# Introduction to Graph Theory

**Topics:**Computational complexity theory, Travelling salesman problem, Graph theory

**Pages:**12 (1431 words)

**Published:**January 23, 2013

An introduction to solution methods

Ecole des Mines de Nantes Master MOST 2012-2013

Olivier Péton

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Problem

Min f ( x )

xS

An optimization problem

S is the solution set that represents all feasible solutions of a problem. f is the objective function that maps S to R. It evaluates each feasible solution. Also called evaluation function or cost function

Minimization = maximization !

max f ( x) min ( f ( x))

xS xS

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Mathematical modeling

1.

Decision Variables x1,…,xn A solution = a value for each variable Objective function min f(x1,…,xn) Constraints g1(x1,…,xn) a g2(x1,…,xn) b g3(x1,…,xn) = c

2.

3.

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Solutions of an optimization problem

What is a feasible solution ? What is an optimal solution ? How many optimal solutions are there for a given problem ?

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Combinatorial Optimization (C.O.) problems

Different fields in Optimization:

Linear Programming Non-Linear Optimization Integer – Mixed Linear Programming Graph / network optimization Routing, Scheduling, Supply Chain,…

Combinatorial optimization studies optimization on finite and discrete domains.

Find the minimum s* of f on a finite set S.

f ( s ) Min f ( s)

sS

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Characteristics of C.O. problems ?

A solution is a combination of values given to different variables. The variables are discrete. The number of possible combination is possibly huge.

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Ex1: The shortest Path

Data: Weighted Directed Graph G = (X,U) 2 distinct vertices s & t

Objective: find a shortest path from s to t

Model: P = {paths from s to t} W = cost of edge (i,j)

ij

f : P

( i , j )

R Wij

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Ex2: the 0-1 knapsack

A mountaineer has a knapsack of capacity b kg and n cans with weights ai and energy content ci, i = 1…n. Which cans can he bring such that the overall energy content of the knapsack is maximal ? Data: two vectors a and c of size n and a number b.

Objective: find a subset of indices such that the overall value is maximal and the overall weight is b. Model: n Max ci xi i 1 n ai xi b i 1 x ,1 n 0

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Ex3: The Travelling Salesman Problem

Given a set of cities to be visited, knowing the travel costs between each city, the Travelling Salesman Problem (TSP) is to find the cheapest way of visiting all of the cities and returning to the starting point.

Optimal tour through 24978 cities in Sweden : 9 (source www.tsp.gatech.edu)

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Ex 3 : The TSP model

Given a complete graph G(V, E), we use a single index e to be associated with an edge eE, and index i to a vertex iV. ce represents the travel cost through edge e. (S) and E(S) denote the set of edges with only one and both endpoints in SV.

Min c e x e

e E

s .t .

e ( i )

x

e

2 S 1

i V S V , S

e E ( S )

x

e

x e 0,1

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Ex 3: Solution set of the TSP

What is the size of the solution set ? (How many possible Hamiltonian circuits in Kn?)

1 (n 1)! 2

The number of the permutations of set (1,…,n) is n! Some permutations are redundant because they represent the same tour. For example:

1

2

3

4

5

6

7 1 2 3 4 5 6 7 2 7 6 5 4 3 2 1

2

n

3

4

5

6

7

1

. . .

7 1 2 3 4 5 6

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Classical Applications of Operations Research

Routing / Transportation problems Production management Scheduling/Planning Problems Logistics network design / facility location Problems Investment planning (advertisement, financial investment, …) Telecommunication: network design, routing, pricing

Air Traffic Management

Rostering / Timetabling Supply chain optimization

Healthcare

Yield management Website: Science of Better

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Main challenges in combinatorial optimisation

Discrete nature of the data : Mathematical Programming...

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