CS 332 : Algorithms - Concept of NP Completeness

CS 332 : Algorithms - Concept of NP Completeness

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  David Luebke 1 06/12/25 CS332: Algorithms NP Completeness
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  David Luebke 2 06/12/25 Review:Dynamic Programming ● When 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?
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  David Luebke 3 06/12/25 Review:Greedy Algorithms ● A 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
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  David Luebke 4 06/12/25 Review:Activity-Selection Problem ● The 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?
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  David Luebke 5 06/12/25 Review:Activity-Selection ● Formally: ■ Given a set S of n activities ○ si = start time of activity i fi = 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?
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  David Luebke 6 06/12/25 Review:Activity-Selection ● Formally: ■ Given a set S of n activities ○ si = start time of activity i fi = 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: si  fk}
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  David Luebke 7 06/12/25 Review:Greedy Choice Property 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.
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  David Luebke 8 06/12/25 Review: TheKnapsack Problem ● The 0-1 knapsack problem: ■ A thief must choose among n items, where the ith item worth vi dollars and weighs wi 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?
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  David Luebke 9 06/12/25 NP-Completeness ●Some problems are 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(nk ) for some constant k ■ Polynomial time: O(n2 ), O(n3 ), O(1), O(n lg n) ■ Not in polynomial time: O(2n ), O(nn ), O(n!)
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  David Luebke 10 06/12/25 Polynomial-TimeAlgorithms ● Are some 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
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  David Luebke 11 06/12/25 NP-CompleteProblems ● 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
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  David Luebke 12 06/12/25 AnNP-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?
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  David Luebke 13 06/12/25 Pand NP ● As 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?
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  David Luebke 14 06/12/25 Nondeterminism ●Think of a 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
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  David Luebke 15 06/12/25 Pand NP ● Summary 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?)
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  David Luebke 16 06/12/25 NP-CompleteProblems ● We will 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(n100 ) time, you’ve proved that P = NP. Retire rich & famous.
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  David Luebke 17 06/12/25 Reduction ●The crux of 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
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  David Luebke 18 06/12/25 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: (x1, x2, …, xn) = (y1, y2, …, yn) where yi = 1 if xi = TRUE, yi = 0 if xi = 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?
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  David Luebke 19 06/12/25 UsingReductions ● If P 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
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  David Luebke 20 06/12/25 ComingUp ● Given one 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
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  David Luebke 21 06/12/25 TheEnd