keep
  the best one. If there is a designated starting
  vertex, rewrite the circuit using that vertex as
  the reference point.

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• 6.8 The Cheapest-Link Algorithm

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Algorithm 4 The Cheapest-Link Algorithm

• Step 1. Pick the cheapest link (edge with
  smallest weight) available. Among all the edges
  of the graph, the cheapest link is edge AC,
  with a cost of 119.

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Algorithm 4 The Cheapest-Link Algorithm

• Step 2. Pick the next cheapest link available
  and mark it. In this case edge CE with a cost of
  120.

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Algorithm 4 The Cheapest-Link Algorithm

• Step 3, 4, , N -1 Continue picking and marking
  the cheapest unmarked link available that does
  not
• (a) close a circuit, or
• (b) create three edges coming out of a single
  vertex.

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Algorithm 4 The Cheapest-Link Algorithm

• The next cheapest link available is edge BC
  (121), but we should not choose BC we would
  have three edges coming out of vertex C.

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Algorithm 4 The Cheapest-Link Algorithm

• The next cheapest link available is AE (133),
  but we cant take AE either-- the vertices A, C,
  and E would form a small circuit.

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Algorithm 4 The Cheapest-Link Algorithm

• The next cheapest link available is BD (150).
  Choosing BD would not violate either of the two
  rules, so we can add it to our budding circuit.

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Algorithm 4 The Cheapest-Link Algorithm

• The next cheapest link available is AD (152)
  and it works just fine.

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Algorithm 4 The Cheapest-Link Algorithm

• Step N. Connect the last two vertices to close
  the red circuit. At this point, we have only one
  way to close up the Hamilton circuit, edge BE.

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Algorithm 4 The Cheapest-Link Algorithm

• The Hamilton circuit can now be described using
  any vertex as the reference point. For A, we
  describe it as A, C, E, B, D, A with a total cost
  of 741.

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Conclusion

• How does one find an optimal Hamilton circuit in
  a complete weighted graph?
• The nearest-neighbor and cheapest-link algorithms
  are two fairly simple strategies for attacking
  TSPs.
• The search for an optimal and efficient general
  algorithm.