NP-Completeness. Dr. B. S. Panda Professor, Department of Mathematics IIT Delhi, Hauz Khas, ND-16

Transcription

1 NP-Completeness Dr. B. S. Panda Professor, Department of Mathematics IIT Delhi, Hauz Khas, ND-16 1

2 Decision Problem A problem whose answer is either yes or no ( 1 or 0) Examples: HC Problem Instance: A graph G=(V,E) Question: Is G Hamiltonian? Planar Graph Problem Instance: A Graph G=(V,E) Question: Is G Planar? 2

3 Optimization Problem A Problem which asks for Max/Min solution Examples: TSP Problem: Instance: A weighted complete graph G=(V,E) Question: Find a HC C of G of minimum cost. Clique Problem: Instance: A graph G=(V,E) Questions: Find a clique of maximum cardinality in G. Other examples: Independent set problem, Matching problem, Shortest path problem, MST problem, Min Dom. Set Problem, 3

4 Decision Version of Optimization Problem Every opt. problem has an underlying decision problem. Introduce a parameter k and ask a question involving less than equal to (greater than equal to k) for max (for Min) problem. Examples: MAX_CP to CDP CDP Instance: A graph G=(V,E) and integer k. Question: Does G contains a clique of size at least k? 4

5 NP Efficient Verification Algorithm An algorithm B is an efficient verification algorithm for a decision problem if the following properties hold B is a polynomial time algorithm that takes two arguments, I and C, where I is an instance of and C is a certificate For every instance I of, I is an yes instance iff there exists a certificate C such that C I k, for some constant k, and B(I,C)=yes. NP={ is a decision problem that admits an efficient verification algorithm} A certificate is a characterizing property. P= ={ is a decision problem that admits a polynomial time algorithm} 5

6 P NP Thm: P NP. Proof: Let P. This implies there exists a polynomial time algorithm A that solves. Consider B(I,C) { return A(I);} Now B is a polynomial time algorithm. ( ) Let I be an yes instance. So, A(I)=yes. Choose C arbitrarily. Now B(I,C)=yes. Let B(I,C)=yes for some guess C with C k. So, A(I)=yes. This implies I is an yes instance. So, B(I,C) is an efficient verifier and hence NP. 6

7 How to show a problem to be in NP? Thm: HC is in NP. Proof: Consider B(G, =(v 1,v 2,,v n )) { for i=1 to n do { if (v i v i+1 E )return(no)}; if (v 1 v n E) return(no); return(yes); } Note that B is an efficient verifier for HC. So, HC NP. 7

8 Polynomial Reduction Problem 1 is polynomially reducible to 2, denoted by 1 P 2, if there exists a function f: I( 1 ) I( 2 ) such that f can be computed in polynomial time I Y(I( 1 )) iff f(i) Y(I( 2 )). Note that P is a transitive relation on Decision problems. Example: ISDP P CDP ISDP P VCDP 8

9 NP Complete A problem is NP-complete if 1. NP 2. P for all NP. A decision problem is NP-hard if every problem in NP is polynomially reducible to, i.e. P for all NP. Note that NP-hardness can be defined for optimization problems as well. 9

10 What does NP-Completeness Provide? NP-Completeness does not Provide a method of obtaining polynomial time algorithms for problems of 2 nd categories. Tell that polynomial time algorithm for these problems do not exist. Does show that some of these problems for which no polynomial time algorithms are known are computationally related. The relation tells that if you could solve any one of these problems then other problems in this class can also be solved in polynomial time. This suggests that these problems are hard to solve. 10

11 Some Properties of NP-complete and NP- Hard problems An NP-complete problem can be solved in polynomial time if and only if all other NPcomplete problems can be solved in polynomial time. If an NP-hard problem can be solved in polynomial time, then all the NP-complete problems can be solved in polynomial time. ALL NP-Complete Problems are NP-Hard. Some NP-Hard problems are not known to be NP-Complete. 11

12 Satisfiability Problem (SAT) Boolean variable: a variable that can take the values 0 or 1. A literal is either a variable or its negation. Formula: consists of literals and the operators and and or (, ) CNF formula: formula that uses And operators and clauses and each clause is constructed using or operators. A formula is satisfiable if there is a truth assignment that assigns truth values T or F to variables such that the formula evaluates to be true. SAT Problem: Given a CNF formula F. Question: Is F satisfiable? 12

13 The 1 st NP-complete Problem Thm ( Cook-Levin) SAT is NP-complete, i.e. every problem in NP is polynomially transformable to SAT. This leads to the following working definition of NP-complete. NP-complete: A problem is NP-complete if 1. NP 2. P for some NP-complete problem. 13

14 Not every NP-Hard Problem is NP-Complete. Halting Problem: Given any arbitrary deterministic Algorithm A and an input I to A, decide whether A will terminate on I. It is well known that this problem is undecidable. So, Halting Problem NP. Claim: SAT Halting. Program A { Try all possible truth assignment of X to check whether F is satisfiable. If yes stop; else enter into an infinite loop; } So, A halts on input X iff F is satisfiable. 14

15 So, if we have a polynomial time algorithm to verify whether A stops for X, we can decide in polynomial time whether F is satisfiable. Hence, SAT Halting. Hence, Halting is NP-Hard but not NP-Complete. 15

16 NP-Complete Graph Problem Thm: CDP is NP-complete Proof: SAT P CDP Let {C 1,C 2,,C k } be the k clauses of an instance of SAT that constitute a formula F. Construct G=(V,E), V={(,i)} is a literal in clause C i }. E={(,i)(,j) i j and is not the complement of }. NP: Choose a set C and check whether it is a clique of size at least k. So, CDP is in NP 16

17 Claim: F is satisfiable iff G has a clique of size at least k. Assume that F is satisfiable. So each clause Ci has a true literal i. Let S={( i,i), 1 i k}. Now S forms a clique of size k. Suppose G has a clique S of size at least k. Assign T to i if ( I,i) is in S. Extend this truth assignment to other variables. Clearly, this truth assignment is a satisfiable truth assignment. Hence CDP is NP-complete. 17

18 NP-complete graph problems Vertex Cover: A subset S of vertices such that every has an end point in S. VCDP: Instance: A Graph G=(V,E) and an integer k. Question: Does G have a VC of size k? Thm: S is a clique of G iff V-S is a vertex cover of G c. Thm: CDP P VCDP and VCDP NP. Cor: VCDP is NP-complete. 18

19 Independent Set Decision Problem Thm: CDP P ISDP and ISDP P. So ISDP is NPcomplete. Thm: 3-SAT P CNDP Proof: Given an instance of 3-SAT, construct a graph G=(V,E) as follows. Assume that n 4.. V={x 1,x 2,,x n } {x c 1,x c 2,,x c n} {y 1,y 2,,y n } {C 1,C 2,,C r } E={x i x c i, 1 i n} {y i y j i j} {y i x j i j} {y i x c j i j} {x i C j x i C j } {x c ic j x c i C j } 19

20 Claim: G is n+1 colorable iff F is satisfiable Note that {yi, 1 i n} forms a clique. So G needs at least n colors. Assume that f(yi)=i. As xi and xi are adjacent, they cannot be given the same colors. Also Xi is adjacent to yj if I j. So xi can get the color i. In that case xi needs a new color n+1. 20

21 Example of SATY CNDP x 1 v x 2 v x 3 (1) -x 3 v -x 4 v x 2 (2) True assignment: x 1 =T x 2 =F x 3 =F x 4 =T E={ (x i, -x i ) 1 i n } { (y i, y j ) i j } { (y i, x j ) i j } { (y i, -x j ) i j } { (x i, c j ) x i c j } { (-x i, c j ) -x i c j }

22 Proof of SATY CNDP Satisfiable n+1 colorable (1) f(y i ) = i (2) if x i = T, then f(x i ) = i, f(-x i ) = n+1 else f(x i ) = n+1, f(-x i ) = i (3)if x i in c j and x i = T, then f(c j ) = f(x i ) if -x i in c j and -x i = T, then f(c j ) = f(-x i ) ( at least one such x i ) 8-22

23 (1) y i must be assigned with color i. (2) f(x i ) f(-x i ) either f(x i ) = i and f(-x i ) = n+1 or f(x i ) = n+1 and f(-x i ) = i (3) at most 3 literals in c j and n 4 at least one x i, x i and -x i are not in c j f(c j ) n+1 (4) if f(c j ) = i = f(x i ), assign x i to T if f(c j ) = i = f(-x i ), assign -x i to T (5) if f(c j ) = i = f(x i ) (c j, x i ) E x i in c j c j is true if f(c j ) = i = f(-x i ) similarly

24 Reference Computers and intractability: A guide to the Theory of NPcompleteness, Garey and Johnson, W.H. Freeman, 1979 Introduction to Algorithms, Cormen,Leiserson,Rivest,Stein, PHI 2009 Concrete Mathematics: A foundation for computer Science, graham, Knuth, and Patashnik, Addisionwesley,1989 Computer Algorithms/C++, Horowitz, Sahni, and Rajasekaran, University Press, 2008 Introduction to Algorithm, Kleinberg and Tardos, Pearson, 2008 Applied and Algorithmic Graph theory, Chatrand and Ollermann, McGraw-Hill,