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The Theory of NP-Completeness

The Theory of NP-Completeness. NP. =. NPC. P. P. X. P=NP. ?. NP: Non-deterministic Polynomial. P: Polynomial. NPC: Non-deterministic Polynomial Complete. P : the class of problems which can be solved by a deterministic p olynomial algorithm.

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The Theory of NP-Completeness

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  1. The Theory of NP-Completeness

  2. NP = NPC P P X P=NP ? NP: Non-deterministic Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete

  3. P: the class of problems which can be solved by a deterministic polynomial algorithm. • NP: the class of decision problem which can be solved by a non-deterministic polynomial algorithm. • NP-hard: the class of problems to which every NP problem reduces. • NP-complete (NPC): the class of problems which are NP-hard and belong to NP.

  4. Decision problems • The solution is simply “Yes” or “No”. • Optimization problem : more difficult Decision problem • E.g. the traveling salesperson problem • Optimization version: Find the shortest tour • Decision version: Is there a tour whose total length is less than or equal to a constant C ?

  5. Nondeterministic algorithms • A nondeterministic algorithm is an algorithm consisting of two phases: guessing and checking. • Furthermore, it is assumed that a nondeterministic algorithm always makes a correct guessing.

  6. Nondeterministic algorithms • They do not exist and they would never exist in reality. • They are useful only because they will help us define a class of problems: NP problems

  7. NP algorithm • If the checking stage of a nondeterministic algorithm is of polynomial time-complexity, then this algorithm is called an NP (nondeterministic polynomial) algorithm.

  8. NP problem • If a decision problem can be solved by a NP algorithm, this problem is called an NP (nondeterministic polynomial) problem. • NP problems : (must be decision problems)

  9. To express Nondeterministic Algorithm • Choice(S) : arbitrarily chooses one of the elements in set S • Failure : an unsuccessful completion • Success : a successful completion

  10. Nondeterministic searching Algorithm : j ← choice(1 : n) /* guess if A(j) = x then success /* check else failure • A nondeterministic algorithm terminates unsuccessfully iff there exist no set of choices leading to a success signal. • The time required for choice(1 : n) is O(1). • A deterministic interpretation of a non-deterministic algorithm can be made by allowing unbounded parallelism in computation.

  11. Problem Reduction • Problem A reduces to problem B (AB) • iff A can be solved by using any algorithm which solves B. • If AB, B is more difficult (B is at least as hard as A) • Note: T(tr1) + T(tr2) < T(B) • T(A)  T(tr1) + T(tr2) + T(B)  O(T(B))

  12. Polynomial-time Reductions • We want to solve a problem R; we already have an algorithm for S • We have a transformation function T • Correct answer for R on x is “yes”, iff the correct answer for S on T(x) is “yes” • Problem R is polynomially reducible to S if such a transformation T can be computed in polynomial time • The point of reducibility: S is at least as hard to solve as R

  13. Polynomial-time Reductions • We use reductions (or transformations) to prove that a problem is NP-complete • x is an input for R; T(x) is an input for S • (R  S) x T(x) Algorithm for S T Yes or no answer Algorithm for R

  14. NPC and NP-hard • A problem A is NP-hard if every NP problem reduces to A. • A problem A is NP-complete (NPC) if A∈NP and every NP problem reduces to A. • Or we can say a problem A is NPC if A∈NP and A is NP-hard.

  15. NP-Completeness • “NP-complete problems”: the hardest problems in NP • Interesting property • If any oneNP-complete problem can be solved in polynomial time, then every problem in NP can also be solved similarly (i.e., P=NP) • Many believe P≠NP

  16. Importance of NP-Completeness • NP-complete problems: considered “intractable” • Important for algorithm designers & engineers • Suppose you have a problem to solve • Your colleagues have spent a lot of time to solve it exactly but in vain • See whether you can prove that it is NP-complete • If yes, then spend your time developing an approximation (heuristic) algorithm • Many natural problems can be NP-complete

  17. Relationship Between NP and P • It is not known whether P=NP or whether P is a proper subset of NP • It is believed NP is much larger than P • But no problem in NP has been proved as not in P • No known deterministic algorithms that are polynomially bounded for many problems in NP • So, “does P = NP?” is still an open question!

  18. Cook’s theorem (1971) NP = P iff SATP • SAT (the satisfiability problem) is NP-complete • It is the first NP-complete problem • Every NP problem reduces to SAT

  19. SAT is NP-complete • Every NP problem can be solved by an NP algorithm • Every NP algorithm can be transformed in polynomial time to an SAT problem (a Boolean formula C) • Such that the SAT problem is satisfiable iff the answer for the original NP problem is “yes” • That is, every NP problem  SAT • SAT is NP-complete

  20. Definition of the satisfiability problem: Given a Boolean formula, determine whether this formula is satisfiable or not. • A literal: xi or -xi • A clause: x1 v x2 v -x3 Ci • A formula: conjunctive normal form C1& C2 & … & Cm

  21. The Satisfiability Problem • The satisfiability problem • A logical formula: x1 v x2 v x3 & - x1 & - x2 the assignment : x1 ← F , x2 ← F , x3 ← T will make the above formula true. (-x1, -x2 , x3) represents x1 ← F , x2 ← F , x3 ← T

  22. If there is at least one assignment which satisfies a formula, then we say that this formula is satisfiable; otherwise, it is unsatisfiable. • An unsatisfiable formula: x1 v x2 & x1 v -x2 & -x1 v x2 & -x1 v -x2

  23. The satisfiability problem • Resolution principle c1 : -x1 v -x2 v x3 c2 : x1 v x4  c3 :-x2 v x3 v x4 (resolvent) • If no new clauses can be deduced  satisfiable -x1 v -x2 v x3 (1) x1 (2) x2 (3) (1) & (2) -x2 v x3 (4) (4) & (3) x3 (5) (1) & (3) -x1 v x3 (6)

  24. The satisfiability problem • If an empty clause is deduced  unsatisfiable - x1 v -x2 v x3 (1) x1 v -x2 (2) x2 (3) - x3 (4)  deduce (1) & (2) -x2 v x3 (5) (4) & (5) -x2 (6) (6) & (3) □ (7)

  25. Nondeterministic SAT • Guessing for i = 1 to n do xi← choice( true, false ) if E(x1, x2, … ,xn) is true Checking then success else failure

  26. Transforming Searching to SAT • Does there exist a number in { x(1), x(2), …, x(n) }, which is equal to 7? • Assume n = 2

  27. Transforming Search to SAT • Does there exist a number in { x(1), x(2), …, x(n) }, which is equal to 7? • Assume n = 2. nondeterministic algorithm: i = choice(1,2) if x(i)=7 then SUCCESS else FAILURE

  28. i=1 v i=2 & i=1 → i≠2 & i=2 → i≠1 & x(1)=7 & i=1 → SUCCESS & x(2)=7 & i=2 → SUCCESS & x(1)≠7 & i=1 → FAILURE & x(2)≠7 & i=2 → FAILURE & FAILURE → -SUCCESS & SUCCESS (Guarantees a successful termination) & x(1)=7 (Input data) & x(2)≠7

  29. CNF (conjunctive normal form) : i=1 v i=2 (1) i≠1 v i≠2 (2) x(1)≠7 v i≠1 v SUCCESS (3) x(2)≠7 v i≠2 v SUCCESS (4) x(1)=7 v i≠1 v FAILURE (5) x(2)=7 v i≠2 v FAILURE (6) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (10)

  30. Satisfiable with the following assignment: i=1 satisfying (1) i≠2 satisfying (2), (4) and (6) SUCCESS satisfying (3), (4) and (8) -FAILURE satisfying (7) x(1)=7 satisfying (5) and (9) x(2)≠7 satisfying (4) and (10)

  31. Searching for 7, but x(1)7, x(2)7 • CNF (conjunctive normal form):

  32. Apply resolution principle:

  33. Searching for 7, where x(1)=7, x(2)=7 • CNF:

  34. The Node Cover Problem • Def: Given a graph G = (V, E), S is the node coverof G if S  V and for ever edge (u, v)  E, (u,v) is incident to a node in S. node cover: {1, 3} {5, 2, 4} • Decision problem:  S  S  K 

  35. How to Prove a Problem S is NP-Complete? • Show S is in NP • Select a known NP-complete problem R • Since R is NP-complete, all problems in NP are reducible to R • Show how R can be poly. reducible to S • Then all problems in NP can be poly. reducible to S (because polynomial reduction is transitive) • Therefore S is NP-complete

  36. Cook • Cook showed the first NPC problem: SAT • Cook received Turing Award in 1982.

  37. Karp • R. Karp showed several NPC problems, such as 3-STA, node (vertex) cover, and Hamiltonian cycle, etc. • Karp received Turing Award in 1985

  38. NP-Completeness Proof: Reduction All NP problems SAT Clique 3-SAT Vertex Cover Chromatic Number Dominating Set

  39. NPC Problems • CLIQUE(k): Does G=(V,E) contain a clique of size k? Definition: • A clique in a graph is a set of vertices such that any pair of vertices are joined by en edge.

  40. NPC Problems • Vertex Cover(k): Given a graph G=(V, E) and an integer k, does G have a vertex cover with k vertices? Definition: • A vertex cover of G=(V, E) is V’V such that every edge in E is incident to some vV’.

  41. NPC Problems • Dominating Set(k): Given an graph G=(V, E) and an integer k, does G have a dominating set of size k ? Definition: • A dominating set D of G=(V, E) is DV such that every vV is either in D or adjacent to at least one vertex of D.

  42. NPC Problems • SAT: Give a Boolean expression (formula) in DNF (conjunctive normal form), determine if it is satisfiable. • 3SAT: Give a Boolean expression in DNF such that each clause has exactly 3 variables (literals), determine if it is satisfiable.

  43. NPC Problems • Chromatic Coloring(k): Given a graph G=(V, E) and an integer k, does G have a coloring for k Definition • A coloring of a graph G=(V, E) is a function f : V  { 1, 2, 3,…, k }  if (u, v)  E, then f(u)f(v).

  44. 0/1 Knapsack problem M(weight limit)=14 best solution: P1, P2, P3, P5(optimal) This problem is NP-complete.

  45. Traveling salesperson problem • Given: A set of n planar points Find: A closed tour which includes all points exactly once such that its total length is minimized. • This problem is NP-complete.

  46. Partition problem • Given: A set of positive integers S Find: S1 and S2 such that S1S2=, S1S2=S, iS1i=iS2 i (partition into S1 and S2 such that the sum of S1 is equal to S2) • e.g. S={1, 7, 10, 9, 5, 8, 3, 13} • S1={1, 10, 9, 8} • S2={7, 5, 3, 13} • This problem is NP-complete.

  47. Art gallery problem NP-complete !!

  48. 3-Satisfiability Problem (3-SAT) • Def:Each clause contains exactly three literals. • (I)3-SAT is an NP problem (obviously) • (II)SAT  3-SAT Proof: (1) One literal L1 in a clause in SAT: In 3-SAT: L1 v y1 v y2 L1 v -y1 v y2 L1 v y1 v -y2 L1 v -y1 v -y2

  49. (2) Two literals L1, L2 in a clause in SAT: In 3-SAT: L1 v L2 v y1 L1 v L2 v -y1 (3) Three literals in a clause: remain unchanged. (4) More than 3 literals L1, L2, …, Lk in a clause: in 3-SAT: L1 v L2 v y1 L3 v -y1 v y2  Lk-2 v -yk-4 v yk-3 Lk-1 v Lk v -yk-3

  50. Example of Transforming a 3-SATInstance to an SAT Instance • An instance S in SAT: x1 v x2 -x3 x1 v -x2 v x3 v -x4 v x5 • The instance S in 3-SAT: x1 v x2 v y1 x1 v x2 v -y1 -x3 v y2 v y3 -x3 v -y2 v y3 -x3 v y2 v -y3 -x3 v -y2 v -y3 x1 v -x2 v y4 x3 v -y4 v y5 -x4 v x5 v -y5 SAT transform 3-SAT S S

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