1 The Theory of NP-Completeness

2 Cook ’ s Theorem (1971) Prof. Cook Toronto U. Receiving Turing Award (1982) Discussing difficult problems: worst case lower bound seems to be in the order of an exponential function NP-complete (NPC) Problems

3 Finding lower bound by problem transformation 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) Since  (A)   (B) +T(tr 1 ) + T(tr 2 ), we have  (B)   (A) –(T(tr 1 ) + T(tr 2 )) We have  (B)   (A) if T(tr 1 ) + T(tr 2 )   (A)

4 The lower bound of the convex hull problem sorting  convex hull A B an instance of A: (x 1, x 2, …, x n ) ↓ transformation an instance of B: {( x 1, x 1 2 ), ( x 2, x 2 2 ), …, ( x n, x n 2 )} assume: x 1 < x 2 < … < x n

5 The lower bound of the convex hull problem If the convex hull problem can be solved, we can also solve the sorting problem, but not vice versa. We have that the convex hull problem is harder than the sorting problem. The lower bound of sorting problem is  (n log n), so the lower bound of the convex hull problem is also  (n log n).

6 NP P NPC NP: Non-deterministic Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete P=NP?

7 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.

8 Nondeterministic algorithms Machines for running nondeterministic algorithms do not exist and they would never exist in reality. (They can only be made by allowing unbounded parallelism in computation.) Nondeterministic algorithms are useful only because they will help us define a class of problems: NP problems

9 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.

10 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)

11 Decision problems The solution is simply “ Yes ” or “ No ”. Optimization problem : harder Decision problem : easier 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 ?

12 Decision version of sorting Given a 1, a 2, …, a n and c, is there a permutation of a i s ( a 1, a 2, …,a n ) such that ∣ a 2 – a 1 ∣ + ∣ a 3 – a 2 ∣ + … + ∣ a n – a n- 1 ∣＜ C ?

13 Decision vs Original Version We consider decision version problem D rather than the original problem O because we are addressing the lower bound of a problem and D ∝ O

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

15 Nondeterministic searching Algorithm : input: n elements and a target element x output: success if x is found among the n elements; failure, otherwise. 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).

16 Relationship Between NP and P It is known P  NP. However, 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 We cannot find a polynomial-time algorithm for many NP problems. But, no NP problem is proved to have exponential lower bound. ( No NP problem has been proved to be not in P.) So, “ does P = NP ? ” is still an open question! Cook tried to answer the question by proposing NPC.

17 NP-complete (NPC) A problem A is NP-complete (NPC) if A ∈ NP and every NP problem reduces to A.

18 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 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

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

20 Proof of NP-Completeness To show that A is NP-complete (I) Prove that A is an NP problem (II) Prove that  B  NPC, B  A  A  NPC Why ? Transitive property of polynomial-time reduction

21 0/1 Knapsack problem Given M (weight limit) and V, is there is a solution with value larger than V? This is an NPC problem. P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 P8P8 Value Weight

22 Traveling salesperson problem Given: A set of n planar points and a value L Find: Is there a closed tour which includes all points exactly once such that its total length is less than L? This is an NPC problem.

23 Partition problem Given: A set of positive integers S Find: Is there a partition of S 1 and S 2 such that S 1  S 2 = , S 1  S 2 =S,  i  S1 i=  i  S2 i (partition S into S 1 and S 2 such that element sum of S 1 is equal to that of S 2 ) e.g. S={1, 7, 10, 9, 5, 8, 3, 13} S 1 ={1, 10, 9, 8} S 2 ={7, 5, 3, 13} This problem is NP-complete.

24 Art gallery problem: *Given a constant C, is there a guard placement such that the number of guards is less than C and every wall is monitored? *This is an NPC problem.

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

26 NP-Completeness Proof: Reduction Vertex Cover Clique3-SAT SAT Chromatic Number Dominating Set All NP problems

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

28 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

29 Some concepts Up to now, none of the NPC problems can be solved by a deterministic polynomial time algorithm in the worst case. It does not seem to have any polynomial time algorithm to solve the NPC problems. The lower bound of any NPC problem seems to be in the order of an exponential function. The theory of NP-completeness always considers the worst case.

30 Caution ! If a problem is NP-complete, its special cases may or may not be of exponential time- complexity. We consider worst case lower bound in NP-complete.

31 Some concepts Not all NP problems are difficult. (e.g. the MST problem is an NP problem.) (But NPC problem is difficult.) If A, B  NPC, then A  B and B  A. Theory of NP-completeness If any NPC problem can be solved in polynomial time, then all NP problems can be solved in polynomial time. (NP = P)

32 Undecidable Problems They cannot be solved by guessing and checking. They are even more difficult than NP problems. E.G.: Halting problem: Given an arbitrary program with an arbitrary input data, will the program terminate or not? It is not NP It is NP-hard (SAT  Halting problem )

33 NP : the class of decision problem which can be solved by a non-deterministic polynomial algorithm. P: the class of problems which can be solved by a deterministic polynomial algorithm. NP-hard: the class of problems to which every NP problem reduces. (It is “at least as hard as the hardest problems in NP.”)NP NP-complete: the class of problems which are NP-hard and belong to NP.

34 The satisfiability (SAT) problem Def : Given a Boolean formula, determine whether this formula is satisfiable or not. A literal : x i or -x i A clause : x 1 v x 2 v -x 3  c i A formula : conjunctive normal form C 1 & c 2 & … & c m

35 The satisfiability (SAT) problem The satisfiability problem The logical formula : x 1 v x 2 v x 3 & - x 1 & - x 2 the assignment : x 1 ← F, x 2 ← F, x 3 ← T will make the above formula true (-x 1, -x 2, x 3 ) represents x 1 ← F, x 2 ← F, x 3 ← T

36 The satisfiability problem satisfiable unsatisfiable 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 : x 1 v x 2 & x 1 v -x 2 & -x 1 v x 2 & -x 1 v -x 2

37 Resolution principle c 1 : -x 1 v -x 2 v x 3 c 2 : x 1 v x 4  c 3 : -x 2 v x 3 v x 4 (resolvent) If no new clauses can be deduced  satisfiable -x 1 v -x 2 v x 3 (1) x 1 (2) x 2 (3) (1) & (2) -x 2 v x 3 (4) (4) & (3) x 3 (5) (1) & (3) -x 1 v x 3 (6) The satisfiability problem x1 cannot satisfy c1 and c2 at the same time, so it is deleted..

38 The satisfiability problem If an empty clause is deduced  unsatisfiable - x 1 v -x 2 v x 3 (1) x 1 v -x 2 (2) x 2 (3) - x 3 (4)  deduce (1) & (2) -x 2 v x 3 (5) (4) & (5) -x 2 (6) (6) & (3) □ (7)

39 Nondeterministic SAT Guessing for i = 1 to n do x i ← choice( true, false ) if E(x 1, x 2, …,x n ) is true Checking then success else failure

40 Transforming the NP searching algorithm to the SAT problem Does there exist a number in { x(1), x(2), …, x(n) }, which is equal to 7? Assume n = 2

41 Transforming searching to SAT 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

42 Transforming searching to SAT 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)

43 Transforming searching to SAT Satisfiable at 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)

44 Searching in CNF with inputs Searching for 7, but x(1)  7, x(2)  7 CNF :

45 Searching in CNF with inputs Apply resolution principle :

46 Searching in CNF with inputs

47 Q&A