1. NP-Completeness Objectives: At the end of the lesson, students should be able to: 1. Differentiate between class P, NP, and NPC 2. Reduce a known NPC problem into another NPC problem 3. Show a given problem is in NPC 4. Solve a problem using an approximate solution.

2. Content • Informal introduction - the classes P, NP and NP-complete - the graph colouring • polynomial reductions • Approximation algorithms - travelling salesman problem

6. Tractable vs intractable Problems that can be solved in poly time are called tractable

7. Undecidable Problem • Decidable (P or NP) • Undecidable problem – e.g Halting Problem Given a computer program and input to it, determine whether the program will halt on that input or continue working indefinitely on it.

8. A Problem in NPC How to show that a problem is in NPC? Prove that no efficient algorithm is likely to exist. Three key concepts for showing whether a problem is in NPC 1.Decision problems vs optimisation problems 2.Polynomial reductions 3.An initial NPC-problem

15. Graph colouring This algorithm finds a k-colouring of G=(V,E), if there is one, and stores it in the array c. Graph-colouring(G,k) { For each v in V c[v] = guess({1,2,…,k}) For each v in V For each w in N(v) if (c[w] ==c[v] return false return true }

17. Other complexity classes • There are many complexity classes that are much harder than NP. • PSPACE. Problems that can be solved using a reasonable amount of memory without regard to how much time the solution takes. • EXPTIME. Problems that can be solved in exponential time. • Undecidable. For some problems, we can prove that there is no algorithm that always solves them, no matter how much time or space is allowed.

23. Some NP-complete problems • Does a given undirected graph have a Hamiltonian cycle? • Traveling salesman problem (TSP) • Graph coloring

26. TSP is in NP • Given an instance to the TSP problem, use as “proposed solution” the sequence of n vertices in the tour e.g. ACDBA, ACDA. • Check that this sequence contains every vertex exactly once. ACDBA is yes, ACDA is no. • Sum up the edge costs and check whether this is at most k. Total cost for ACDBA = 35+12+10+20 = 77. This can all be done in polynomial-time, ths TSP ε NP.

27. Complete Graphs • there is an edge “between” every possible tuple of vertices. |e| = C(n,2) = n. (n-1)/2

29. Reduction of HC to TSP G has a Hamiltonian cycle ↔ G’ has a tour of cost at most 0 Suppose G has a Hamiltonian cycle h. Then each edge in h has cost 0 in G’. Thus h is a tour of cost 0 in G’. Suppose G’ has a tour h’ of cost at most 0. As all edges in G’ have cost 0 or 1, cost(h’) = 0, hence all edges in h’ have cost 0. Therefore, h’ is a hamiltonian cycle of G.

32. TSP

38. Question ?????

39. Summary 1. Problems fall into class P, NP, and NPC. 2. Reduction of a known NPC problem into another NPC problem (Hcycle≤p TSP) 3. Shown a given problem is in NPC (TSP) 4. Solved a problem using an approximate solution. Last updated: 2/11/2010.