1. Computation Basics & NP-Completeness
박상준

2. 컴퓨터로 문제풀기 Computational Efficiency? 컴퓨팅 How the Analyzing goes?
Problem Solving
Running Time
함수 -> 기본적인 스텝의 수
How the Analyzing goes?

3. Insersion-Sort
The running time of the algorithm is the sum of running times for each statements executed

4. Running time
T(n) is expressed as an2+bn+c for constants a,b,c; it is thus a quadratic function of n.

5. O-notation
O(g(n))={f(n): there exist positive constants c and n0 such that 0 <=f(n)<= cg(n) for all n >= n0 }
It is upper bound on the worst-case running time
an2+bn+c=O(n2 )
We say “ The running time is O(n2 ) ”

6. Polynomial-time algorithms
An algorithms that ,on inputs of size n their worst-case running time is O(nk) for some constant k.
Complexity class P : the Set of decision problems that are solvable in polynomial time

7. NP-Completeness
Although problem O(n100) looks intractable, there are very few practical problems that require such a high-degree polynomial time order
P ≠ NP ? No one knows
For simplicity, the theory of NP-completeness restricts attention to decision problems: those having a yes/no solution

8. NP-Complete Problems

9. Hamiltonian cycles
Graph G=(V,E) V:vertex(정점) E:edge(간선)

10. Traveling Salesman Problems (TSP)

11. Steiner Tree
Instance: Graph G=(V,E), subset R⊆V, positive integer K <= |V|-1.
Question: Is there a subtree of G that includes all the vertices of R and that contains on more than K edges?

12. The Maximum Clique Problem
Clique : in undirected graph G=(V,E), a subset V’⊆V of vertices, each pair of which is connected by an edge in E
Size of a clique is the number of vertices it contains.
It is Exist, CLIQUE={<G,k>:G is a graph with a clique of size k}?

13. The Vertex-cover Problem
To find a vertex cover of minimum size in a given graph