13- np-complete-problems-i.pdf

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CS 306 DAA, IIITDM Jabalpur 11/7/20

NP-Complete Problems - I

The Story of Sissa and Moore

Lesson of Exponential Growth

Put one grain of rise in the first square of the chessboard, two in the second, four in the third, eight in theforth, . And so on.

18,446,744,073,709,551,615 grains of rice

Exponential Growth

colonies of bacteria,

cells in a fetus,

transistor density in chips, the computer speed(Moores Law)


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Moores Law

Looks like providing a disincentive for

developing polynomial algorithms ?

if an algorithm is exponential, why not wait it out

until Moore's law makes it feasible

NOT Really!!!

Claim:

Moores Law

Claim: a huge incentive for developing

polynomial algorithms

Why?

Boolean Satisfiability (SAT) complexity is O(2n)

Sorting complexity is O(n logn)

Year Variables (SAT) Sorting

1975 25 100

1985 31 100

1995 38 100

2005 45 100


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The Story of Sissa and Moore

The Morale of the Story:

Exponential algorithms make polynomially slow progress,

while polynomial algorithms advance exponentially fast

Exponential expansion is not sustainable!

Bacterial colonies run out of food

Electronic chips hit the atomic scale

Moore's law will stop doubling the speed of our computers within

a decade or two

The progress will depend on algorithmic ingenuity, orsome other new idea

Problem Solving

Shortest paths, MSTs, LCS, MCM, and manyothers

We know that we can solve them efficiently, though

we are searching for a solution (path, tree, matching,etc.) from among an exponential population ofpossibilities

The quest for efficient algorithms

is about finding clever ways to bypass this process ofexhaustive search, using clues from the input in orderto dramatically narrow down the search space


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So far So Good!

Algorithmic techniques that defeat the specter

of exponentiality:

Greedy algorithms, Dynamic programming, Divide

and conquer, Linear Programming

Now, the time has come to meet the quest's

most embarrassing and persistent failures

For some other search problems, the fastest

algorithms we know for them are all exponential

Difficult Problems

If a problem has an O(nk) time algorithm

(where n being the input size and k is a

constant), then we class it as having

polynomial time complexity and as being

efficiently solvable

If there is no known polynomial time

algorithm, then the problem is classed as

intractable, or difficult, or Hard Problem.


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Euler vs. Rudrata Tours

The dividing line is not always obvious.

Consider two apparently similar problems:

Euler's problem

(often characterized as the Bridges of Knigsberg -

a popular 18th Century puzzle)

Rudrata's (Hamiltonian) problem

(after the great Irish mathematician who

rediscovered it in the 19th century)

The Eulers Problem

The 18th century German city of Knigsberg

was situated on the river Pregel

The puzzle asks whether it is possible to take a

tour through the park, crossing each bridge

exactly onceA

DB

C


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The Eulers Problem

An exhaustive search requires starting at every

possible point and traversing all the possible

paths from that point - an O(n!) problem.

Eulerian Tour

Euler showed that an Eulerian Tour existed iff

1. it is possible to go from any vertex to any other by

following the edges (the graph must be connected),

and

2. every vertex must have an even number of edges

connected to it, with at most two exceptions (which

constitute the starting and ending points).


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Eulers Tour vs. Rudratas Tour

Thus we now have a O(n) problem todetermine whether a path exists

We can now easily see that the Bridges ofKnigsberg does not have a solution

A quick inspection shows that it does have aHamiltonian path

However there is no known efficient

algorithm for determining whether aRudratas tour exists.

The Rudratas Problem

Can one visit all the squares of the chessboard,

without repeating any square, in one long walk that

ends at the starting square and at each step makes a

legal knight move?

Also known as

The Knight Tour Problem


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The Rudratas Problem

The General Problem:

given a graph, find a cycle that visits each vertex

exactly onceor report that no such cycle exists Now known as Hamiltonian Cycle

Class P and NP

Euler's problem lies in the class P: problems

solvable in Polynomial time

Hamilton's problem is believedto lie in class

NP (Non-deterministic Polynomial)


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Class P and NP

What is a Search Problem?

any proposed solution can be quickly checked forcorrectness (in Polynomial Time)

A search problem is specified by an algorithm C thattakes two inputs, an instance I and a proposedsolution S, and runs in time polynomial in (I), i.e.

We say S is a solution to I if and only ifC (I, S) = true

All search problems are NP

The class of all search problems that can besolved in polynomial time is denoted P

The Famous Problem: P NP ?

The Common Belief is that P NP

Otherwise, we need no Mathematician!!!

However, NO PROOF!


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Hard Problems and Easy Problems

NP Complete Problems

If P NP then we know that P is GOOD!

What about the problems on the left side of

the table?

all, in some sense, exactly the same problem

we show that these problems are the hardest

search problems in NP, so that

if even one of them has a polynomial time

algorithm, then every problem in NP has a

polynomial time algorithm


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NP-Complete Problems

There is a special class of problems in NP: the

NP-complete problems.

All the problems in NP are efficiently reducible

to them.

By efficiently, we mean in polynomial time, so

the termpolynomially reducible provides a

more precise definition

Reduction

The Hardest Search Problems:

A search problem is NP-complete if all other

search problems reduce to it


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The Space of NP (Assuming P NP)

Reduction

A B (A is Polynomial Time Reducible to B)

Efficient Problems: If B is efficient then C is alsoefficient

Hard Search Problems (NP Complete): If A is hardthen B is also hard

So if A is NP-complete,

a new search problem B is also NP-complete,simply by reducing A to B

This is how the set of NP-Complete problems hasgrown since then


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Satisfiability (SAT)

A Problem of great importance

chip testing

computer design

image analysis

software engineering, and others

A canonical hard problem

Satisfiability

An Example

a collection of clauses, each consisting of the disjunction of

several literals,

a literal is either a Boolean variable (such as x) or the

negation of one

The SAT Problem:

given a Boolean formula in conjunctive normal form, either

find a satisfying truth assignment of the Boolean variables

or else report that none exists.


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Satisfiability

Interestingly

1-SAT and 2-SAT are in P

3-SAT is NP-Complete

A generalized SAT can be reduced to 3-SAT

Travelling Salesperson (TSP)

Another famous hard problem (NP-C)

Nave complexity is O(Factorial (n-1))

A DP solution (still exponential) is follows


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TSP

A DP Solution

Sub-Problems?

let C (S, j) be the length of the shortest path

visiting each node in S exactly once, starting at 1

and ending at j

For |S| > 1, we define

Travelling Salesperson A DP Solution

The Algorithm

Time Complexity?


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Reductions Again

RUDRATA (s, t) PATH RUDRATA CYCLE ?

RUDRATA CYCLE problem: given a graph, is

there a cycle that passes through each vertex

exactly once?

RUDRATA (s, t)- PATH problem: given a graph

with two vertices s and t, and we want a path

starting at s and ending at t that goes through

each vertex exactly once

RUDRATA PATH RUDRATA CYCLE


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RUDRATA PATH RUDRATA CYCLE

We have to consider both the cases

When the instance of RUDRATA CYCLE has a

solution

When the instance of RUDRATA CYCLE does not

have a solution

More on NP-Completeness

See the next lecture


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Summary

Polynomial Time Complexity Problems which have solutions withtime complexity O(nk) where k is a constant are said to havepolynomial time complexity.

Class P Set of problems which have solutions with polynomial timecomplexity.

Non-deterministic Polynomial (NP) A problem which can be solvedby a series of guessing (non-deterministic) steps but whose solutioncan be verified as correct in polynomial time is said to lie in class NP.

Eulerian Path Path which traverses each arc of a graph exactly once A P-time problem

Hamiltonian Path Path which passes through each node of a graphexactly once An NP-Complete problem

NP-Complete Problems Set of problems which are all related toeach other in the sense that if any one of them can be shown to bein class P, all the others are also in class P.

References

Chapter 8, Algorithms by S. Dasgupta, C.H.

Papadimitriou, and U.V. Vazirani

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