Approximation Algorithms TSP

Approximation Algorithms
for NP – Hard Problems
Dr. P. Subathra
Professor
Dept. of IT
KAMARAJ College of Engg. & Tech.
(AUTONOMOUS)
Madurai
1

• We discuss a different approach to handling
difficult problems of combinatorial
optimization
• Traveling salesman problem and the knapsack
problem
• The decision versions of these problems are
NP-complete.
• The optimization versions fall in the class of
NP-hard problems
2

• There are no known polynomial-time
algorithms for these problems
• So how do we handle them?
• If an instance of the problem in question is
very small
– exhaustive-search algorithm
– dynamic programming technique
– branch-and-bound technique (comparatively
larger problems can be solved)
3

• There is a radically different way of dealing
with difficult optimization problems:
• solve them approximately by a fast algorithm.
• applications where a good but not necessarily
optimal solution will suffice
• approximation algorithms run based on some
problem-specific heuristic.
4

• A heuristic is a common-sense rule drawn
from experience rather than from a
mathematically proved assertion.
5

How accurate this approximation is?
• The accuracy of an approximate solution sa to
a problem of minimizing some function f by
the size of the relative error of this
approximation,
where s ∗ is an exact solution to the problem.
6

How accurate this approximation is?
• accuracy ratio
• Minimization Problem
• Maximization Problem
• The closer r(sa) is to 1, the better the
approximate solution is.
7

POLYNOMIAL-TIME APPROXIMATION
ALGORITHM
DEFINITION
• A polynomial-time approximation algorithm is said to
be a c-approximation algorithm, where c ≥ 1, if the
accuracy ratio of the approximation it produces does
not exceed c for any instance of the problem in
question:
• The best (i.e., the smallest) value of c is called the
performance ratio of the algorithm and denoted RA.
8

POLYNOMIAL-TIME APPROXIMATION
ALGORITHM
• We would like to have approximation
algorithms with RA as close to 1 as possible.
• Some approximation algorithms have infinitely
large performance ratios (RA=∞).
9

Travelling Salesman Problem
10

• There are more than a dozen approximation
algorithms that solve TSP in a reasonable
amount of time producing an approximate
solution.
• Example:
– Greedy Algorithms for the TSP
– Minimum-Spanning-Tree–Based Algorithms
– Local Search Heuristics
11

• Nearest-neighbor algorithm
• Multifragment-heuristic algorithm
• Twice-around-the-tree algorithm
• Christofides Algorithm
• 2-opt algorithms
• Lin-Kernighan algorithms
12

Approximation Algorithm for TSP
Greedy Algorithms for the TSP
13

Greedy Algorithms for the TSP:
Nearest-neighbor algorithm
• Algorithm:
Step 1 Choose an arbitrary city as the start.
Step 2 Repeat the following operation until
all the cities have been visited:
go to the unvisited city nearest the one
visited last (ties can be broken arbitrarily).
Step 3 Return to the starting city.
14

With a as the starting vertex,
the nearest-neighbor algorithm
yields the tour (Hamiltonian
circuit) sa:
a − b − c − d − a of length 10.
15

An optimal solution can be s*:
a − c − d − b − a of length 8.
16

The accuracy ratio of this
approximation is
• If the last links weight is ∞,
then the cost of the solution
will be infinity.
• Hence, RA =∞for this
algorithm
17

Greedy Algorithms for the TSP:
Multifragment-heuristic algorithm
• It is problem of finding a minimum-weight
collection of edges in a given complete
weighted graph so that all the vertices have
degree 2.
18

• Step 1
– Sort the edges in increasing order of their weights. (Ties
can be broken arbitrarily.)
– Initialize the set of tour edges to be constructed to the
empty set.
• Step 2
– Repeat this step n times, where n is the number of cities in
the instance being solved:
– add the next edge on the sorted edge list to the set of tour
edges, provided this addition does not create a vertex of
degree 3 or a cycle of length less than n;
– otherwise, skip the edge.
• Step 3
– Return the set of tour edges.
19

• Sorted Edges as per weights:
{(a,b), (c,d), (b,c), (a,c), (b,d), (a,d)}
{ 1 , 1 , 2 , 3 , 3 , 6 }
• Initial Tour Set = {}
• Inset edge to Tour Set = {(a,b)} - OK
• Inset edge to Tour Set = {(c,d)} – OK
• Inset edge to Tour Set = {(b,c)} - OK
• Inset edge to Tour Set = {(a,c)} – NO
• Inset edge to Tour Set = {(b,d)} – NO
• Inset edge to Tour Set = {(a,d)} - OK
20

• This method is applied to Eucledian Graphs
which follows the below conditions:
– triangle inequality d[i, j ]≤ d[i, k]+ d[k, j] for any
triple of cities i, j, and K
– symmetry d[i, j ]= d[j, i] for any pair of cities i and j
• Accuracy ratio satisfy the following inequality
21

Minimum-Spanning-Tree–Based Algorithms
• There are approximation algorithms for the
traveling salesman problem that exploit a
connection between Hamiltonian circuits and
spanning trees of the same graph.
• Removing an edge from a Hamiltonian circuit
yields a spanning tree, so we use Min Span Tree
to produce a TSP tour
22

Twice-around-the-tree algorithm
• Step 1 Construct a minimum spanning tree of the graph
corresponding to a given instance of the traveling salesman
problem.
• Step 2 Starting at an arbitrary vertex, perform a walk around the
minimum spanning tree recording all the vertices passed by. (DFS
traversal.)
• Step 3 Scan the vertex list obtained in Step 2 and eliminate from it
all repeated occurrences of the same vertex except the starting one
at the end of the list. The vertices remaining on the list will form a
Hamiltonian circuit, which is the output of the algorithm.
23

Initial Graph Minimum Spanning Tree
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Minimum Spanning Tree DFS - a, b, c, b, d, e, d, b, a.
25

DFS - a, b, c, b, d, e, d, b, a. Eliminating the Second
Occurrences
a – b – c – d – e - a
26

Initial Graph TSP : Cost = 4+6+10+7+12 = 39
27

Christofides Algorithm
• It also uses a minimum spanning tree like twice-
around-the-tree algorithm
• Eulerian circuit exists in a connected multigraph if and
only if all its vertices have even degrees.
• The Christofides algorithm obtains such a multigraph
by adding to the graph the edges of a minimum-weight
matching of all the odd-degree vertices in its minimum
spanning tree.
• Then the algorithm finds an Eulerian circuit in the
multigraph and transforms it into a Hamiltonian circuit
28

Christofides Algorithm
Algorithm
Step 1. Find a minimum spanning tree T
Step 2. Find a minimum matching M for the odd-
degree vertices in T
Step 3. Add M to T
Step 4. Find an Euler tour
Step 5. Cut short
29

Christofides Algorithm – Step 1
Initial Graph Minimum Spanning Tree
30

• Odd degree edges:
a, b, c, e
• Possible matching pairs
with cost are :
(a,b) & (c,e) = 4+11 = 15
(a,c) & (b,e) = 8+ 9 = 17
(a,e) & (b,c) = 12+ 6 = 18
• Minimum of it is:
(a,b) & (c,e) = 4+11 = 15
31

• Odd degree edges:
a, b, c, e
• Possible matching pairs
with cost are :
(a,b) & (c,e) = 4+11 = 15
(a,c) & (b,e) = 8+ 9 = 17
(a,e) & (b,c) = 12+ 6 = 18
• Minimum of it is:
(a,b) & (c,e) = 4+11 = 15
32

Minimum Spanning Tree Matching Pair Added to MST
33

Matching Pair Added to MST Euler Tour / Euler Circuit
a – b- c – e – d – b - a
34

Initial Graph Cut Short - Eliminate Repetition
a – b- c – e – d – b - a
a – b - c – e – d - a
Hamiltonian Circuit
35

Local Search Heuristics
• Good approximations to optimal tours can be
obtained by iterative-improvement algorithms,
which are also called local search heuristics.
36

LOCAL SEARCH HEURISTICS
• These algorithms start with some initial tour
• On each iteration, the algorithm explores a
neighborhood around the current tour by replacing a
few edges in the current tour by other edges.
• If the changes produce a shorter tour, the algorithm
makes it the current tour and continues by exploring its
neighborhood in the same manner
• otherwise, the current tour is returned as the
algorithm’s output and the algorithm stops
37

LOCAL SEARCH HEURISTICS
2-OPT ALGORITHMS
• The 2-opt algorithm works by deleting a pair
of nonadjacent edges in a tour and
reconnecting their endpoints by the different
pair of edges to obtain another Tour
• This operation is called the 2-change.
38

2-OPT ALGORITHM
• Choice 1
40

2-OPT ALGORITHM
• Choice 2
41

2-OPT ALGORITHM
• Choice 3
42

2-OPT ALGORITHM
• Choice 4
43

3-OPT ALGORITHMS
• 2 different options
44

Lin-Kernighan algorithm
• The Lin-Kernighan algorithm is a variable-opt
algorithm: its move can be viewed as a 3-opt
move followed by a sequence of 2-opt moves.
45

Approximation Algorithms for TSP
46

Approximation Algorithms for TSP
47

APPLICATIONS OF TSP
• Circuit Board and VLSI – chip fabrication
• X-ray crystallography
• Genetic Engineering
48

Approximation Algorithms TSP

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