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# Lecture 26 181 Lecture 26 181 Notes | EduRev

## : Lecture 26 181 Lecture 26 181 Notes | EduRev

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Lecture 26

181

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Lecture 26

181

193

Coping with NP-Completeness

Is your real problem a special subcase?

E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto  3- vs 2-
coloring

E.g. you only need planar graphs, or degree 3 graphs, …?

Guaranteed approximation good enough?

E.g. Euclidean TSP within 2 * Opt in poly time

Fast enough in practice (esp. if n is small),

E.g. clever exhaustive search like backtrack, branch &
bound, pruning

Heuristics – usually a good approximation and/or
usually fast

Page 3

Lecture 26

181

193

Coping with NP-Completeness

Is your real problem a special subcase?

E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto  3- vs 2-
coloring

E.g. you only need planar graphs, or degree 3 graphs, …?

Guaranteed approximation good enough?

E.g. Euclidean TSP within 2 * Opt in poly time

Fast enough in practice (esp. if n is small),

E.g. clever exhaustive search like backtrack, branch &
bound, pruning

Heuristics – usually a good approximation and/or
usually fast

194

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3
4
6
4
7
2
5
8
Example:

b = 34

NP-complete problem: TSP

Input: An undirected graph
G=(V,E) with integer edge
weights, and an integer b.

Output: YES iff there is a
simple cycle in G passing
through all vertices (once),
with total cost = b.

Page 4

Lecture 26

181

193

Coping with NP-Completeness

Is your real problem a special subcase?

E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto  3- vs 2-
coloring

E.g. you only need planar graphs, or degree 3 graphs, …?

Guaranteed approximation good enough?

E.g. Euclidean TSP within 2 * Opt in poly time

Fast enough in practice (esp. if n is small),

E.g. clever exhaustive search like backtrack, branch &
bound, pruning

Heuristics – usually a good approximation and/or
usually fast

194

5
3
4
6
4
7
2
5
8
Example:

b = 34

NP-complete problem: TSP

Input: An undirected graph
G=(V,E) with integer edge
weights, and an integer b.

Output: YES iff there is a
simple cycle in G passing
through all vertices (once),
with total cost = b.

195

€
lim
n?8
NN
OPT
?8
TSP - Nearest Neighbor Heuristic

NN Heuristic –go to nearest unvisited vertex

Fact: NN tour can be about (log n) x opt, i.e.  (above example is not that bad)

Page 5

Lecture 26

181

193

Coping with NP-Completeness

Is your real problem a special subcase?

E.g. 3-SAT is NP-complete, but 2-SAT is not; ditto  3- vs 2-
coloring

E.g. you only need planar graphs, or degree 3 graphs, …?

Guaranteed approximation good enough?

E.g. Euclidean TSP within 2 * Opt in poly time

Fast enough in practice (esp. if n is small),

E.g. clever exhaustive search like backtrack, branch &
bound, pruning

Heuristics – usually a good approximation and/or
usually fast

194

5
3
4
6
4
7
2
5
8
Example:

b = 34

NP-complete problem: TSP

Input: An undirected graph
G=(V,E) with integer edge
weights, and an integer b.

Output: YES iff there is a
simple cycle in G passing
through all vertices (once),
with total cost = b.

195

€
lim
n?8
NN
OPT
?8
TSP - Nearest Neighbor Heuristic

NN Heuristic –go to nearest unvisited vertex

Fact: NN tour can be about (log n) x opt, i.e.  (above example is not that bad)

198

2x Approximation to EuclideanTSP

A TSP tour visits all vertices, so contains a spanning tree, so
TSP cost is > cost of min spanning tree.

Find MST

Find “DFS” Tour

Shortcut

TSP = shortcut < DFST = 2 * MST < 2 * TSP

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