Approximation Algorithm1(近似算法(一))(Introduction to Algorithms, 算法导论，CLRS)学习笔记

Approximation Algorithm

1. Approximation ratio

Cost: the size of the solution, for example, in vertex cover, it’s the size of the cover; in TSP, it’s the total distance.

Since the approximation algorithm, the cost we have is always greater than the optimal solution.

2. Vertex Cover

• Running time: $O(|V|+|E|)$, loop over all edges and all vertices.

• 

  $|A|$: the number of edges chosen by the algorithm, and since one edge covers two vertices, so $|A|=|C|/2$. And since this is an approximation algorithm, and at least one of vertices chosen each time must be in $C$, thus $|C|$ must be greater than the edges chosen by the algorithm.

3. Traveling Salesperson

3.1 Algorithm

• Construct a minimum spanning tree, whose sum of edge weights is as small as possible.

• Remember to skip duplicates;

• Proof to the theorem:

  • Since the distance in MIS is less than the distance in TSP; $c(T)\le c(H^{\ast })$

  • the sum of weights in Euler tour is twice the sum in MIS, since we go through each vertex twice; $c(W)=2c(T)$

  • the sum of weight returned by the algorithm is less than the sum of Euler tour, since we will skip duplicates: $c(H)\le c(W)$

  • Above all, we have: $c(H)\le 2c(H^{\ast })$.

4. Set Cover

Find a set of subsets, which can cover the big set.

4.1 Greedy Algorithm

Intuition: each time pick the remaining largest subset, until the set is covered.

$|X|$: the size of the big set.

• $c_x:$ if element $x$ is covered for the first time by $S_i$<