10
10.1
Polynomial reductions and N P -completeness
introduction
Our main objective in the rst sections has been the development of ecient algorithms
for the solutions of combinatorial optimization pr
Exercises Section 7
1. Consider a 3/4/G/B -scheduling problem. For each job the table below gives the
orders in which the machines are used.
J1
J2
J3
12
41
24
34
32
31
Verify if the following schedule
9
9.1
Algorithms and Complexity
Introduction
The minimum spanning tree problem (MST) can be solved by exhaustively examining all
spanning trees and choosing the best. Since there are nn2 spanning tree
Exercises week 10
1. Consider the scheduling problem n/1/nT : Given ready times r1 , . . . , rn , due dates
d1 , . . . , dn , processing times p1 , . . . , pn and an integer k , does there exist a sch
Exercises Section 9
1. Give as good as possible an upperbound for the complexity of the following two
algorithms in terms of the size N of the input.
a. Johnsons algorithm for the n/2/F/Cmax schedulin
Exercises Scheduling Independent Tasks
1. Apply the discussed approximation algorithms to the following instance of SIT (2):
29, 28, 27, 26, 23, 20, 17, 13, 7, 7, 3.
2. Prove that the optimization pro
12
12.1
Scheduling Independent Tasks
Approximation Algorithms
Approximation algorithms are algorithms that produce a feasible solution for some optimization problem that is not necessarily optimal. Of
7
Scheduling: Introduction
Scheduling is about the optimal planning of processing of a number of jobs through a
number of machines. Such a planning is called a schedule and is a precise description of
Exercises 14. Routing and Loading
1. Consider the instance of the vehicle routing problem with one depot
(0) six customers (1, . . . , 6) and the following symmetric cost matrix:
0
1
2
3
4
5
6
0
28
21
14
14.1
Routing and Loading
Vehicle Routing Problem
We consider the distribution problem in which vehicles based on a central facility (depot)
are required to visit - during a given time period - geog
Exercises Section 6
1. Solve the following knapsack problem
programming.
object
value
weight
with knapsack capacity 8, by use of dynamic
12345
47354
53221
2. Give an ILP formulation of the knapsack pr
Exercises Section 5
1. Consider the following ILP problem. Maximize x2 subject to
2kx1 + x2 2k,
2kx1 + x2 0,
x1 , x2 Z .
Z
a. Show that multiplying the inequalities with 2k 1 and 2k + 1 leads to a cut
3
3.1
Flows; costs and demands
Costs
Given a directed graph D = (V, A) with capacities, a source r and a sink s. Suppose
that for each arc a A there are costs k (a) for one unit of ow f (a) on a. So t
2
2.1
Flows and cuts
Max-ow min-cut
Consider a directed graph D = (V, A) (also called digraph or network) with two distinguished nodes r and s, called source and sink respectively. Each arc a A has a
1
1.1
Trees and paths
Minimum spanning trees
Consider an undirected graph (V, E ) with n nodes, for which every edge e E has a
weight w (e) R (also called length or cost ). The problem is to nd a span
4
4.1
LP and ILP
LP relaxations
Consider again the problem of nding a maximum ow f in a digraph D = (V, A) with
arc capacities c(a), a source r and a sink s. This can be formulated as:
Maximize
subjec
11
11.1
Flow-shop Problems
Introduction
In this section we discuss ow-shop problems of type n/m/F/Fmax with all ready times
equal to 0. We remind the reader that for m 3 there exists an optimal permut
6
The Knapsack Problem
6.1
Problem Formulation
In this chapter we discuss the knapsack problem to illustrate the Branch and Bound algorithm
of the previous chapter and to introduce the technique of Dy
Exercises Section 1
1. Given a complete graph with vertex set cfw_A, B, C, D, E, F and the following weights
on the edges.
ABCDEF
A 0 6 9 11 5 9
B603652
C930344
D 11 6 3 0 5 6
E554508
F924680
a. Dete
Exercises Section 3
1. Consider the directed graph given below (Attention: not all arcs point from left to
right.) Numbers proceeded by a $-sign indicate the costs, and the other numbers
give the capa
Exercises Section 2
1. Consider the network (= directed graph) below. Numbers indicate the capacities
of the links (= arcs). The thick links all carry a ow of size 1, and on all other
links the ow has
Exercises Section 4
1. Given a network (V, A) with weights on the nodes. Describe the problem of nding
a closure with maximum weight as an ILP (integer linear programming) problem.
2. Show that the fo
5
5.1
Solution Algorithms for ILP
A Cutting-Plane Algorithm
We saw that some integer linear programming problems (namely those that have a matrix that is totally unimodular) are not harder than the co
13
13.1
The Traveling Salesman Problem
Introduction
In this chapter we concentrate on one specic N P -hard combinatorial optimization problem: the traveling salesman problem (TSP). We discuss the comp