Computation Basics & NP-Completeness

1. Computation Basics &NP-Completeness 박상준

2. 컴퓨터로 문제풀기 • Computational Efficiency? • 컴퓨팅 • 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