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Network Optimization Problems: Models and AlgorithmsPowerPoint Presentation

Network Optimization Problems: Models and Algorithms

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Network Optimization Problems:Models and Algorithms

- In this handout:
- Approximation Algorithms
- Traveling Salesman Problem

Classes of discrete optimization problems

- Class 1 problems havepolynomial-time algorithms for solving the problems optimally.
Ex.: Min. Spanning Tree problem

- For Class 2problems (NP-hard problems)
- No polynomial-time algorithm is known;
- And more likely there is no one.
Ex.: Traveling Salesman Problem

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

Zalgo / Zopt α , (for a minimization problem)

where Zalgo is the cost of the algorithm output,

Zopt is the cost of an optimal solution.

- α is called the approximation guarantee (or factor) of the algorithm.

Some Characteristics of Approximation Algorithms

- Time-efficient (sometimes not as efficient as heuristics)
- Don’t guarantee optimal solution
- Guarantee good solution within some factor of the optimum
- Rigorous mathematical analysis to prove the approximation guarantee
- Often use algorithms for related problems as subroutines
Next we will give

an approximation algorithm for TSP.

An approximation algorithm for TSP

Given an instance for TSP problem,

- Find a minimum spanning tree (MST) for that instance.
(using the algorithm of the previous handout)

- To get a tour, start from any node and traverse the arcs of MST by taking shortcuts when necessary.
Example:

Stage 1 Stage 2

red bold arcs form a tour

start from this node

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) :
cuw cuv + cvw 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.

v

w

u

Approximation guarantee for the algorithm (proof)

Optimal MST sol-n

Optimal TSP sol-n

A tree obtained from the tour

- First let’s compare the optimal solutions of MST and TSP for any problem instance G=(V,E), c: E R .
- Idea: Get a tour from Minimum spanning tree without increasing its cost too much (at most twice in our case).

(*)

Cost (Opt. TSP sol-n)

Cost (of this tree)

Cost (Opt. MST sol-n)

≥

≥

6

3

1

4

2

Approximation guarantee for the algorithm (proof)red bold arcs form a tour

- 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 f1, …, fk

(or simply coincides with a tree arc)

start from this node

6

3

1

4

2

Approximation guarantee for the algorithm (proof)red bold arcs form a tour

- Based on triangle inequality,
c(e) c(f1)+…+c(fk)

E.g, c15 c13 + c35

c23 c23

- But each tree (thin) arc
is shortcut exactly twice. (**)

E.g., tree arc 3-5 is shortcut by tour arcs 1-5 and 5-6 .

- The following chain of inequalities concludes the proof,
by using the facts we obtained so far:

start from this node

Performance of TSP algorithms in practice

- A more sophisticated algorithm (which again uses the MST algorithm as a subroutine) guarantees a solution within factor of 1.5 of the optimum (Christofides).
- For many discrete optimization problems, there are benchmarks of instances on which algorithms are tested.
- For TSP, such a benchmark is TSPLIB.
- On TSPLIB instances, the Christofides’ algorithm outputs solutions which are on average 1.09 times the optimum.
For comparison, the nearest neighbor algorithm outputs solutions which are on average 1.26 times the optimum.

- A good approximation factor often leads to good performance in practice.

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