Algorithms and Complexity Theory. Chapter 8: Introduction to Complexity. Computer Science - Durban - September 2005
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1 Algorithms and Complexity Theory Chapter 8: Introduction to Complexity Jules-R Tapamo Computer Science - Durban - September 2005 Contents 1 Introduction Dynamic programming Heuristics: Travelling Salesperson problem Polynomial Problems Examples of Polynomial Problems Examples of non Polynomial Problems Decision Problems Examples decision Problem Polynomial reduction Example of Polynomial reductions HC P T SP Properties of Polynomial reductions N P Class N P Structure N P-complete Problems How to Show that a problem is N P-complete Propositional Calculus Some Known N P-complete problems P = N P?
2 2 1 INTRODUCTION 1 Introduction A paradigm is a model or example of environment and methodology in which systems and softwares are developed and operated. Algorithmic schema use to design an algorithmic solution for a given problem. - Divide-and-conquer (Mergesort, Convex hull, etc). - Dynamic Programming (Matrix-Chain-Multiplication, All pairs Shortest Path,etc.) - etc. An heuristic is an approximation model, based on domain knowledge from a particular application, that give guidance in the solution of problem. - Travelling Salesperson problem The complexity classes allow to classify problems and to give common algorithmic solution to some problems using transformations called adequate reduction. 1.1 Dynamic programming - Is an example of paradigm - suppose we want to multiply a sequence of n matrices M 1 M 2... M n It is possible to give a method to compute using parenthesis: M 1 M 2 M 3 M 4 can be computed using one the five following schemas : 1. M1(M2(M3M4)) 2. M1((M2M3)M4) 3. (M1M2)(M3M4) 4. (M1(M2M3))M4 5. ((M1M2)M3)M4. For n matrices there are T (n) = 1 n ( 2(n-1) n-1 ways to compute M 1 M 2... M n Each way has a cost: the number of scalar multiplications required to compute the product. The cost depends on the schema. We want to minimize the cost of the multiplication. From the chapter on Dynamic programming we know that to compute: ) M 1 M 2... M n we can use an O(n 3 ) time and O(n 2 ) space algorithm to compute the best way to do it. 1.2 Heuristics: Travelling Salesperson problem The salesperson wants to minimize the total travelling cost (cost, or distance) required to visit all the cities( n cities ) in a territory, and returns to the starting point. Means given a complete, weighted graph and an integer k, is there a hamiltonian cycle with total weight at most k. - It is an exponential solution O(2 n ). Search algorithms, backtracking, branch-and-bound, etc. - one prefers to compute efficiently the approached solution: Compute the Minimum Spanning Tree
3 3 And then Depth-First-Search the minimum spanning tree then cancel multiple occurrences of cities and create short-cut. 2 Polynomial Problems When can we say that an algorithm is efficient? Will we require that the time complexity be an O(n 3 ), O(n 2 ) or yet O(n)? There is not absolute answer to this question, but some significant limitations we can point out! An exponential complexity (O(c n ) for n > 1) is almost always excessive. We will then consider as acceptable all function with polynomial complexity, means O(n k ) for a given k. In other words, an efficient algorithm will be a polynomial algorithm. The class P (Polynomial) is the set of problems which can be solved by deterministic polynomial algorithm. 2.1 Examples of Polynomial Problems EXAMPLE Quicksort of n items - product of 2 n-digits integers - etc. 2.2 Examples of non Polynomial Problems EXAMPLE Travelling Salesperson Problem (TSP). Given a set of n cities, a distance D, and for every pair of cities, the distance d(v i, v j ) from the city v i to the city v j. The salesperson wants to minimize the total traveling cost (cost, or distance) required to visit all the cities( n cities ) in a territory, and returns to the starting point. Means to find a permutation v p1, v p2,..., v pn such that 1 i<n d(v pi, v pi+1 ) + d(v pn, v p1 ) D There is a straightforward algorithm for the problem: - compute the sum for all the possible permutation. but, the number of permutation is equal to n!. and then the algorithm is not polynomial.
4 4 4 POLYNOMIAL REDUCTION EXAMPLE Hamiltonian cycle (HC) In an undirected graph is a simple cycle that passes through every vertex exactly once. The problem is to determine if there exists an hamiltonian cycle in a graph G=(V,E). Means we have to find a permutation v p1, v p2,..., v pn of vertices such that (v pi, v pi+1 ) E, i 1 i < n and (v pn, v p1 ) E. As for TSP to try all the permutations is an algorithm which can allow us to solve the problem. But this algorithm is not polynomial. It is possible to build a relation between the two problems. 3 Decision Problems A Decision Problem is a question that has two possible answers, Yes and No. The question is about some input. A Problem Instance is the combination of the problem and a specific input. Usually the statement has two part: 1. Instance Description part defines the information expected in input. 2. The question part states the actual Yes-or-No question; the question contains variables defined in the instance description. 3.1 Examples decision Problem EXAMPLE TSP: Given a complete, Weighted graph and an integer D, Is there an Hamiltonian cycle with total weight at most D? EXAMPLE HC : Does a given Undirected Graph have a Hamiltonian cycle? 4 Polynomial reduction Let P 1 and P 2 be two decision problems. A polynomial reduction from P 1 to P 2 is a function, f, from the input set for the decision problem P 1 into the input set for the decision problem Q such that the following hold: 1. f can be computed in polynomial time. 2. f computes from an instance x of problem P 1 an instance f(x) of problem P 2 such that the instance x is true if and only if the instance f(x) is true. A problem P 1 is polynomially reducible to P 2 if there exists a polynomial reduction ( transformation) from P 1 to P 2. The notation is P 1 P P Example of Polynomial reductions There exists a polynomial transformation from HC to TSP. To establish this property, describe the polynomial algorithm which transforms an instance of HC to instance of TSP which is true if and only if the instance of HC is true. Let an instance of HC be defined by G = (V,E). The transformation of this instance is the instance of TSP (C, d(v i, v j ), D) describe as follows:
5 4.2 HC P T SP 5 C is the set of cities, C = V the distances { are given by 1 if (vi, v d(v i, v j ) = j ) E 2 if (v i, v j ) / E the constant D is the number of cities ie : D = V This transformation is polynomial because: - it takes the list of vertices to create the list of cities - determines the distances between every pair of cities from the description of the edges of graph. Each of those two operations can be achieved in polynomial time. 4.2 HC P T SP Suppose first the instance of TSP we have build is true. If there exists a traversal of cities of length D = V, this traversal can only contains path between cities, means with length 1. Indeed, as each distance is 1 or 2 and the traversal contains V paths, to have length V each path must have length 1. But each path between cities with length 1 corresponds to an edge of graph G defining the instance of HC. Then a traversal of cities with length D corresponds to a hamiltonian cycle of G. If the instance of HC is true, there exists an hamiltonian cycle in G. Using the corresponding instance in TSP, the cycle defines a traversal of cities using only paths with length 1 and then the total length is V. Thus the TSP instance we will obtain is then true. 4.3 Properties of Polynomial reductions Lemma 1 if P 1 P P 2 then Proof if P 2 P then P 1 P if P 1 / P then P 2 / P if P 2 P then P 1 P If there exists a polynomial algorithm to solve P 2, as an instance of P 1 can be reduce to an instance of P 2 then it exists an polynomial algorithm to solve P 1. if P 1 / P then P 2 / P If P 1 / P, if we conclude that P 2 P then from the first part we obtain P 1 P which is a contradiction. Lemma 2 The polynomial transformations are transitive: if P 1 P P 2 and P 2 P P 3 then P 1 P P 3 Proof The transformation from P 1 to P 3 is obtained by composition of the transformations from P 1 to P 2 and the transformation from P 2 to P 3.
6 6 5 N P CLASS Two problems are polynomially equivalent, notation P 1 P P 2, if and only if P 1 P P 2 and P 2 P P 1 Example HC and TSP are polynomially equivalent 5 N P Class The class N P (Non Deterministic Polynomial) is the set of problems which can be solved by non deterministic polynomial algorithm. Non deterministic allow to choose an instruction within a set of instructions such that algorithm converges to the solution if it exists. HC and TSP are in N P. HC is N P the following non deterministic algorithm solves the problem: The algorithm generates in a non deterministic manner a permutation of the vertices of graph. The number of steps of this phase is proportional to the number of vertices of graph, which is polynomial. then the algorithm verifies that the permutation computed in the first part is an hamiltonian cycle, and the answer is true of false. 5.1 N P Structure Lemma 3 The N P class contains P ( P N P) Proof Immediate, because deterministic polynomial algorithms are particular cases of Non deterministic polynomial algorithms. 5.2 N P-complete Problems A problem P is N P-hard if every Q in N P is reducible to P (Q P P). A problem P is N P-complete if: 1. P N P 2. P is N P - hard Theorem If any N P-complete problem is in P, then P = N P Proof if P P and P is N P-complete = P N P and Q N P,Q P P. = Q N P, Q P = N P P (1) (1) and Lemma 3 = P = N P
7 5.3 How to Show that a problem is N P-complete How to Show that a problem is N P-complete To prove Q is N P-complete choose some known N P-complete problem P and reduce P to Q, 1. If P is N P-complete, all problem R N P are reducible to P; that is R P P 2. Show P P Q 3. Then all problems R N P satisfy R P Q by transitivity of reductions 4. Therefore Q is N P-complete. 5.4 Propositional Calculus Here we are interested with Conjunctive Normal Form. A Conjunctive Normal Form is an expression of form: E 1 E 2... E i... E k where E i is a clause. a clause is a disjunction of propositional variables or a negation of propositional variables, means x 1 x 2... x j... x l where x j is a literal. A literal is an expression with form p or p Example (p p) p (q r s t) (s t) An Expression is Satisfiable if there exists at least on interpretation function for which it is true. An Interpretation function is the one which assigns truth values to the propositional variables. The satisfiability problem is : Given a formula of propositional calculus in conjunctive normal form is it satisfiable? It is denoted SAT-problem. 5.5 Some Known N P-complete problems Cook s Theorem The SAT-Problem is N P-complete. Graph Coloring : Given an undirected Graph G=(V,E) and a positive integer k, is there a coloring of G using at most k colors? Knapsack Problem : Let S be a set of integers and t an integer, is there a subset A S such that x A x = t? TSP HC P = N P? Big question in Computer Science When we find an N P-complete problem, we try to use an heuristics, means approximations. And we have to give bounds of those approximations. Usually we want to have an polynomial heuristics. there are theoretical limits of computers. There is a Lack of practical and efficient algorithms.
8 8 5 N P CLASS Bibliography 1. Cormen T. H., Leiserson C.E. and Rivest R. L., Introduction to Algorithms, McGraw-Hill, Baase S. and Gelder A., Computer Algorithms: Introduction to Design and Analysis, Addison Wesley, Lewis H. R. and Papadimitriou C.H., Elements of the Theory of Computation, Prentice Hall International,1981.
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