1. Hard Problems Introduction to NP
Lecture 22
Hard Problems
Introduction to NP

2. Key Words Decision problem Optimization problem
Non-deterministic algorithm
NP-hard problem
NP-complete problem
Intractable problems
Inherently difficult problems

3. Traveling Salesman Problem
Given: weighted complete graph G=(V,E,W)
Tour: visits all vertices exactly once
Cost of the tour: sum of weights of traversed edges
Optimization Problem: Find Optimal (Min Cost) Tour
Decision Problem: given a value w.
Is there a tour of weight < w?

4. 0-1 Knapsack Problem Given: n items weighting w1, w2, …, wn lbs
valued at p1, p2, …, pn $’s
a knapsack of capacity W
Optimization Problem:
Find maximum total of profit of items that can be placed in the knapsack
Decision Problem:
Given also a profit P
Find if there is a set of items of total weight <W with profit >P

5. Can associate a DECISION PROBLEM to any OPTIMIZATION PROBLEM

6. Graph Coloring Problem
Given: a graph G=(V,E)
Coloring of G: coloring of vertices so that adjacent vertices get different colors
Optimization Problem: Find a coloring of G with minimum number of colors
Decision Problem: given a number k of colors.
Is there a coloring with < k colors?

7. Clique Problem Given: a graph G=(V,E) Clique: complete subgraph
Optimization Problem: Find the largest clique
Decision Problem: given a number k.
Is there a clique of size >= k colors?

8. Hamilton Cycle Problem
Given: a graph G=(V,E)
Hamilton Cycle: a cycle visiting every vertex exactly once (without repetitions)
Decision Problem: Is there a Hamilton cycle?

9. Traveling Salesman Problem
Given: weighted complete graph G=(V,E,W)
Tour: visits all vertices exactly once
Cost of the tour: sum of weights of traversed edges
Optimization Problem: Find Optimal (Min Cost) Tour
Decision Problem: given a value w.
Is there a tour of weight < w?

10. Circuit Satisfiability
Logic gates: AND, OR, NOT
See how they operate:
Given: a circuit built up with logic gates, with n inputs x1, x2, … , xn and one output
The circuit is satisfied for some input values of 0/1 associated to x1, x2, … , xn if output of the circuit is 1.
Decision Problem: is there an assignment of 0/1 values to x1, x2, … , xn so that the circuit is satisfied?

11. Formula Satisfiability
Logic operators: AND, OR, NOT
Variables: x1, x2, … , xn
Boolean formula: ((x1 AND x2) OR (NOT x3 ))
Given: a boolean formula F with n variables x1, x2, … , xn
The formula is satisfied for some input values of 0/1 associated to x1, x2, … , xn if it evaluates to 1.
Decision Problem: is there an assignment of 0/1 values to x1, x2, … , xn so that the formula F is satisfied?

12. CNF Formula Satisfiability
CNF= Conjunctive Normal Form: conjunction of disjunctions ((.. OR..) AND (.. OR..) AND (..))
AND: conjunction
OR: disjunction
Boolean formula: ((x1 AND x2) OR (NOT x3 ))
Given: a boolean formula F in CNF with n variables x1, x2, … , xn
Decision Problem: is there an assignment of 0/1 values to x1, x2, … , xn so that the CNF formula F is satisfied?

13. 3SAT: Satisfiability of CNF Formulas with 3 literals per disjunction
Instance of 3SAT:
( (NOT x1) AND x2 AND x3) OR (x1 AND x3 AND x4) )
Given: a boolean formula F in CNF with n variables x1, x2, … , xn, with 3 literacls per clause
Decision Problem: is there an assignment of 0/1 values to x1, x2, … , xn so that the formula F is satisfied?

14. Hardness of these Problems
None of the problems presented before is known to have a polynomial time algorithm
But they all have a common property: that if a solution is given, it is easy to verify it

15. Verification Problems
Given a decision problem: Does there exist x such that a property holds?
Verification Problem: given an object y, claimed to be a solution to the decision problem, verify that it is indeed the case

16. Examples Verify that a tour is a TSP tour of weight < w
Verify that a given coloring with k colors is valid
Verify that a given complete subset of k vertices is a clique of G
Verify that a certain assignment of 0/1 values satisfies a boolean circuit/formula
Verify that a certain subset of items fits into a knapsack and has profit larger than P

17. Can associate a VERIFICATION PROBLEM to any DECISION PROBLEM

18. It is easy to see that the VERIFICATION PROBLEM for
Traveling Salesman Problem
0-1Knapsack
Graph Coloring
Clique
Circuit Satisfiability
Hamilton Tour
is in P (can be done in polynomial time)

19. General “algorithm” for such problems:
Guess a candidate for a solution
Verify if it is a solution
If you are lucky, you’re done
Else you might have to generate all possible candidate solutions to find one, or say that none exists

20. Relating the Decision Problem to the Verification Problem
TSP: is there …. so that …
“a tour”: object can be described in polynomial space
“it has a weight < w”: property can be verified in polynomial time

21. Complexity Class NP
Class of all decision problems that can be described as:
is there …. (a candidate solution describable in polynomial space)
so that … (property that can be verified in polynomial time holds)?
NP= nondeterministic polynomial time

22. MOST FAMOUS OPEN PROBLEM in THEORETICAL COMPUTER SCIENCE

23. P = NP?

24. What if ?
TRUE: then all the previously listed problems would have polynomial time algorithms
FALSE: one should stop looking for polynomial time algorithms for these problems, since none would exist (deep theorem, due to S. Cook – done in 353 Complexity Theory). This would imply a lower bound on the complexity of all of these problems.

25. NP-complete problems
If you proved that there is no polynomial time algorithm for any of them, you proved that there is no polynomial time algorithm for all the problems in NP
If you proved there exists a polynomial time algorithm for any of them, then you proved there is a polynomial time algorithm for all of the problems in NP.

26. NP-Complete Problems TSP Hamilton Cycle Graph Coloring 0-1 Knapsack
CNF Formula Satisfiability, etc
are as HARD as any other problem in NP