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Introduction An algorithm with O(n3) complexity isn’t bad because it can still be run for fairly large inputs in a reasonable amount of time. In this chapter, we are concerned with problems with exponential complexity. Tractable and intractable problems Study the class of problems for which no reasonably fast algorithms have been found, but no one can prove that fast algorithms do not exist. 2

Our Old List of Problems Sorting Searching Shortest paths in a graph Minimum spanning tree Traveling salesman problem Knapsack problem Towers of Hanoi 3

TractabilityAn algorithm solves the problem inpolynomial time if its worst-case timeefficiency belongs to O(p(n)) where p(n) is apolynomial of the problem’s input size n.Problems that can be solved in polynomialtime are called tractable.Problems that cannot be solved in polynomialtime are called intractable. 4

Classifying a Problem’s ComplexityIs there a polynomial-time algorithm that solves the problem?Possible answers: yes no because it can be proved that all algorithms take exponential time because it can be proved that no algorithm exists at all to solve this problem don’t know, but if such algorithms were to be found, then it would provide a means of solving many other problems in polynomial time 5

Types of ProblemsOptimization problem: construct a solution thatmaximizes or minimizes some objective functionDecision problem: A question that has twopossible answers, yes and no. Example: Hamiltonian circles: A Hamiltonian circle in an undirected graph is a simple circle that passes through every vertex exactly once. The decision problem is: Does a given undirected graph have a Hamiltonian circle? 6

Some More Problems (1)Many problems will have decision and optimization versions Traveling salesman problem Optimization problem: Given a weighted graph, find Hamiltonian cycle of minimum weight. Decision problem: Given a weighted graph and an integer k, is there a Hamiltonian cycle with total weight at most k? Knapsack : Suppose we have a knapsack of capacity W (a positive integer) and n objects with weights w1, …, wn, and values v1, …, vn (where w1, …, wn, and v1, …, vn are positive integers) Optimization problem: Find the largest total value of any subset of the objects that fits in the knapsack (and find a subset that achieves the maximum value) Decision problem: Given k, is there a subset of the objects that fits in the knapsack and has total value at least k? 7

Some More Problems (2)Bin packing: Suppose we have an unlimited number of bins each ofcapacity one, and n objects with sizes s1, …, sn, where 0 ≤ si≤ 1( siare rational numbers) Optimization problem: Determine the smallest number of bins into which the objects can be packed(and find an optimal packing). Decision problem: Given, in addition to the inputs described, an integer k, do the objects fit in k bins?Graph coloring: coloring: assign a color to each vertex so that adjacent vertices are not assigned the same color. Chromatic number: the smallest number of colors needed to color G.We are given an undirected graph G = (V, E) to be colored. Optimization problem: Given G, determine the chromatic number . Decision problem: Given G and a positive integer k, is there a coloring of G using at most k colors? If so, G is said to be k-colorable. 8

The class PP : the class of decision problems that can be solved in O(p(n)), where p(n) is a polynomial on n.Why use the existence of a polynomial time bound as the criterion? if not, very inefficient nice closure properties Given a computer program machine independent in a strong sense and an input to it, determine whether the program will halt on thatWhat is the solvability of a decision problem? input or continue working Solvable/decidable in polynomial time indefinitely on it. Solvable/decidable but intractable Unsolvable/undecidable problems: e.g., the halting problem No polynomial algorithm has been found, nor has the impossibility of such an algorithm been proved. 9

The class NPInformally, NP is the class of decision problems for which a given proposed solution for a given input can be checked quickly(in polynomial time) to see if it really is a solution.Formally, NP: the class of decision problems that can be solved by nondeterministic polynomial (NP) algorithms A nondeterministic algorithm: a two-stage procedure that takes as its input an instance I of a decision problem and does the following “guessing” stage: An arbitrary string S is generated that can be thought of as a guess at a solution for the given instance (but may be complete gibberish as well) “verification” stage: A deterministic algorithm takes both I and S as its input and check if S is a solution to instance I, (outputs yes if s is a solution and outputs no or not halt at all otherwise) 10

The class NP A nondeterministic algorithm solves a decision problem if and only if for every yes instance of the problem it returns yes on some execution. A nondeterministic algorithm is said to be polynomially bounded if there is a polynomial p such that for each input of size n for which the answer is yes, there is some execution of the algorithm that produces a yes output in at most p(n) steps. 11

Example: graph coloring Nondeterministic graph coloring First phase: generate a string s, a list of characters, c1c2…cq ,which the second phase interpret as a proposed coloring solution Second phase: interpret the above characters as colors to be assigned to the vertices ci vi 12

Example: CNF Satisfiability The problem: Given a boolean expression expressed in conjunctive normal form(CNF), can we assign values true and false to variables to satisfy the CNF ? This problem is in NP. Nondeterministic algorithm: Guess truth assignment Make the formula true Check assignment to see if it satisfies CNF formula Example:(A⋁¬B ⋁ ¬C ) ⋀ (¬A ⋁ B) ⋀ (¬ B ⋁ D ⋁ E ) ⋀ (F ⋁ ¬ D) Truth assignments: A B CD E F1. 0 1 1 0 1 0 literals2. 1 0 0 0 0 13. 1 1 0 0 0 14. … 13

The Relationship between P & NP (1)P ⊆ NP (I ∈ P I ∈ NP): Every decision problem solvable by a polynomial time deterministic algorithm is also solvable by a polynomial time nondeterministic algorithm. To see this, observe that any deterministic algorithm can be used as the checking stage of a nondeterministic algorithm. If I ∈ P, and A is any polynomial deterministic algorithm for I, we can obtain a polynomial nondeterministic algorithm for I merely by using A as the checking stage and ignoring the guess. Thus I ∈ P implies I ∈ NP 14

The Relationship between P & NP (2) ?P = NP (NP ⊆ P?) Can the problems in NP be solved in polynomial time? A tentative view NP P 15

NP-Completeness (1)NP-completeness is the term used to describe decisionproblems that are the hardest ones in NP If there were a polynomial bounded algorithm for an NP- complete problem, then there would be a polynomial bounded algorithm for each problem in NP. Examples Hamiltonian cycle Traveling salesman Knapsack Bin packing Graph coloring Satisfiability 16

Informal Definition of NP-Completeness Informally, an NP-complete problem is a problem in NP that is as difficult as any other problem in this class, because by definition, any other problem in NP can be reduced to it in polynomial time. A decision problem D1 is said to be polynomially reducible to a decision problem D2 if there exists a function t that transforms instances of D1 to instances of D2 such that 1. t maps all yes instances of D1 to yes instances of D2 and all no instances of D1 to all no instances of D2. 2. t is computable by a polynomial-time algorithm.If D2 is polynomially solvable, then D1 can also be solved in polynomial time. 17

An Example of Polynomial Reductions Problem P: Given a sequence of Boolean values, does at least one of them have the value true? Problem Q: Given a sequence of integers, is the maximum of the integers positive? Transformation T: t(x1, x2, …, xn) = (y1, y2, …, yn), where yi = 1 if xi = true, and yi = 0 if xi = false. 18

Another Example of Polynomial ReductionsFor example a Hamiltonian circuit problem is polynomially reducible to the decision version of the traveling salesman problem. Hamiltonian circuit problem(HCP): Does a given undirected graph have a Hamiltonian cycle? Traveling salesman problem(TSP): Given a weighted graph and an integer k, is there a Hamiltonian cycle with total weight at most k? Transformation t: Map G, a given instance of the HCP to a weighted complete graph G’, an instance of the TSP by assigning 1 as the edge weight to each edge in G and adding an edge of weight 2 between any pair of nonadjacent vertices in G. Let k be n, the total vertex number in G. 19

Formal Definition of NP-CompletenessA decision problem D is said to be NP-complete if 1. It belongs to class NP. 2. Every problem in NP is polynomially reducible to D.The class of NP-complete problems is called NPC. NPCook’s theorem (1971): discover the first NP-complete NP-completeproblem, CNF-satisfiability problem.Show a decision problem is NP-complete Show that the problem is NP P Show that a known NP-complete problem can be transformed to the problem in question in polynomial time. (transitivity of polynomial reduction)Practical value of NP-completeness 20

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