COMP 382. Unit 10: NP-Completeness

Transcription

1 COMP 382 Unit 10: NP-Completeness

2 Time complexity 1 log n n n 2 n 3 2 n n n

3 Space complexity 1 log n n n 2 n 3 2 n n n

4 Complexity theory Focus on decidability (yes/no) problems

5 What is P? Deterministic, Sequential, Polynomial Time

6 Example problems in P Is x a multiple of y? Is x prime? Are x and y relatively prime? Is the edit distance between x and y less than 5? Is there a path between u and v of length less than 10? Most problems mentioned so far in class.

7 Understanding non-determinism: Guess & check define SubsetSum(input: set of integer, returns boolean subset = Guess() value: integer) return sum(subset) == value Two views of semantics: Oracle makes correct guesses Guess forks for all possibilities at once. Search succeeds if check succeeds for any guess.

8 Understanding non-determinism: Given a certificate, check it define SubsetSum_Certifier (input: set of integer, value: integer, witness: set of integer) returns boolean return subset(witness, input) && sum(witness) == value Original inputs Proof / Certificate / Witness Only needs to verify inputs on which SubetSum would return True.

9 Can simulate non-determinism Previously discussed ND TMs non-deterministic choice of transition Simulate guess & check: Generate all possible guesses one at a time Check Backtrack

10 What is NP? Non-deterministic, Sequential, Polynomial Time ND TM is polynomial time Check phase is polynomial time Includes all of P Includes lots not known to be in P

11 Example problems in NP: SAT Given a Boolean formula, is there a satisfying value assignment? a b b d c b c Deterministic algorithm to solve? Deterministic algorithm to check? What is appropriate certificate?

12 Example problems in NP: Directed Hamiltonian Cycle Given a directed graph, does it have a simple cycle through every vertex? B E A C D F Deterministic algorithm to solve? Deterministic algorithm to check? What is appropriate certificate?

13 Example Problems in NP: Composite Given an integer, is it composite? Deterministic algorithm to solve? Deterministic algorithm to check? What is appropriate certificate?

14 Many open questions L=P? P=BPP? P=NP? P=PSPACE? NP=co-NP? BPP=NEXP?

15 If P=NP Many hard problems would actually be in P. We d have a path to finding polynomial algorithms for them. Many other complexity open questions would be answered. Someone wins $1M from Clay Math. Institute and becomes famous. If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in creative leaps, no fundamental gap between solving a problem and recognizing the solution once it's found. -- Scott Aaronson

16 A reduces to B in polynomial time: A P B Map instance of one problem to instance(s) of another in P-time. define A(args): // Polynomial calls to subroutine B(), plus // Polynomial other steps. return Prove correctness: Problem A says yes exactly when this implementation of A says yes.

17 A reduces to B in polynomial time: A P B Typical: Map instance of one problem to instance of another in P-time. define A(a_args): b_args = // Polynomial steps. return B(b_args) Prove correctness: Problem A says yes exactly when problem B says yes on its constructed inputs.

18 Using P-reductions: Which is true? A. If B P and A P B, then A P. B. If A P and A P B, then B P.

19 Using P-reductions: Which is true? A. If B P and A P B, then A P. B. If A P and A P B, then B P.

20 NP-Complete A problem Y NP is NPC if for all problems X in NP, X P Y. Theorem: If Y is NPC, then Y P, iff P=NP. Proof: Y is NP-Hard ( ) By assumption, Y NP. If P=NP, then Y P. ( ) Suppose Y P. Let X be any problem in NP. Since X P Y, then X P. So, NP P. Since we already know P NP, then P=NP.

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22 First NPC problem Circuit-SAT Previously said the equivalent Boolean-formula version is in NP.

23 First NPC problem Circuit-SAT For all problems X in NP, X P Circuit SAT: X solvable by a ND TM in O f n steps, where f is polynomial. Goal: Given input x, produce a circuit that is satisfiable iff x X. Circuit s inputs: Hardcoded x Unknown certificate c, of f x bits

24 First NPC problem Circuit-SAT: Example constructed circuit

25 General NPC Proof Strategy Show Y NP. Pick some NPC problem X. Prove X P Y. In theory, the choice of X is irrelevant. In practice, it is very important.

26 3-CNF SAT is NPC Given a Boolean formula in 3-CNF, is there a satisfying value assignment? a b c a c d b c d b c d In NP Just a special case of SAT.

27 Which do we need to show? A. Circuit SAT P 3 CNF SAT B. 3 CNF SAT P Circuit SAT

28 Circuit SAT P 3 CNF SAT Given a circuit, construct an equivalent 3-CNF formula: 1. Label wires. 2. Construct 3-CNF clauses for each gate. 3. AND all the pieces together. a b c c = a b a b c a b c a b c a b c a b c a b = c F F F T F F T F F T F F F T T T T F F F T F T T T T F F T T T T

29 Independent set is NPC Given an undirected graph and integer k, does the graph have a subset of k vertices which are not adjacent to each other? Deterministic algorithm to check? What is appropriate certificate?

30 3 CNF SAT P IndependentSet Given 3-CNF formula, construct a graph with a k-independent set iff the formula is satisfiable. 1. Form a triangle for each clause. 2. Link complementary nodes. 3. Let k = the number of clauses. a b c a b d b c d a d e a a b a b c b d c d d e

31 Vertex cover is NPC Given a graph and an integer k, is there a set S of vertices, such that S k and every edge has an endpoint in S. Deterministic algorithm to check? What is appropriate certificate?

32 IndependentSet P VertexCover Given graph G, use the same G. Claim: S is an independent set of G iff V S is a vertex cover for G.

33 IndependentSet P VertexCover: Proof Claim: S is an independent set of G iff V S is a vertex cover for G. Let S be any independent set. Consider arbitrary edge u, v. u S or v S u V S or v V S V S covers u, v

34 IndependentSet P VertexCover: Proof Claim: S is an independent set of G iff V S is a vertex cover for G. Let V S be any vertex cover. Consider arbitrary u S and v S. u, v E Thus, no two nodes in S are linked by an edge. So, S is an independent set.

35 Set cover is NPC Given a set U of elements, a collection of subsets of U, and an integer k, does there exist a collection of at most k of these subsets whose union is U?

36 Set cover example U = 1,2,3,4,5,6,7, k = 2 S 1 = 3,7 S 2 = 3,4,5,6 S 3 = 1 S 4 = 2,4 S 5 = 5 S 6 = 1,2,6,7 What is a set cover?

37 VertexCover P SetCover Given graph and k, construct U, a collection of subsets, and k that have a set cover iff the graph has a vertex cover. U = E, Subsets = e E: e is incident to v, k = k f a e b e 3 e 2 7 e 6 e 4 e 1 e 5 e d c U = 1,2,3,4,5,6,7, k = 2 S a = 3,7, S b = 2,4 S c = 3,4,5,6, S d = 5 S e = 1, S f = 1,2,6,7

38 Directed Hamiltonian cycle is NPC Given a directed graph, does it have a simple cycle through every vertex? B E A C D F Previously saw in NP.

39 3 CNF SAT P DirHamCycle Given a 3-CNF formula, construct a directed graph that has a HamCycle iff the formula is satisfiable.

40 3 CNF SAT P DirHamCycle: Step 1 x 1 2 n Ham. cycles x 2 One per variable x 3

41 3 CNF SAT P DirHamCycle: Step 2 One pair per clause 2k + 2 nodes

42 3 CNF SAT P DirHamCycle: Step 3 One per clause x 1 x 2 x 3 Positive literal: left-to-right Negative literal: right-to-left x 1 x 2 x 3

43 3 CNF SAT P DirHamCycle: Proof Claim: Formula is satisfiable iff graph has Hamiltonian cycle. Suppose formula has satisfying assignment x. Define Hamiltonian cycle: If x i = 1, traverse row i from left to right. If x i = 0, traverse row i from right to left. For each clause j, there will be at least one row i in which the cycle goes in the correct direction to splice clause node j into the cycle.

44 3 CNF SAT P DirHamCycle: Proof Claim: Formula is satisfiable iff graph has Hamiltonian cycle. Assume constructed graph G has Hamiltonian cycle. Cycle must traverse each row i. Set x i = 1 iff this cycle traverses row i left to right. Cycle must use each clause node j. Must enter and leave via a pair of edges to/from same row. This pair of edges must go in same direction that cycle traverses the row. So each clause is satisfied by the truth assignment.

45 3-Coloring is NPC Given an undirected graph, does there exist a way of coloring the nodes with three colors, so that no two adjacent nodes have the same color? Deterministic algorithm to check? What is appropriate certificate?

46 3 CNF SAT P 3 Color Given 3-CNF formula, create a graph that is 3-colorable iff the formula is satisfiable.

47 3 CNF SAT P 3 Color: Step 1 T F B Ensures each variable has one literal colored like T, one like F. x 1 x 1 x 2 x 2 x 3 x 3 x n x n One pair per variable

48 3 CNF SAT P 3 Color: Step 2 T F B Ensures each clause has at least one literal colored like T. x 1 x 2 x 3 x 1 x 2 x 3 For each clause:

49 3 CNF SAT P 3 Color: Gadget cases 0 T F B Not 3-colorable

50 3 CNF SAT P 3 Color: Gadget cases 1 T F T F T F B B B

51 3 CNF SAT P 3 Color: Gadget cases 2 T F T F T F T F T F B B B B B T B F T B F T B F T B F

52 3 CNF SAT P 3 Color: Gadget cases 3 T F T F T F T F B B B B

53 Subset sum is NPC Given a set of natural numbers and a number S, is there a subset that adds up to S? Note: Reduction must be polynomial in the encoding size. Deterministic algorithm to check? What is appropriate certificate? U = 9, 2,3,6,7,8,12,35 S = 16

54 3 CNF SAT P SubsetSum Given 3-CNF formula, create a set and total that has a subset sum iff the formula is satisfiable. a b c a b d b c d a b c d C 1 C 2 C 3 Set: a ,000,110 a 1 1,000,000 b ,010 b ,101 c ,100 c ,001 d 1 1 1,001 d 1 1 1, S ,111,444

55 3 CNF SAT P SubsetSum Given 3-CNF formula, create a set and total that has a subset sum iff the formula is satisfiable. Exactly one literal picked for each variable. Exactly one number of each pair picked for sum. a b c a b d b c d a b c d C 1 C 2 C 3 Set: a ,000,110 a 1 1,000,000 b ,010 b ,101 c ,100 c ,001 d 1 1 1,001 d 1 1 1, S ,111,444

56 3 CNF SAT P SubsetSum Given 3-CNF formula, create a set and total that has a subset sum iff the formula is satisfiable. Picking literal satisfies corresponding clauses. a b c a b d b c d a b c d C 1 C 2 C 3 Set: a ,000,110 a 1 1,000,000 b ,010 b ,101 c ,100 c ,001 d 1 1 1,001 d 1 1 1, S ,111,444

57 3 CNF SAT P SubsetSum Given 3-CNF formula, create a set and total that has a subset sum iff the formula is satisfiable. Must pick at least one row/number to satisfy each clause. C i s total from the top is 1, 2, or 3 the bottom rows/numbers take up the slack but aren t sufficient alone. a b c a b d b c d a b c d C 1 C 2 C 3 Set: a ,000,110 a 1 1,000,000 b ,010 b ,101 c ,100 c ,001 d 1 1 1,001 d 1 1 1, S ,111,444

58 Common Reduction Strategies Simple mapping/equivalence IndependentSet P VertexCover Special to general case VertexCover P SetCover Complicated encodings Circuit SAT P 3 CNF SAT 3 CNF SAT P IndependentSet 3 CNF SAT P DirHamCycle 3 CNF SAT P 3 Color 3 CNF SAT P SubsetSum (general to special case)

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60 Exercise: Hitting set is NPC Given a set U of elements, a collection of subsets, and integer k, does there exist a subset of U of size k that overlaps with all the given subsets?

61 Exercise: Efficient recruiting is NPC Suppose you are helping organize a summer camp. The camp is supposed to have at least one counselor who is skilled at each of the n sports covered by the camp (baseball, volleyball, ). They have received job applicants from m potential counselors. For each of the n sports, there is a subset of the m applicants qualified in that sport. The question: For a given number k < m, is it possible to hire at most k counselors and have at least one counselor qualified in each sport?

62 Exercise: Zero-weight cycle is NPC Given a directed graph with weighted edges, is there a simple cycle of total weight 0?