1. Lecture 30. P, NP and NP Complete Problems 1

2. Recap Data compression is a technique to compress the data represented either in text, audio or image form. Two important compress techniques are lossy and lossless compression. LZW is the foremost technique for general purpose data compression due to its simplicity and versatility. LZW compression uses a code table, with 4096 as a common choice for the number of table entries. 2

3. Optimization & Decision Problems Decision problems – Given an input and a question regarding a problem, determine if the answer is yes or no Optimization problems – Find a solution with the “best” value Optimization problems can be cast as decision problems that are easier to study – E.g.: Shortest path: G = unweighted directed graph Find a path between u and v that uses the fewest edges Does a path exist from u to v consisting of at most k edges? 3

4. Algorithmic vs Problem Complexity The algorithmic complexity of a computation is some measure of how difficult is to perform the computation (i.e., specific to an algorithm) The complexity of a computational problem or task is the complexity of the algorithm with the lowest order of growth of complexity for solving that problem or performing that task. – e.g. the problem of searching an ordered list has at most lgn time complexity. Computational Complexity: deals with classifying problems by how hard they are. 4

5. Computational Complexity Theory In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given computational problem. The relative computational difficulty of computable functions is the subject matter of computational complexity. Complexity theory analyzes the difficulty of computational problems in terms of many different computational resources. Example: looking up something in a dictionary has only logarithmic complexity because a double sized dictionary only has to be opened one time more (e.g. exactly in the middle - then the problem is reduced to the half). 5

6. Complexity Classes A complexity class is the set of all of the computational problems which can be solved using a certain amount of a certain computational resource. The complexity class P is the set of decision problems that can be solved by a deterministic machine in polynomial time. This class corresponds to an intuitive idea of the problems which can be effectively solved in the worst cases. The complexity class NP is the set of decision problems that can be solved by a non-deterministic machine in polynomial time. This class contains many problems that people would like to be able to solve effectively. All the problems in this class have the property that their solutions can be checked effectively. 6

7. Complexity Classes (taxonomy) Decision Problem Undecidable Type 0 (recursively enumerable) Decidable PSPACE Type 1 (context sensitive) PSPACE-Complete Co-NP NP NP-Complete P P-Complete Type 2 (context free) Type 3 (regular) Strict subset relationship Set equality is unknown 7

8. Deterministic (Turing) Machine Deterministic or Turing machines are extremely basic symbol-manipulating devices which — despite their simplicity — can be adapted to simulate the logic of any computer that could possibly be constructed. They were described in 1936 by Alan Turing. Though they were intended to be technically feasible, Turing machines were not meant to be a practical computing technology, but a thought experiment about the limits of mechanical computation; thus they were not actually constructed. Studying their abstract properties yields many insights into computer science and complexity theory. Turing machines capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or 'mechanical procedure'. 8

9. Nondeterministic (Turing) Machine In theoretical computer science, a non- deterministic Turing machine (NTM) is a Turing machine whose control mechanism works like a non-deterministic finite automaton. An ordinary (deterministic) Turing machine (DTM) has a transition function that, for a given state and symbol under the tape head, specifies three things: – the symbol to be written to the tape – the direction (left or right) in which the head should move – the subsequent state of the finite control An NTM differs in that the state and tape symbol no longer uniquely specify these things - many different actions may apply for the same combination of state and symbol. 9

10. Complexity Class P P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. P is often taken to be the class of computational problems which are "efficiently solvable" or "tractable“. Problems that are solvable in theory, but cannot be solved in practice, are called intractable. There exist problems in P which are intractable in practical terms; for example, some require at least n 1000000 operations. P is known to contain many natural problems, including the decision versions of linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P. 10

11. Complexity Class NP In computational complexity theory, NP ("Non-deterministic Polynomial time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. It is the set of problems that can be "verified" by a deterministic Turing machine in polynomial time. All the problems in this class have the property that their solutions can be checked effectively. This class contains many problems that people would like to be able to solve effectively, including – the Boolean satisfiability problem (SAT) – the Hamiltonian path problem (special case of TSP) – the Vertex cover problem. 11

12. Complexity Class NP-Complete In complexity theory, the NP-complete problems are the most difficult problems in NP ("non-deterministic polynomial time") in the sense that they are the ones most likely not to be in P. If one could find a way to solve any NP-complete problem quickly (in polynomial time), then they could use that algorithm to solve all NP problems quickly. At present, all known algorithms for NP-complete problems require time that is super polynomial in the input size. To solve an NP-complete problem for any nontrivial problem size, generally one of the following approaches is used: – Approximation – Probabilistic – Special cases – Heuristic 12

13. Complexity Class NP-Complete (cont) Some well-known problems that are NP- complete are: – Boolean satisfiability problem (SAT) – N-puzzle – Knapsack problem – Hamiltonian cycle problem – Traveling salesman problem – Subgraph isomorphism problem – Subset sum problem – Clique problem – Vertex cover problem – Independent set problem – Graph coloring problem – Minesweeper 13

14. Complexity Classes P and NP 14

15. Class of “P” Problems Class P consists of (decision) problems that are solvable in polynomial time Polynomial-time algorithms – Worst-case running time is O(n k ), for some constant k Examples of polynomial time: – O(n 2 ), O(n 3 ), O(1), O(n lg n) Examples of non-polynomial time: – O(2 n ), O(n n ), O(n!) 15

16. Tractable/Intractable Problems Problems in P are also called tractable Problems not in P are intractable or unsolvable – Can be solved in reasonable time only for small inputs – Or, can not be solved at all Are non-polynomial algorithms always worst than polynomial algorithms? - n 1,000,000 is technically tractable, but really impossible - n log log log n is technically intractable, but easy 16

17. Example of Unsolvable Problem Turing discovered in the 1930’s that there are problems unsolvable by any algorithm. The most famous of them is the halting problem – Given an arbitrary algorithm and its input, will that algorithm eventually halt, or will it continue forever in an “infinite loop?” 17

18. Examples of Intractable Problems 18

19. Intractable Problems Can be classified in various categories based on their degree of difficulty, e.g., – NP – NP-complete – NP-hard Let’s define NP algorithms and NP problems 19

20. Nondeterministic and NP Algorithms Nondeterministic algorithm = two stage procedure: 1) Nondeterministic (“guessing”) stage: generate randomly an arbitrary string that can be thought of as a candidate solution (“certificate”) 2) Deterministic (“verification”) stage: take the certificate and the instance to the problem and returns YES if the certificate represents a solution NP algorithms (Nondeterministic polynomial) verification stage is polynomial 20

21. Class of “NP” Problems Class NP consists of problems that could be solved by NP algorithms – i.e., verifiable in polynomial time If we were given a “certificate” of a solution, we could verify that the certificate is correct in time polynomial to the size of the input Warning: NP does not mean “non-polynomial” 21

22. Hamiltonian Cycle Given: a directed graph G = (V, E), determine a simple cycle that contains each vertex in V – Each vertex can only be visited once Certificate: – Sequence:  v 1, v 2, v 3, …, v |V|  hamiltonian not hamiltonian 22

23. Is P = NP? Any problem in P is also in NP: P  NP The big (and open question) is whether NP  P or P = NP – i.e., if it is always easy to check a solution, should it also be easy to find a solution? Most computer scientists believe that this is false but we do not have a proof … P NP 23

24. NP-Completeness (informally) NP-complete problems are defined as the hardest problems in NP Most practical problems turn out to be either P or NP-complete. Study NP-complete problems … P NP NP-complete 24

25. Summary Decision problems – Given an input and a question regarding a problem, determine if the answer is yes or no Optimization problems – Find a solution with the “best” value NP-complete - means problems that are 'complete' in NP, i.e. the most difficult to solve in NP NP-hard - stands for 'at least' as hard as NP (but not necessarily in NP); NP-easy - stands for 'at most' as hard as NP (but not necessarily in NP); NP-equivalent - means equally difficult as NP, (but not necessarily in NP); 25

26. In next Lecture In next lecture, we will revised the first fifteen lectures. 26