CSCI 2670 Introduction to Theory of Computing

1. CSCI 2670Introduction to Theory of Computing November 23, 2004

2. Agenda • Today • Section 7.3 • Next week • Section 7.4 November 23, 2004

3. Announcement • Office hours today 2:30 – 3:30 • No homework this week • No more graded homework • I will assign problems next week, but you won’t have to turn them in • Quiz next Wednesday (last quiz!) November 23, 2004

4. Last week • Big-O and small-o notation • Computational complexity depends on • Type of machine • Encoding • The class P • Algorithms that can be solved in O(nk) time on a single-tape deterministic TM using reasonable encoding November 23, 2004

5. Solving vs. verifying • What if we can’t solve the problem in O(nk) time? • Given a problem and a potential solution, can we verify the solution is correct? November 23, 2004

6. Example • The bin-packing problem • Given a set of n items with weights w1, w2, …, wn, and k bins that can hold a maximum weight of 1, can we place these items the bins? • There is no known O(nk) solution to this problem • What if we have a potential solution • b1, b2, …, bn 1 bi  k • Can we verify it in O(nk) time? November 23, 2004

7. Verifier M = “On input <w1, …, wn, b1, …, bn, k> • Initialize s1, s2, …, sk to 0 • For i = 1, …, n • if bi {1, 2, …, k} reject • sb_i = sb_i + wi • if sb_i > 1 reject • Next i • Accept November 23, 2004

8. The class NP Definition: A verifier for a language A is an algorithm V, where A={w|V accepts <w,c> for some string c} The string c is called a certificate of membership in A. Definition: NP is the class of languages that have polynomial-time verifiers. November 23, 2004

9. Why NP? • The N in NP stands for non-deterministic • Any language in NP can be non-deterministically solved in polynomial time using the verifier • Guess the certificate • Verify November 23, 2004

10. Other problems in NP • The vertex cover problem • Given a graph G = <V,E> and a number k in N, does there exist a subset V’ of V such that • |V’| = k • For every (u,v)E, either uV’ or vV’ • There is no known polynomial solution to this problem November 23, 2004

11. Vertex cover • Can we verify vertex cover in polynomial time? • Yes • What should the certificate be? • The subset V’ • How do we verify? • Check |V’| = k • Test that each (u,v)E has uV’ or vV’ • Takes O(|E|×k) time November 23, 2004

12. Group problem 1 • Find a verifier for the subset-sum problem • Given a finite set S  N and a target t  N, does there exist a subset S’  S such that the sum of all elements in S’ is equal to t? November 23, 2004

13. Group problem 2 • Find a verifier for the traveling-salesman problem • Given a weighted graph G (i.e., a graph where each edge has a associated weight) and a distance d, does there exist a cycle through the graph that visits each vertex exactly once (except for the start/end vertex) and has a total distance d? November 23, 2004

14. Group problem 3 • Find a verifier for the set partition problem • Given a set S of number, does there exist a subset A of S such that the sum of all elements in A is equal to the sum of all elements in S-A? November 23, 2004

15. Group problem 4 • Find a verifier for the independent set problem • Given a graph G = <V,E> and a number k, does there exist a subset V’ of V such that |V’|=k and no edge in E connects 2 elements of V’? November 23, 2004

16. Group problem 5 • Find a verifier for the graph 3-colorability problem • Given a graph G = <V,E> is there a way of assigning the colors red, blue and green to the vertices of G such that no edge connects two nodes of the same color? November 23, 2004

17. Group problem 6 • Find a verifier for the cyclic ordering problem • Given a finite set A and a collection C of three-tuples of distinct elements of A (i.e., for each (x,y,z)C, the elements x, y, and z are all in A and they are all different), is there a ono-to-one mapping f:A→{1, 2, …, |A|} such that for each (a,b,c) in C, f(a) < f(b) < f(c)? November 23, 2004

18. Gobble Gobbble!