Friday, April 2nd Test # 2

Includes:

• Greedy Algorithms

• Nondeterministic algorithms, NP, NP-Completeness

NP-Complete

A problem npc is NP-complete if:

• npc is in NP

• Every other problem prob in NP can be transformed in polynomial time into npc. (NP-Hard)

Transformation:

npcprobPolynomial transformation

solution

How to Proof that a Problem is NP-Complete?

• First problem was hard to proof: Conjunctive Normal Form (Cook, 1971)

• Every problem q afterwards is “easier”:

Show that q is in NPFind a problem, npc, that is NP-complete and show that npc can be transformed in polynomial time into q

npc

MSTTSP

Shortest Path

Sorting …Knapsack

q

What Does this Thing “Polynomial Transformation” Means?

A problem A can be transformed in polynomial time into a problem B if we can write a polynomial algorithm that transforms A into B such that when we solve B, we also solve A

A B

Example:

A: Find a Maximum Spanning Tree of an input graph GB: Find a Minimum Spanning Tree of an input graph G

Transformation from A into B:Multiply the weight of each edge in G by -1

Conjunctive Normal Form

A conjunctive normal form (CNF) is a Boolean expression consisting of one or more disjunctive formulas connected by an AND symbol (). A disjunctive formula is a collection of one or more (positive and negative) literals connected by an OR symbol ().

Example: (a) (¬ a ¬b c d) (¬c ¬d) (¬d)

Problem (CNF-satisfaction): Give an algorithm that receives as input a CNF form and returns Boolean assignments for each literal in form such that form is true

Example (above): a true, b false, c true, d false

Cook Theorem (1971)

The CNF-satisfaction satisfaction is NP-completeThe CNF-satisfaction satisfaction is NP-complete

(Vague) Idea of The Proof (I)

Computer Memory

Program … <instruction> ….

Program … <instruction> ….

State1: S1 State2: S2

S1

S1S2

S2

A computation: S1, S2, S3, …, Sm

(Vague) Idea of The Proof (II)

Computer Memory

Program … <instruction> ….

Program … <instruction> ….

Statej

Sj

Sj

Sj can be represented as a logic formula Fj

The computation can be represented S1, S2, S3, …, Sm as(F1 F2 … Fm), which is transformed into a CNF

The Issue with the “Cheating”

phaseII(C: path, min: int )//input: C a guessed solution//output: true iff C is a TSP

If |C| < n then return falseVisited {}for i =1 to n do { (u,v) = C[i] if v in Visited then return false else Visited Visited + {v}}return cost(C) = min

How can we know the minimum cost in advance?

•The concept of nondeterministic algorithms is not for solving actual problems.

•This concept is useful for determine properties about the problems (e.g., NP-completeness)

•When transforming problems into other problems we formulate problems as decision problems

Decision ProblemsA decision problem is one that has a Yes or No answer

Standard Formulation Decision Problem Formulation

Find the position of 3 in the array A[1..N]

Is the element 3 in the array A[1..N]?

Find the minimum number in the array A[1..N]

Is m the minimum number in the array A[1..N]?

Find a minimum spanning tree for a graph G

Is there a minimum spanning tree G of cost m?

Find a Traveling Salesman Solution for a graph G

Is there a Traveling salesman solution for G of cost m?

Note that the nondeterministic algorithms return True or False

Circuit-sat (I)

A Boolean combinatorial circuit consists of one or more Boolean components connected by wires such that there is one connected component (i.e., there are no separate parts) and the circuit has only one output. Boolean components:

xy x y

xy x y

x ¬x

Circuit-sat (II)

Circuit-sat: Given a Boolean combinatorial circuit, find a Boolean assignment of the circuit’s input such that the output is true

xy

z

Homework• Show that Circuit-sat is NP-complete

Show that Circuit-sat is in NP Show that CNF can be polynomial transformed into Circuit-sat

(a) (¬a ¬b c d) (¬c ¬d) (¬d) (Section 010)( a ¬b) (c ¬b) (d ¬b) b (Section 011)

Consider the formulas:

•Section 010: Formulate the Hamiltonian circuit as a decision problem show that Graph Coloring is in NP (pseudocode!)

•Section 011: Formulate Graph Coloring as a decision problem Show that the Hamiltonian Circuit is in NP (pseudocode!)

a. For which assignments is the formula true?b. Draw the formula in an equivalent circuitc. Explain in words how the transformation is done

CNF Circuit-SAT