Design and Analysis of Algorithms

Transcription

1 CSE 101, Winter 2018 Design and Analsis of Algorithms Lecture 15: NP-Completeness Class URL:

2 From Application to Algorithm Design Find all maimal regularl-spaced, collinear subsets of given pointset Lower bound? Naïve algorithm? Simpler variants?

3 What is an EFFICIENT Algorithm? The algorithms we have studied so far have runtimes that are tpicall bounded b some polnomial in the size of the input We call these efficient algorithms But for man problems, no polnomial-time algorithm is known We suspect that man of these problems can never be solved efficientl

4 Boolean Satisfiabilit Conjunctive normal form (CNF) = Product of Sums A Boolean epression S in CNF is the product (and) of several sums (or) S ( z) ( z) ( z)

5 Boolean Satisfiabilit Conjunctive normal form (CNF) = Product of Sums A Boolean epression S in CNF is the product (and) of several sums (or) S ( z) ( z) ( z) A Boolean epression is satisfiable if there eists an assignment of 0 s and 1 s to its variables such that the epression evaluates to 1.

6 The Satisfiabilit Problem (SAT) ( z) ( ) ( z) ( z ) ( z) SAT(F): Given a Boolean formula F in CNF, is there a satisfing assignment of truth values to the variables? This is a Decision question: Answer is YES or NO If F has n variables, there are 2 n different truth assignments Answer in above case:?

7 A Problem is a Language Decision problem: A problem whose answer (to an instance) is either es or no Eample: RodCutIncome(R,k) Can income of $k be obtained from RodCut instance R? A decision problem can be viewed as a languagerecognition problem Is (R,k) RodCutIncome? U: the set of possible inputs to the decision problem L U is the set of inputs which ield es L: the language corresponding to the problem

8 A Problem is a Language PA3 Optimization Decision

9 A Problem is a Language Decision problem: A problem whose answer is either es or no Eample: TSP(G,k): Does there eist a TSP tour in graph G that has total edge length k? A decision problem can be viewed as a languagerecognition problem Is (G,k) TSP? is w TSP? U: the set of possible inputs w to the decision problem L U is the set of inputs which ield es L: the language corresponding to the problem

10 A Problem is a Language G: w: (1,2,7); (1,3,5); (1,4,20); (2,3,2); (3,4,3); #15# TSP(G,15) = Does there eist a TSP tour in G with cost 15? YES w L TSP TSP(G, 50) = NO w L TSP

11 Polnomial-Time Reducibilit Let L 1 and L 2 be two languages from the input spaces U 1 and U 2 Definition: L 1 is polnomiall reducible to L 2 if there eists a polnomial-time algorithm that converts each input u 1 U 1 to another input u 2 U 2 such that u 1 L 1 iff u 2 L 2 Recall:

12 Polnomial-Time Reducibilit SORT reduces to CHULL What is u 1 U 1? What is u 2 U 2? such that u 1 L 1 iff u 2 L 2

13 Polnomial-Time Reducibilit L 1 is polnomiall reducible to L 2 if there eists a polnomial-time algorithm that converts each input u 1 U 1 to another input u 2 U 2 such that u 1 L 1 iff u 2 L 2 Instance of A = u 1 Pol-time Transform Instance of B = u 2 Decider of B YES NO YES NO Decider of A Note: polnomial-time in size of input u 1 also polnomial in the size of u 1. (Wh?) size of u 2 is

14 Polnomial-Time Reducibilit Definition: X is polnomial-time reducible to Y, written X P Y, if there eists an algorithm for solving X that would be polnomial if we took no account of the time needed to solve arbitrar instances of Y. Eamples of Decision questions: for convenience, require n > 2 X(n): Is_It_Prime? Y(n): Is_Largest_Factor_Other_Than_Itself_Greater_Than_Or_Equ al_to_2? Z(n,k): Is_Number_Of_Distinct_Factors_Greater_Than_k? Claim 1: X P Y Claim 2: X P Z

15 Polnomial-Time Reducibilit Definition: X is polnomial-time reducible to Z, written X P Z, if there eists an algorithm for solving X that would be polnomial if we took no account of the time needed to solve arbitrar instances of Z. Eample: X(n): Is_It_Prime? Z(n,k): Is_Number_Of_Distinct_Factors_Greater_Than_k? Claim: X P Z Instance of = u 1 Pol-time Transform Instance of = u 2 Decider of _ YES NO YES NO Decider of _

16 Polnomial-Time Reducibilit Definition: X is polnomial-time reducible to Y, written X P Y, if there eists an algorithm for solving X that would be polnomial if we took no account of the time needed to solve arbitrar instances of Y. Eample: X(n): Is_It_Prime? Y(n): Is_Largest_Factor_Other_Than_Itself_Greater_Than_Or_Equal_To_2? Claim: X P Y Instance of = u 1 Pol-time Transform Instance of = u 2 Decider of _ YES NO YES NO Decider of _

17 Observations About Pol-Time Reducibilit L 1 P L 2 and L 2 P L 3 L 1 P L 3 L 1 P L 2 and L 2 P L 1 L 2, L 1 are polnomiall equivalent

18 Observations About Pol-Time Reducibilit L 1 P L 2 and L 2 P L 3 L 1 P L 3 L 1 P L 2 and L 2 P L 1 L 2, L 1 are polnomiall equivalent Where this is going Solvable problems have polnomial-time algorithms Some problems (TSP, SAT, etc.) are hard ( non-solvable ) The hard problems have equivalent difficult: the are all polnomiall equivalent. If we can solve an one of them in polnomial time, then we can solve them all in polnomial time

19 Checkpoint Suppose A reduces to B. i.e., A is pol-time reducible to B Instance of A = u 1 Pol-time Transform Instance of B = u 2 Decider of B YES NO YES NO Decider of A What can we conclude: If B is eas A must be eas If A is eas NOTHING If A is hard B must be hard If B is hard NOTHING

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21 The Classes P and NP Classes of Decision Problems P: Problems for which there eists a deterministic polnomial-time algorithm NP: Problems for which there eists a nondeterministic polnomial-time algorithm What is non-deterministic polnomial-time? Solutions are small (== polnomial-size) Guessing is free, but solutions must be checkable in polnomial time Succinct Certificate P NP, but whether P = NP is not known Most people believe P!= NP, which would impl P NP

22 NP-Hard, NP-Complete NP-Hard: Problem X is NP-hard if ever problem in NP is polnomiall reducible to X NP-Complete: Problem X is NP-complete if: X belongs to NP, and X is NP-hard Also, Y is NP-complete if Y NP and some NPcomplete problem X is polnomiall reducible to Y NP-Complete problems are the hardest problems in NP

23 Picture of P, NP, NP-Hard, NP-Complete

24 Picture of P, NP, NP-Hard, NP-Complete

25 SAT is NP-Complete (Cook, 1971) Cook s theorem (1971): The SAT problem is NP-complete Also, Levin (1972) Once we know an NP-complete problem, proving other problems NP-complete becomes easier A new problem Y is NP-complete if: (1) Y is in NP (2) SAT or an other NP-complete problems, is polnomiall reducible to Y I.e., reduce from SAT to Y

26 NP-Complete Problems 0, 0 Number Partition: Given a set of numbers S = { 1, 2,, n }, is there a disjoint partition of S into S 1, S 2 (i.e., where S 1 S 2 = S, and S 1 S 2 = ) such that the sum of numbers in S 1 = sum of numbers of S 2? k-tolerant Number Partition: Given a set of numbers S = { 1, 2,, n } and a number k, is there a disjoint partition of S into S 1, S 2 (i.e., where S 1 S 2 = S, and S 1 S 2 = ) such that sum of numbers in S 1 - sum of numbers of S 2 k? Task: Reduce k-tolerant Number Partition to Number Partition

27 NP-Complete Problems 0, 0 Number Partition: Given a set of numbers S = { 1, 2,, n }, is there a disjoint partition of S into S 1, S 2 (i.e., where S 1 S 2 = S, and S 1 S 2 = ) such that the sum of numbers in S 1 = sum of numbers of S 2? k-tolerant Number Partition: Given a set of numbers S = { 1, 2,, n } and a number k, is there a disjoint partition of S into S 1, S 2 (i.e., where S 1 S 2 = S, and S 1 S 2 = ) such that sum of numbers in S 1 - sum of numbers of S 2 k? Task: Reduce k-tolerant Number Partition to Number Partition

28 Two Parts To An NP-Completeness Proof Problem is in NP A known NP-complete problem reduces to it

29 NP-Complete Problems 0, 0 Reduce k-tolerant Number Partition to Number Partition

30 NumPart, Bin-Packing NumPart(S): Given a set of numbers S = { 1, 2,, n } with 0 < i 1, there a disjoint partition of S into S 1, S 2 (i.e., where S 1 S 2 = S, and S 1 S 2 = ) such that i S 1 = j S 2? Bin-Packing(L,B): Given a list of items L = { 1, 2,, n } with 0 < i 1, and a number B, can L be packed into B unit-capacit bins? Given that NumPart is NP-complete, prove that Bin-Packing is NP-complete

31 NP-Complete Problems #1, #2, #3 The following five problems are NP-complete (Manber, Introduction to Algorithms: A Creative Approach) Definition: A verte cover of G =( V, E) is V V such that ever edge in E is incident to some v V. VC(G,k): Given undirected G = (V, E) and integer k, does G have a verte cover with k vertices? CLIQUE(G,k): Does G contain a clique of size k? 3COLOR(G): Is G properl colorable with 3 colors?

32 NP-Complete Problems #1, #2, #3 VC(G,k): Given undirected G = (V, E) and integer k, does G have a verte cover with k vertices? CLIQUE(G,k): Does G contain a clique of size k? 3COLOR(G): Is G properl colorable with 3 colors?

33 NP-Complete Problems #4, #5 Definition: A dominating set D of G = (V, E) is D V such that ever v V is either in D or adjacent to at least one verte of D DS(G,k): given G, k, does G have a dominating set of size k? 3SAT(F): Given Boolean epression F in CNF such that each clause has eactl 3 variables, is F satisfiable?

34 NP-Completeness Proofs: Reductions All NP problems Clique 1 Verte Cover SAT 4 3 3SAT 5 3-Colorabilit 2 Dominating Set To prove NP-completeness of a new problem: Prove that the problem belongs to NP Show polnomial reducibilit from a known NPcomplete problem This slide shows the order in which we will develop some classic reductions

35 NP-Completeness Proof: Verte Cover (VC) Problem: Given undirected G = (V, E) and integer k, does G have a verte cover with k vertices? Theorem: The VC problem is NP-complete. Proof: Approach: Reduction from CLIQUE I.e., given that CLIQUE is NP-complete, we will prove that VC is NP-complete VC is in NP. This is true since we can guess a cover of size k and check it in polnomial time Goal: Transform arbitrar CLIQUE instance into VC instance such that CLIQUE answer is es iff VC answer is es

36 NP-Completeness Proof: Verte Cover (VC) Claim: CLIQUE(G, k) has same answer as VC ( G, n-k), where n = V. Observe: If C = (u, F) clique in G, then v-u is VC in G.

37 NP-Completeness Proof: Verte Cover(VC) Observe: If D is a VC in G, then G has no edge between vertices in V-D. So, we have that a k-clique in G a size (n-k) VC in G Can transform in polnomial time.

38 NP-Completeness Proof: DS Definition: A dominating set D of G = (V, E) is D V such that ever v V is either in D or adjacent to at least one verte of D Problem: given G, k does G have a dominating set of size k? Theorem: DS is NP-complete Proof: (reduction from VC) DS NP Given instance (G,k) of VC, make ever edge of G into a triangle and answer DS.

39 NP-Completeness Proof: DS G: v w G : z u vz G has DS D of size k iff G has VC C of size k. Without loss of generalit, assume DV, D is a DS ever edge hits some verte in D D is a VC in G C is a VC new and old vertices dominated v z vu vw zu w u uw

40 NP-Completeness Proof: 3SAT 3SAT: Given a Boolean epression in CNF such that each clause has eactl 3 variables, determine satisfiabilit. Theorem: 3SAT is NP-Complete Proof: 3SATNP Let E be an arbitrar instance of SAT arbitrar clause of E For k 4 )... ( 2 1 k C ) (... ) ( ) ( ) ( ' k k k C

41 NP-Completeness Proof: 3SAT C is satisfiable iff C is satisfiable: C = 1 at least one i = 1 C = 1 C = 1 at least one i = 1 can set i s so that C = 1 ) ( ) ( ) ( ) ( ' ) ( ) ( ) ( ' ) ( C C C C

42 NP-Completeness Proof: CLIQUE Problem: Does G=(V,E) contain a clique of size k? Theorem: Clique is NP-Complete. (reduction from SAT) Idea: Make column of k vertices for each clause with k variables. No edge within a column. All other edges present ecept between and

43 NP-Completeness Proof: CLIQUE Eample: G = G has m-clique (m is the number of clauses in E), iff E is satisfiable. (Assign value 1 to all variables in clique) ) ( ) ( ) ( z z z E z z z