1. Approximation algorithms
Given minimization problem (e.g. min vertex cover, TSP,…) and an efficient algorithm that always returns some feasible solution.
The algorithm is said to have approximation ratio  if for all instances, cost(sol. found)/cost(optimal sol.)  

2. Approximation algorithms
Given maximization problem (e.g. MAXSAT, MAXCUT) and an efficient algorithm that always returns some feasible solution.
The algorithm is said to have approximation ratio  if for all instances, cost(optimal sol.)/cost(sol. found)  

3. General design/analysis trick
Our approximation algorithm often works by constructing some relaxation providing a lower bound and turning the relaxed solution into a feasible solution without increasing the cost too much.
The LP relaxation of the ILP formulation of the problem is a natural choice. We may then round the optimal LP solution.

4. Not obvious that it will work….

5. Min weight vertex cover
Given an undirected graph G=(V,E) with non- negative weights w(v) , find the minimum weight subset C ⊆ V that covers E.
Min vertex cover is the case of w(v)=1 for all v.

6. ILP formulation Find (xv)v ∈ V minimizing  wv xv so that xv ∈ Z
For all (u,v) ∈ E, xu + xv ≥ 1.

7. LP relaxation Find (xv)v ∈ V minimizing  wv xv so that xv ∈ R
For all (u,v) ∈ E, xu + xv ≥ 1.

8. Relaxation and Rounding
Solve LP relaxation.
Round the optimal solution x* to an integer solution x: xv = 1 iff x*v ≥ ½.
The rounded solution is a cover: If (u,v) ∈ E, then x*u + x*v ≥ 1 and hence at least one of xu and xv is set to 1.

9. Quality of solution found
Let z* =  wv xv* be cost of optimal LP solution.
 wv xv ≤ 2  wv xv*, as we only round up if xv* is bigger than ½.
Since z* ≤ cost of optimal ILP solution, our algorithm has approximation ratio 2.

10. Relaxation and Rounding
Relaxation and rounding is a very powerful scheme for getting approximate solutions to many NP-hard optimization problems.
In addition to often giving non-trivial approximation ratios, it is known to be a very good heuristic, especially the randomized rounding version.
Randomized rounding of x ∈ [0,1]: Round to 1 with probability x and 0 with probability 1-x.

11. Min set cover
Given set system S1, S2, …, Sm ⊆ X, find smallest possible subsystem covering X.

12. Min set cover vs. Min vertex cover
Min set cover is a generalization of min vertex cover.
Identify a vertex with the set of edges adjacent to the vertex.

13. Greedy algorithm for min set cover

14. Approximation Ratio
Greedy-Set-Cover does not have any constant approximation ratio Even true for Greedy-Vertex-Cover!
Quick analysis: Approximation ratio ln(n)
Refined analysis: Approximation ratio Hs where s is the size of the largest set and Hs = 1/1 + 1/2 + 1/ /s is the s’th harmonic number.
s may be small on concrete instances. H3 = 11/6 < 2.

15. Approximation Schemes
Some optimization problems can be approximated very well, with approximation ratio 1+ for any >0.
An approximation scheme takes an additional input, >0, and outputs a solution within 1+ of optimal.

16. PTAS and FPTAS
An approximation scheme is a Polynomial Time Approximation Scheme, if for every fixed >0, the algorithm runs in polynomial time in the input length n.
An approximation scheme is a Fully Polynomial Time Approximation Scheme, if the algorithm runs in time polynomial in n and in 1/.

17. Knapsack problem
Given n items with weights w1,...,wn , values v1,...,vn and weight limit W,
fit items within weight limit maximizing total value.

18. FPTAS for Knapsack
From tutorials: We have a pseudo-polynomial time algorithm for Knapsack in time O(n2V), where V is largest value.
We use this in step 4.