lecture 27

lecture 27

• 1.
  CS 332: AlgorithmsNP Completeness
• 2.
  Review: Dynamic ProgrammingWhen applicable: Optimal substructure: optimal solution to problem consists of optimal solutions to subproblems Overlapping subproblems: few subproblems in total, many recurring instances of each Basic approach: Build a table of solved subproblems that are used to solve larger ones What is the difference between memoization and dynamic programming? Why might the latter be more efficient?
• 3.
  Review: Greedy AlgorithmsA greedy algorithm always makes the choice that looks best at the moment The hope: a locally optimal choice will lead to a globally optimal solution For some problems, it works Yes: fractional knapsack problem No: playing a bridge hand Dynamic programming can be overkill; greedy algorithms tend to be easier to code
• 4.
  Review: Activity-Selection ProblemThe activity selection problem : get your money’s worth out of a carnival Buy a wristband that lets you onto any ride Lots of rides, starting and ending at different times Your goal: ride as many rides as possible Naïve first-year CS major strategy: Ride the first ride, when get off, get on the very next ride possible, repeat until carnival ends What is the sophisticated third-year strategy?
• 5.
  Review: Activity-Selection Formally:Given a set S of n activities s i = start time of activity i f i = finish time of activity i Find max-size subset A of compatible activities Assume activities sorted by finish time What is optimal substructure for this problem?
• 6.
  Review: Activity-Selection Formally:Given a set S of n activities s i = start time of activity i f i = finish time of activity i Find max-size subset A of compatible activities Assume activities sorted by finish time What is optimal substructure for this problem? A: If k is the activity in A with the earliest finish time, then A - { k } is an optimal solution to S’ = { i  S : s i  f k }
• 7.
  Review: Greedy ChoiceProperty For Activity Selection Dynamic programming? Memoize? Yes, but… Activity selection problem also exhibits the greedy choice property: Locally optimal choice  globally optimal sol’n Them 17.1: if S is an activity selection problem sorted by finish time, then  optimal solution A  S such that {1}  A Sketch of proof: if  optimal solution B that does not contain {1}, can always replace first activity in B with {1} ( Why? ) . Same number of activities, thus optimal.
• 8.
  Review: TheKnapsack Problem The 0-1 knapsack problem : A thief must choose among n items, where the i th item worth v i dollars and weighs w i pounds Carrying at most W pounds, maximize value A variation, the fractional knapsack problem : Thief can take fractions of items Think of items in 0-1 problem as gold ingots, in fractional problem as buckets of gold dust What greedy choice algorithm works for the fractional problem but not the 0-1 problem?
• 9.
  NP-Completeness Some problemsare intractable : as they grow large, we are unable to solve them in reasonable time What constitutes reasonable time? Standard working definition: polynomial time On an input of size n the worst-case running time is O( n k ) for some constant k Polynomial time: O(n 2 ), O(n 3 ), O(1), O(n lg n) Not in polynomial time: O(2 n ), O( n n ), O( n !)
• 10.
  Polynomial-Time Algorithms Aresome problems solvable in polynomial time? Of course: every algorithm we’ve studied provides polynomial-time solution to some problem We define P to be the class of problems solvable in polynomial time Are all problems solvable in polynomial time? No: Turing’s “Halting Problem” is not solvable by any computer, no matter how much time is given Such problems are clearly intractable, not in P
• 11.
  NP-Complete Problems The NP-Complete problems are an interesting class of problems whose status is unknown No polynomial-time algorithm has been discovered for an NP-Complete problem No suprapolynomial lower bound has been proved for any NP-Complete problem, either We call this the P = NP question The biggest open problem in CS
• 12.
  An NP-Complete Problem:Hamiltonian Cycles An example of an NP-Complete problem: A hamiltonian cycle of an undirected graph is a simple cycle that contains every vertex The hamiltonian-cycle problem: given a graph G, does it have a hamiltonian cycle? Draw on board: dodecahedron, odd bipartite graph Describe a naïve algorithm for solving the hamiltonian-cycle problem. Running time?
• 13.
  P and NPAs mentioned, P is set of problems that can be solved in polynomial time NP ( nondeterministic polynomial time ) is the set of problems that can be solved in polynomial time by a nondeterministic computer What the hell is that?
• 14.
  Nondeterminism Think ofa non-deterministic computer as a computer that magically “guesses” a solution, then has to verify that it is correct If a solution exists, computer always guesses it One way to imagine it: a parallel computer that can freely spawn an infinite number of processes Have one processor work on each possible solution All processors attempt to verify that their solution works If a processor finds it has a working solution So: NP = problems verifiable in polynomial time
• 15.
  P and NPSummary so far: P = problems that can be solved in polynomial time NP = problems for which a solution can be verified in polynomial time Unknown whether P = NP (most suspect not) Hamiltonian-cycle problem is in NP : Cannot solve in polynomial time Easy to verify solution in polynomial time ( How? )
• 16.
  NP-Complete Problems Wewill see that NP-Complete problems are the “hardest” problems in NP: If any one NP-Complete problem can be solved in polynomial time… … then every NP-Complete problem can be solved in polynomial time… … and in fact every problem in NP can be solved in polynomial time (which would show P = NP ) Thus: solve hamiltonian-cycle in O( n 100 ) time, you’ve proved that P = NP . Retire rich & famous.
• 17.
  Reduction The cruxof NP-Completeness is reducibility Informally, a problem P can be reduced to another problem Q if any instance of P can be “easily rephrased” as an instance of Q, the solution to which provides a solution to the instance of P What do you suppose “easily” means? This rephrasing is called transformation Intuitively: If P reduces to Q, P is “no harder to solve” than Q
• 18.
  Reducibility An example:P: Given a set of Booleans, is at least one TRUE? Q: Given a set of integers, is their sum positive? Transformation: (x 1 , x 2 , …, x n ) = (y 1 , y 2 , …, y n ) where y i = 1 if x i = TRUE, y i = 0 if x i = FALSE Another example: Solving linear equations is reducible to solving quadratic equations How can we easily use a quadratic-equation solver to solve linear equations?
• 19.
  Using Reductions IfP is polynomial-time reducible to Q, we denote this P  p Q Definition of NP-Complete: If P is NP-Complete, P  NP and all problems R are reducible to P Formally: R  p P  R  NP If P  p Q and P is NP-Complete, Q is also NP-Complete This is the key idea you should take away today
• 20.
  Coming Up Givenone NP-Complete problem, we can prove many interesting problems NP-Complete Graph coloring (= register allocation) Hamiltonian cycle Hamiltonian path Knapsack problem Traveling salesman Job scheduling with penalities Many, many more
• 21.
  The End