Introduction to Graph Theory

Operations research
An introduction to solution methods

Ecole des Mines de Nantes Master MOST 2012-2013

Olivier Péton

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Problem

Min f ( x ) xS  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)) xS xS

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Mathematical modeling
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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

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Solutions of an optimization problem

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

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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) sS - 5-

Characteristics of C.O. problems ?
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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