Algorithms

1
Algorithms

• Unit 16. Approximation Algorithms
• Chih-Hung Wang

• The Vertex-cover Problem
• The Traveling-salesman Problem
• Others

2
Approximation Algorithm

• We say an approximation algorithm for the problem
  has a ratio bound of if for any input size
  n, the cost C of the solution produced by the
  approximation algorithm is within a factor of
  of the C of the optimal solution
• This definition applies for both minimization and
  maximization problems.

3
The Vertex-cover Problem
4
Example
5
Proof
6
The Traveling-salesman Problem
Triangle inequality
7
Approximation Method
8
Example
9
Proof
Triangle inequality
10
The General TSP (1)

• If we drop the assumption that the cost function
  c satisfies the triangle inequality, good
  approximate tours cannot be found in polynomial
  time unless PNP.

11
The General TSP (2)
12
The Set-covering Problem
13
Example
14
A Greedy Approximation Algorithm for Set-cover
15
Analysis
16
The Subset-sum Problem

• The optimization problem
• Find a subset of x1, x2, xn whose sum is as
  large as possible but not larger than t.
• An exponential-time algorithm

17
Exact-Subset-Sum
18
Example
19
A Fully Polynomial-time Approximation Scheme
20
Trim
?(m)
21
Approx-subset-sum
22
Example (1)
23
Example (2)
24
Example (3)
25
Analysis

• Theorem 35.8
• APPROX_SUBSET_SUM is a fully polynomial
  approximation scheme for the subset-sum problem.