1. P, NP, and NP-Complete
Suzan Köknar-Tezel

2. CSC 551 Design and Analysis of Algorithms
Giving credit where credit is due
These lecture notes are based on slides by
Dr. Cusack
Dr. Leubke
Dr. Goddard
And Wikipedia
I have modified them and added new slides

3. Tractability Some problems are intractable
As they grow large, we are unable to solve them in reasonable time
What is reasonable time? Standard working definition: polynomial time
On input of size n, the worst-case running time is (nk)
Polynomial time: (n2), (n3), (1) (nlgn)
Not in polynomial time: (2n), (nn), (n!)

4. Polynomial-Time Algorithms
Are some problems solvable in polynomial time?
Of course: every algorithm we’ve studied provides polynomial-time solution to some problem
Are all problems solvable in polynomial-time? No
Most are optimization problems

5. NP-completeness Near the end of 60’s
Growing list of problems with no efficient solution
Remarkable discovery:
many of these problems were interrelated
If one solved in polynomial time, all will be solved in polynomial time
NP-completeness

6. Our new focus: showing a given problem cannot be solved efficiently!

7. Optimization/Decision Problems
Optimization problems
They ask: “What is the optimal solution to problem X?”
Examples
0-1 Knapsack
Fractional Knapsack
Minimum Spanning Tree
Decision problems
They ask: “Is there a solution to problem X with property Y?”
Example
Does graph G have a MST of weight ≤ W?

8. Optimization/Decision Problems
An optimization problem tries to find an optimal solution
A decision problem tries to answer a yes/no question
Many problems will have decision and optimization versions
E.g: MST
Optimization: find an MST of graph G
Decision: does graph G have a MST of weight < W?

9. Our new focus: showing a given problem cannot be solved efficiently!
We will phrase optimization problems as decision problems
If the decision version (which is intuitively easier to solve) cannot be solved effciiently, the optimization version also cannot be solevd effciiently.

10. The Class P
P: the class of decision problems that have polynomial-time deterministic algorithms
That is, they are solvable in (nk)
A deterministic algorithm is (essentially) one that always computes the correct answer
Sample problems in P
Fractional knapsack
MST
Single-source shortest path
Sorting

11. The Class NP
NP (nondeterministic polynomial-time): the class of decision problems that are verifiable in polynomial time.

12. Sample Problems in NP Fractional Knapsack MST
Single-source shortest path
Sorting
Others?
Knapsack
Hamiltonian Cycle Problem
Satisfiability (SAT)
Conjunctive Normal Form (CNF) SAT
3-CNF SAT

13. Hamiltonian Cycle Problem (HCP)
A hamiltonian-cycle of an undirected graph is a simple cycle that contains every vertex exactly once and returns to the starting vertex
The Hamiltonian-Cycle Problem: given an undirected graph G, does it have a hamiltonian cycle?
The HCP is in NP
A solution is easy to verify in polynomial time (how?) 13

14. The MILLION DOLLAR Question
Does P = NP?
This is literally a million dollar question
The Millennium Problems
The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has designated 7 problems as Millennium Problems
Each problem has a prize of $1,000, to whomever solves it
P = NP is one of the seven problems

15. P and NP What do we mean when we say a problem is in P?
A: A solution can be found in polynomial time
What do we mean when we say a problem is in NP?
A: A solution can be verified in polynomial time
What is the relation between P and NP?
A: P  NP, but no one knows whether P = NP 15

16. NP-Complete Problems NPC problems are the “hardest” problems in NP
If any one NPC problem can be solved in polynomial time…
… then every NPC problem can be solved in polynomial time…
… and in fact every problem in NP can be solved in polynomial time (which would show that P = NP)
Thus: solve the hamiltonian-cycle in (n100) time, you’ve proved that P = NP. Retire rich & famous.

17. Reduction The crux of NP-Completeness is reducibility
Informally, a problem P can be reduced to another problem Q if any instance of P can be “easily rephrased” as an instance of Q, the solution to which provides a solution to the instance of P
What do you suppose “easily” means?
This rephrasing is called transformation
Intuitively: If P reduces to Q easily, P is “no harder to solve” than Q

18. Reducibility An example:
P: Given a set of Booleans, is at least one TRUE?
Q: Given a set of integers, is their sum positive?
Transformation: (x1, x2, …, xn) (y1, y2, …, yn) where yi = 1 if xi = TRUE, yi = 0 if xi = FALSE

19. Using Reductions
If P is polynomial-time reducible to Q, we denote this P ≤P Q
Definition of NP-Complete: A decision problem Q is NPC iff
Q  NP (What does this mean?)
Every problem in NP is polynomial-time reducible to Q
I.e. R ≤P Q for every R  NP

20. NP-Hard Problems Definition: P is NP-Hard iff
Every problem in NP is polynomial-time reducible to P
I.e. R ≤P P for every R  NP
So, an alternative definition for NPC:
P is NPC iff P is NP-Hard and P  NP
Important: For a problem to be NP-hard it does not have to be in class NP 20

21. The Relationship between P, NP, and NP-Hard
If there is a polynomial algorithm for any NP-hard problem
Then there are polynomial algorithms for all problems in NP
And hence P = NP
If P  NP, then NP-hard problems have no solutions in polynomial time
If P = NP, it does not resolve whether the NP-hard problems can be solved in polynomial time

22. The Relationship between P, NP, and NP-Hard
NPC P NP
NP-Hard
P = NP = NPC
possibilities
If P  NP
If P = NP

23. Proving NP-Completeness
Steps to prove a problem Q is NPC?
Prove Q  NP
Pick a known NPC problem P
Reduce P to Q
Describe a transformation that maps instances of P to instances of Q, such that “yes” for Q = “yes” for P
Prove the transformation works
Prove it runs in polynomial time
NP-hard

24. Why reduction to single NPC problem P is okay?
If P is NPC, then every problem in NP is polynomial-time reducible to P
Transitivity: If P can be polynomial reducible to Q, then every problem in NP is polynomial-time reducible to Q
Hence, If P ≤P Q and P is NPC, Q is also NPC
This is the key idea for today

25. Why Prove NP-Completeness?
Though nobody has proven that P  NP, if you prove a problem NP-Complete, most people accept that it is probably intractable
Therefore it can be important to prove that a problem is NP-Complete
Don’t need to come up with an efficient algorithm
Can instead work on approximation algorithms

26. NPC Problems We will now look at some NPC problems
In some cases we will also look at the transformations

27. Hamiltonian Cycle Problem
A hamiltonian-cycle of an undirected graph is a simple cycle that contains every vertex exactly once and returns to the starting vertex
The Hamiltonian-Cycle Problem: given an undirected graph G, does it have a hamiltonian cycle?
The HCP is in NPC

28. Traveling salesman problem (TSP)
The well-known traveling salesman problem:
Optimization variant: a salesman must travel to n cities, visiting each city exactly once and finishing where he begins. How to minimize travel time?
Model as complete graph with cost c(i,j) to go from city i to city j
How would we turn this into a decision problem?
A: ask if  a TSP with cost <= k

29. Hamiltonian Cycle  TSP
The steps to prove TSP is NP-Complete:
Prove that TSP  NP (Argue this)
Reduce the undirected hamiltonian cycle problem to the TSP
So if we had a TSP-solver, we could use it to solve the hamilitonian cycle problem in polynomial time
How can we transform an instance of the hamiltonian cycle problem to an instance of the TSP?
Can we do this in polynomial time?

30. Transformation: Hamiltonian Cycle  TSP
Let G = (V,E) be a graph with n nodes
Ham. cycle problem
Construct an instance of TSP as follows:
Let G’ = (V,VxV) be a complete graph on the vertices of G
The edge weights c(u,v) are 1 if the edge (u,v) is in E and 2 otherwise
The bound k is n where n is the number of vertices in V

31. Transformation: Hamiltonian Cycle  TSP
No Ham. Cycle 1 u v u v 1 1 1 2 x w x w 2
The best tour has c of 5, which is
greater than n, so NO ham. cycle

32. Transformation: Hamiltonian Cycle  TSP
Ham. Cycle Present 1 u v u v 1 1 1 1 x w x w 2
The best tour has c of 4, which is
equal to n, so  ham. cycle

33. Transformation: Hamiltonian Cycle  TSP
A hamiltonian cycle in G is a tour in G’ with cost n.
If there are no hamiltonian cycles in G, any tour in G’ must cost at least n+1
So G has a hamiltonian cycle iff G’ has a traveling salesperson tour of weight n
Is this a polynomial-time transformation?

34. The Satisfiability (SAT) Problem
Given a boolean expression on n variables, can we assign values such that the expression is TRUE?
Ex: ((x1x2)  ((x1x3)  x4))  x2
Simple enough but no known deterministic polynomial time algorithm exists
Easy to verify in polynomial time
This is the first NPC problem
Proven by Stephen Cook in 1971

35. Conjunctive Normal Form (CNF)
Even if the form of the boolean expression is simplified, no known polynomial time algorithm exists
Literal: An occurrence of a boolean or its negation
A boolean formula is in CNF if it is an AND of clauses, each of which is an OR of literals
Ex: (x1   x2)  (x1  x3  x4)  (x5 )
3-CNF: Each clause has exactly 3 distinct literals
Ex: (x1  x2  x3 )  (x1  x3  x4)  (x5  x3  x4 )
Notice: TRUE if at least one literal in each clause is true

36. 3SAT Problem
Satisfiability of Boolean formulas in 3-CNF form (the 3SAT Problem) is NP-Complete
Proof: Nope
The reason we care about the 3-CNF problem is that it is relatively easy to reduce to others
Thus by proving 3SAT NP-Complete we can prove many seemingly unrelated problems NP-Complete

37. 3SAT  Clique What is a clique of a graph G?
A: a subset of vertices fully connected to each other, i.e. a complete subgraph of G
The clique problem: how large is the maximum-size clique in a graph?
Can we turn this into a decision problem?
A: Yes, we call this the k-clique problem
Is the k-clique problem within NP?

38. 3SAT  Clique What should the reduction do?
A: Transform a 3-CNF formula to a graph, for which a k-clique will exist (for some k) iff the 3-CNF formula is satisfiable

39. Transformation: 3SAT  Clique
The reduction:
Let B = C1  C2  …  Ck be a 3-CNF formula with k clauses, each of which has 3 distinct literals
For each clause put a triple of vertices in the graph, one for each literal
Put an edge between two vertices if they are in different triples and their literals are consistent, meaning not each other’s negation

40. Transformation: 3SAT  Clique
B = (x  y  z)  (x  y  z )  (x  y  z ) y x z x x y y z z

41. Transformation: 3SAT  Clique
Prove the reduction works:
If B has a satisfying assignment, then each clause has at least one literal (vertex) that evaluates to 1
Picking one such “true” literal from each clause gives a set V’ of k vertices. V’ is a clique (Why?)
If G has a clique V’ of size k, it must contain one vertex in each triple (clause) (Why?)
We can assign 1 to each literal corresponding with a vertex in V’, without fear of contradiction

42. Transformation: 3SAT  Clique
B = (x  y  z)  (x  y  z )  (x  y  z ) y x z
Assume the satisfying
assignment is:
x = 1
y = 1
z = 0
So we know we have
chosen k vertices,
and those k vertices
must be a clique x x y y z z

43. Transformation: 3SAT  Clique
B = (x  y  z)  (x  y  z )  (x  y  z ) y x z
Assume there is
a clique of size 3
Make the assignment:
x = 1
y = 1
z = 1
and we know that
each triple MUST
contain one of these
and thus B is satisfied x x y y z z

44. General Comments
Literally hundreds of problems have been shown to be NP-Complete
Some reductions are profound, some are comparatively easy, many are easy once the key insight is given

45. Other NP-Complete Problems
Subset-sum: Given a set of integers, does there exist a subset that adds up to some target T?
0-1 knapsack: when weights not just integers
Hamiltonian path: Obvious
Graph coloring: can a given graph be colored with k colors such that no adjacent vertices are the same color?
Etc…

46. Review: P and NP What do we mean when we say a problem is in P?
A: A solution can be found in polynomial time
What do we mean when we say a problem is in NP?
A: A solution can be verified in polynomial time
What is the relation between P and NP?
A: P  NP, but no one knows whether P = NP

47. Review: NP-Complete
What, intuitively, does it mean if we can reduce problem P to problem Q?
P is “no harder than” Q
How do we reduce P to Q?
Transform instances of P to instances of Q in polynomial time s.t. Q: “yes” iff P: “yes”
What does it mean if Q is NP-Hard?
Every problem PNP p Q
What does it mean if Q is NP-Complete?
Q is NP-Hard and Q  NP

48. Review: Proving Problems NP-Complete
How do we usually prove that a problem R is NP-Complete?
A: Show R NP, and reduce a known NP-Complete problem Q to R