Network Optimization Problems - Discrete Modeling and Optimization - Lecture Slides, Slides of Discrete Structures and Graph Theory

Download Network Optimization Problems - Discrete Modeling and Optimization - Lecture Slides and more Slides Discrete Structures and Graph Theory in PDF only on Docsity!

Algorithms for Network

Optimization Problems

This handout:

• Minimum Spanning Tree Problem
• Approximation Algorithms
• Traveling Salesman Problem

Terminology of Graphs

• A graph (or network) consists of

– a set of points

• a set of lines connecting certain pairs of the points.

The points are called nodes (or vertices ).

The lines are called arcs (or edges or links ).

• Example:

Terminology of Graphs:

Cycles, Connectivity and Trees

• A path that begins and ends at the same node is called a cycle.

Example:

• Two nodes are connected if there is a path between them.
• A graph is connected if every pair of its nodes is connected.
• A graph is acyclic if it doesn’t have any cycle.
• A graph is called a tree if it is connected and acyclic.

Example :

Minimum Spanning Tree Problem

• Given : Graph G=(V, E), |V|=n

Cost function c: E → R.

• Goal : Find a minimum-cost spanning tree for V

i.e., find a subset of arcs E* ⊆ E which

connects any two nodes of V

with minimum possible cost.

• Example:

e

b

c

d

a

e

b

c

d

a

G=(V,E)

Min. span. tree: G=(V,E)

Red bold arcs are in E*Docsity.com

The algorithm applied to our example

• Initialization: Select node a to start.

Its closest node is node b. Connect nodes a and b.

• Iteration 1: There are two unconnected node closest to a

connected node: nodes c and d

(both are 3 units far from node b).

Break the tie arbitrarily by

connecting node c to node b.

e

b

c

d

a

e

b

c

d

a

Red bold arcs are in E*;

thin arcs represent potential links.

The algorithm applied to our example

• Iteration 2 : The unconnected node closest to a connected

node is node d (3 far from node b). Connect nodes b and d.

• Iteration 3: The only unconnected node left is node e. Its

closest connected node is node c

(distance between c and e is 4).

Connect node e to node c.

• All nodes are connected. The bold

arcs give a min. spanning tree.

e

b

c

d

a

e

b

c

d

a

Three main directions to solve

NP-hard discrete optimization problems:

• Integer programming techniques
• Heuristics
• Approximation algorithms
• We gave examples of the first two methods for TSP.
• In this handout,

an approximation algorithm for TSP.

Definition of

Approximation Algorithms

• Definition : An α-approximation algorithm is a

polynomial-time algorithm which always produces a

solution of value within α times the value of an

optimal solution.

That is, for any instance of the problem

Z

algo

/ Z

opt

≤ α , ( for a minimization problem )

where Z

algo

is the cost of the algorithm output,

Z

opt

is the cost of an optimal solution.

• α is called the approximation guarantee (or factor )

of the algorithm.

An approximation algorithm for TSP

Given an instance for TSP problem,

1. Find a minimum spanning tree (MST) for that instance.

(using the algorithm of the previous handout)

2. To get a tour, start from any node and traverse the arcs of

MST by taking shortcuts when necessary.

Example:

Stage 1 Stage 2

start from

this node

red bold arcs

form a tour

Docsity.com

Approximation guarantee

for the algorithm

• In many situations, it is reasonable to assume that triangle

inequality holds for the cost function c: E → R defined on

the arcs of network G=(V,E) :

c

uw

≤ c

uv

+ c

vw

for any u, v, w ∈V

• Theorem:

If the cost function satisfies the triangle ineqality,

then the algorithm for TSP

is a 2-approximation algorithm.

w

v

u

Approximation guarantee

for the algorithm (proof)

 The algorithm

• takes a minimum spanning tree
• starts from any node
• traverse the MST arcs

by taking shortcuts when necessary

to get a tour.

 What is the cost of the tour compared to the cost of MST?

• Each tour (bold) arc e is a shortcut

for a set of tree (thin) arcs f

1

, …, f

k

(or simply coincides with a tree arc)

start from

this node

red bold arcs

form a tour

Approximation guarantee

for the algorithm (proof)

• Based on triangle inequality,

c(e) ≤ c(f

1

)+…+c(f

k

E.g, c

15

≤ c

13

• c

35

c

23

≤ c

23

• But each tree (thin) arc

is shortcut exactly twice. ()**

E.g., tree arc 3-5 is shortcut by tour arcs 1-5 and 5-.

 The following chain of inequalities concludes the proof,

by using the facts we obtained so far:

start from

this node

red bold arcs

form a tour

2 cost(MST) 2 cost(optimal TSP)

cost(our tour) c(e) 2 c(e)

by(*)

thinarcse

byΔineq.,(**)

boldarcs e

= ⋅ ≤ ⋅

= ≤ ⋅

∑ ∑