1. MIT and James Orlin © 2003 1 NP-hardness

2. MIT and James Orlin © 2003 2 Moving towards complexity theory Linear Programming is viewed as easy and Integer Programming is viewed as hard” Next, we address some theoretical ways of characterizing easy vs. hard problems Often referred to as the theory of NP- completeness or NP-hardness

3. MIT and James Orlin © 2003 3 Overview of complexity How can we show a problem is efficiently solvable? –We can show it constructively. We provide an algorithm and show that it solves the problem efficiently. How can we show a problem is not efficiently solvable? –How do you prove a negative? –This is the aim of complexity theory, which is the topic of today’s lecture. –The approach today is non-standard in that it covers half of the usual definitions of an intro to complexity

4. MIT and James Orlin © 2003 4 What do we mean by a problem? C onsider maximize 3x + 4y subject to 4x + 5y  23 x  0, y  0 This is an “instance” of linear programming. When we say the linear programming problem, we refer to the collection of all instances. Similar, the integer programming problem (or integer programming) refers to the collection of all instances of integer programming The traveling salesman problem refers to all instances of the traveling salesman problem etc.

5. MIT and James Orlin © 2003 5 Instances versus problem Complexity theory addresses the following problem: when is a problem hard? Note: it does not deal with the question of whether any instance is hard.

6. MIT and James Orlin © 2003 6 General Fact As problem instances get larger, the time to solve the problem grows. But how fast? We say that a problem is solvable in polynomial time if there is a polynomial p( ) such that the time to solve a problem of size n is at most p(n).

7. MIT and James Orlin © 2003 7 Example: Sorting a list of n items by using a greedy approach 439121411710824534912141171082453491214117108245 3 45 7 3491214111082457 8 3491214111024578 910 34912141124578910 11 1434912142457891011 122414 Greedy sorting: for i = 1 to n, choose the ith least item on the list and put it in position i.

8. MIT and James Orlin © 2003 8 A quick analysis of greedy sorting 3491214111082457 8 Suppose that there are n items. How many have to be scanned to find the next largest item? How much time does it take to insert the next largest item in the correct place? Claim: the running time is at most 100 n 2. This is a polynomial time algorithm. The best possible time for sorting is around n log n.

9. MIT and James Orlin © 2003 9 General Fact Examples: –Finding an word in a dictionary with n entries. Time  log n, depending on assumptions. Polynomial time –Sorting n items time  n log n Polynomial time –Finding the shortest path from s to t time  n 2. Polynomial time –Complete enumeration of a binary integer program on n variables: time > 2 n. Exponential time (not polynomial time)

10. MIT and James Orlin © 2003 10 Running times as n grows

11. MIT and James Orlin © 2003 11 Overview of next few slides Easy problems: running time is guaranteed to grow slower than a polynomial in the size of the input. Hard problems are everything else. Next: what do we mean by the size of the input?

12. MIT and James Orlin © 2003 12 Polynomial Time Algorithms For any instance I of a problem, let S(I) be the number of inputs. –Examples. For an integer programming instance, S  m x n –For a capital budgeting instance, S  n. –What would the size of a TSP on n cities be? Let M(I) be the largest integer in the data. –We assume that integers are expressed in binary. –Consider the problem of determining whether a number M is prime. It’s size grows as log M. (The size is not just 1). Size(I) is the number of digits to represent I. S(I) + log M(I)  Size(I)  S(I)  log M(I)

13. MIT and James Orlin © 2003 13 Polynomial time algorithms The algorithm A for problem X runs in polynomial time if the number of steps taken by A on any instance I is bounded by a polynomial in size(I). Equivalently, there is some polynomial p, so that for every instance I, the number of steps taken by A is less than p(S(I) + log M(I))). e.g., Suppose the number of steps is less than 1000  size(I) 7  1000  [S(I)  log M(I)] 7  1000  [S(I) + log M(I)] 14 Interesting fact: everyone in complexity talks about the size of the problem, but almost no one cares about measuring it precisely.

14. MIT and James Orlin © 2003 14 On Polynomial Time Algorithms We consider a problem X to be “easy” or efficiently solvable, if there is a polynomial time algorithm A for solving X. We let P denote the class of problems solvable in polynomial time. Some more problems that are in the class P include: –Linear Programming –The assignment problem and transportation problem and minimum cost flow problem –Finding a topological order –Finding a critical path –Finding an Eulerian cycle

15. MIT and James Orlin © 2003 15 Which are polynomial time algorithms? (answer with partner). To determine whether M is prime, one can divide M by every integer less than M. This takes at most M divisions. Dijkstra’s algorithm solves a shortest path problem in at most 100 n 2 steps. The number of steps taken by one variant of the simplex algorithm on the minimum cost flow problem is 1000 n log n pivots. Linear programming can be solved by a technique called the ellipsoid algorithm in at most n log (M) iterations, where each iteration takes at most 1000 n 3 steps.

16. MIT and James Orlin © 2003 16 Can integer programming be solved in polynomial time? FACT: every algorithm that has ever been developed for integer programming takes exponential time. Hundreds of very smart researchers have tried to come up with polynomial time algorithms for integer programming, and failed. It is generally believed that there is no polynomial time algorithm for integer programming. Complexity theory: deals with proving that integer programming is hard.

17. MIT and James Orlin © 2003 17 Hard problems in practice What can you say to your manager if he or she hands you a problem that is too difficult for you to solve. (adapted from Garey and Johnson)

18. MIT and James Orlin © 2003 18 I cant’ find an efficient algorithm. I guess I’m too dumb.

19. MIT and James Orlin © 2003 19 I cant’ find an efficient algorithm, because no such algorithm is possible

20. MIT and James Orlin © 2003 20 I can’t find an efficient algorithm, but neither can these famous researchers.

21. MIT and James Orlin © 2003 21 David Johnson George Dantzig Richard Karp Michael Garey Ralph Gomory Alan Turing Kurt Gödel John Von Neumann Steven Cook

22. MIT and James Orlin © 2003 22 The class NP-easy Consider an optimization problem X, in which for any instance I, the goal is to find a feasible solution x for I with maximum value f I (x). (or minimum) We say that X is NP-Easy if there is a polynomial p( ) with the following properties: For every instance I of X 1.There is an optimal solution x for I such that size(x) < p(size( I )). “There is a small sized optimum solution” 2.For any proposed solution y, one can evaluate whether y is feasible in fewer than p(size( I )+ size(y)) steps. “One can efficiently check feasibility” 3.The number of steps to evaluate f(y) is fewer than p(size( I )+ size(y)). “One can efficiently evaluate the objective function.”

23. MIT and James Orlin © 2003 23 The housing problem 400 students applied in the lottery for a wonderful new dorm that holds 100 students. You have a list of pairs of incompatible students. –no two incompatible students are in the list of 100 students chosen for the dorm –Is there an efficient procedure for finding the list of 100 students.

24. MIT and James Orlin © 2003 24 The dorm problem is the independent set problem Create a node for each student Create an arc between two students if they are incompatible A set S of nodes of a graph G = (N, A) are independent if no two nodes of S are adjacent. What is the largest size independent set in G? 1 2 3 5 6 8 4 7 9 10 2 6 This problem is NP-easy

25. MIT and James Orlin © 2003 25 0-1 Integer Programming is NP-easy Checking whether 0-1 integer programming is NP-easy. 1.There is an optimal solution x for I such that size(x) < p(size( I )). “There is a small sized optimum solution” --every solution is an n-vector of 0’s and 1’s 2.For any proposed solution y, one can evaluate whether y is feasible in fewer than p(size( I )+ size(y)) steps. “One can efficiently check feasibility” --evaluating whether a 0-1 vector is feasible means checking each constraint 3.The number of steps to evaluate f(y) is fewer than p(size( I )+ size(y)). “One can efficiently evaluate the objective function.” –evaluating f(x) is to evaluate cx for some linear function c.

26. MIT and James Orlin © 2003 26 Some More NP-easy Problems TSP Is there is a small sized optimum solution? Can one check feasibility efficiently? Can one evaluate the obective function efficiently?

27. MIT and James Orlin © 2003 27 Question of the millennium: is there a polynomial time algorithm for all NP-easy problems? If you can successfully answer this question, you will get $1,000,000. Millennium Prize

28. MIT and James Orlin © 2003 28 On NP-easy problems Theorem. If problem X is NP-easy, and if Y is a special case of X, then Y is NP-easy. Example. 0-1 integer programming is NP-easy Capital budgeting is a special case of 0-1 integer programming. Therefore Capital budgeting is NP- easy. “If a problem is easier than an NP-easy problem it is NP-easy.”

29. MIT and James Orlin © 2003 29 Other problems that are NP-easy Set cover problem (fire station problem) Capital budgeting problem Determining the largest prime number less than n that divides integer n –solutions are any numbers that divide n –size of any “solution x” is log x < log n –A solution can be checked as a divisor in polynomial time Also, any problem that is a special case of an NP- easy problem is NP-easy. –So determining if a number is prime is NP-easy

30. MIT and James Orlin © 2003 30 On NP-easy optimization problems Almost any optimization problem that you see will ever want to solve will be NP-easy. It’s a challenge to find optimization problems that are not NP-easy. The next two slides illustrate problems that are not NP-easy.

31. MIT and James Orlin © 2003 31 A problem that is not NP-easy. Problem: INPUT: an integer n Optimization: find the smallest integer N such that N > n, and both N and N+2 are primes. It is possible that the size of optimum solution, which is log N, is exponentially large in the size of the problem instance, which is log n. So, this violates condition 1: the size of the optimum solution may be exponential in the size of the problem instance.

32. MIT and James Orlin © 2003 32 Another Problem that is not NP-easy Consider an integer program: minimize cx + dy subject to Ax + By = b x, y binary n-vectors (that is, n components) We say that y is a “blocking n-vector” if it is a binary n- vector, and there is no feasible solution to Ax + By = b with x binary. What is the least cost blocking n-vector? But to check whether y is a feasible blocking n-vector requires the solution of an integer program. This violates condition 2 since checking whether a solution is feasible can take exponential time.

33. MIT and James Orlin © 2003 33 NP-easy Almost every optimization problem that you will ever see is NP-easy. Question. Can the NP-easy problems be solved in polynomial time? This is a very famous unsolved problem in mathematics. It is often represented as “Does P = NP?” Amazing Fact 1: If 0-1 integer programming can be solved in polynomial time, then every other NP-easy problem can be solved in polynomial time. Amazing Fact 2: If the traveling salesman problem (or the capital budgeting problem, or the independent set problem) can be solved in polynomial time, then every other NP-easy problem can be solved in polynomial time.

34. MIT and James Orlin © 2003 34 The class NP-hard An oracle function is a “black box” for solving an optimization problem. An oracle function for integer programming would take an integer programming instance as input and produce a solution in 1 time unit. Let X be an optimization problem. We say that X is NP-hard if every NP-easy problem can be solved in polynomial time if one is permitted to use an oracle function for X. Theorem: 0-1 integer programming is NP-hard.

35. MIT and James Orlin © 2003 35 NP-equivalence and other classes We say that a problem is NP-equivalent if it is both NP hard and NP-easy. NP-hard NP-easy NP-equivalent NP-complete P

36. MIT and James Orlin © 2003 36 On NP-hardness Theorem. If problem X is NP-hard, and if X is a special case of Y, then Y is NP-hard. Example. 0-1 integer programming is NP-hard 0-1 integer programming is a special case of integer programming. Therefore, integer programming is NP-hard. “If a problem is harder than an NP-hard problem it is NP-hard.”

37. MIT and James Orlin © 2003 37 Some examples of NP-hard problems Traveling Salesman Problem Capital Budgeting Problem (knapsack problem) Independent Set Problem Fire Station Problem (set covering) 0-1 Integer programming Integer Programming Project management with resource constraints and thousands more

38. MIT and James Orlin © 2003 38 Proving that a problem is hard “To prove that problem X is hard, find a problem Y that you know is hard, and show that problem X is easier than Y” To prove that a problem X is NP-hard, start with a “similar” NP-hard problem Y. Then show that Y can be solved in polynomial time if one permits X to be used as a subroutine and counting each solution of x as taking 1 step.

39. MIT and James Orlin © 2003 39 On Proving NP-hardness results Suppose that we know that the problem of determining a hamiltonian cycle is NP-hard. We will show that the problem of finding an a hamiltonian path is also NP-hard. A hamiltonian cycle is a cycle that passes through each node exactly once. A hamiltonian path is a path that includes every node of G. Proof technique: Start with any instance of the hamiltonian cycle problem. We denote this instance as G = (N, A). Transformation proofs (these are standard). Create an instance G’ = (N’, A) for the hamiltonian path problem from G with the following property: there is a hamiltonian path in G’ if and only if there is a hamiltonian cycle in G.

40. MIT and James Orlin © 2003 40 A transformation 1 121 0 22 The original network The transformed network: node 1 of the original network was split into nodes 1 and 21, and nodes 0 and 22 were connected to the split nodes.

41. MIT and James Orlin © 2003 41 Claim 1: If there is a hamiltonian cycle in the original graph then there is a hamiltonian path in the transformed graph. 1 A Hamiltonian Cycle. Connect one to node 1, and the other to node 21. Add in arcs (0,1) and (21, 22). 121 0 22 Take the two arcs in G incident to the node 1.

42. MIT and James Orlin © 2003 42 Claim 1: If there is a hamiltonian path in the transformed graph then there is a hamiltonian cycle in the original graph. 1 Delete the two arcs (0, 1) and (21, 22). Then take the other arcs in G’ incident to 1 and 21, and make them incident to node 1 in G. 121 0 22 A Hamiltonian Path

43. MIT and James Orlin © 2003 43 Proofs of NP-hardness transformations have two parts Original instance I, and transformed instance I ’. Part 1. An optimal (or feasible) solution for I induces an optimal (or feasible) solution for I ’. Part 2. An optimal (or feasible) solution for I ’ induces and optimal (or feasible) solution for I. Formulating problems as integer programs illustrates the type of transformation. Note: transformation can be difficult to develop. Great reference: Garey and Johnson 1979.

44. MIT and James Orlin © 2003 44 Summary on complexity theory. Polynomial time algorithms NP-easy, NP-hard, NP-equivalent Proving hardness results. NP-hard NP-easy NP-equivalent NP-complete P