ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS

1. ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Instructor: Dr. Gautam Das Lecture 3 January 29, 2009 Class notes by Alexandra Stefan

2. Review (see previous lecture): • optimization problem, decision problem, verification problem; • P, NP; • polynomial time reduction; • Overview of lecture: • New definitions: • Decision problem as a set, complement of a decision problem, Co-NP, NP-Complete • Example problems used or mentioned: • HCP, TSP, SP, PRIME, SATISFIABILITY (HCP = Hamiltonian Cycle, TSP = Traveling Salesman, SP = Shortest Path ) • Other things to remember: • Pay attention to the proper formulation of what constitutes the input of each problem. • It is not always easy to produce a certificate for a decision problem.

3. Definitions • NP • set of problems whose verification version can be solved in polynomial time. • Imporatnt: You need to have a certificate • Decision problems as sets: • A decision problem takes an input and produces an ‘Yes’ or ‘No’ output. • Any decision problem, X, can be defined as the set of inputs for which the answer is ‘Yes’. • The complement of a decision problem: • Given a decision problem X, the complement of X is the set of inputs for which the answer is ‘No’. • Co-NP = set of decision problems whose complement problem is in NP. • NP-Complete: A problem X is NP-Complete if: • X is NP • Any NP problem Y can be reduced in polynomial time to X. (“X should be the hardest problem in it’s class”) (Intuition: finding an NP-Complete problem is like finding the largest number in a set of numbers)

4. NP-Complete NP NP Co-NP P P

5. SP – Shortest Path • SP • Input: G (graph) ,s (vertex), t (vertex), x (value) • Output: • ‘Yes’ if there exists a path from s to t of length less or equal to x. • ‘No’, otherwise. • SP is in NP: • Certificate: list of vertices • Easy to check if the certificate is a valid path, to compute it’s length and compare it with x. • SP is in Co-NP: • Certificate: nothing • use Dijkstra to get the shortest path and it’s length, y. This runs in polynomial time. • Compare x with y. • Set version of SP: • The set of tuples: < G (graph) ,s (vertex), t (vertex), x (value) > for which the answer is ‘Yes’.

6. Hamiltonian Cycle Problem (HCP) • Definition: • Input: unweighted graph • Output: • Yes, if there exists a cycle that visits all vertices exactly once • No, otherwise • Examples Graph: Does it have a Hamiltonian cycle? Answer: No Yes

7. Hamiltonian Cycle Problem (HCP) • Is HCP in NP ? • Yes • A certificate is a list of vertices • It is easy to verify that the certificate constitutes a valid cycle in the graph and that it covers all vertices • Is HCP in Co-NP ? • yes • Since HCP is in NP

8. NP-Complete • How do you show that a problem is NP-Complete? • You have to show that any problem in NP can be reduced to it. • First problem that was shown to be NP-Complete • is the SATISFIABILITY problem. • It is known as Cook’s Theorem. • Once we have an NP-Complete problem, X, to show that another NP problem, Y, is NP-Complete, we only need to show that X reduces in polynomial time to Y.

9. Y X NP-Complete If X is NP-Complete and X reduces in polynomial time to Y then any problem can be reduced in polynomial time to Y by First reducing it to X and then reducing X to Y. (Here we used the property of polynomials of being closed under multiplication.)

10. Solver of X (using reduction of X to Y) Input for X reducer Solver of Y answer X Y X Y Universe of inputs for X Universe of inputs for Y Polynomial reduction 3 equivalent ways of saying that X reduces in polynomial time to Y: Set mapping after reduction

11. NP Co-NP P PRIME • PRIME • Input: number n (of m bits) • Output: Is n a prime number or not? • First believed to be NP. • First shown to be both in NP and Co-NP. • Then shown to be in P. PRIMES PRIMES