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Ragesh Jaiswal

CSE, IIT Delhi

CSL 356: Analysis and Design of

Algorithms

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Topics

Greedy Algorithms

Divide and Conquer

Dynamic Programming

Network Flow

Computational intractability

Other topics: Randomized algorithms, Computational

Geometry, Number-theoretic algorithms etc.

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Computational Intractability

Introduction

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Computational Intractability Efficient Algorithms: All algorithms that run in time

polynomialin the input size. We will call them polynomialtime algorithms.

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Question: Given a problem, does there exist an efficient

algorithm to solve the problem?

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There are a lot of problems arising in various fields for which

this question is still unresolved.

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Question: Are these problems relatedin some manner? Are

there certain aspects that are common to all these problems?

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Question: Are these problems relatedin some manner? Are

there certain aspects that are common to all these problems?

Question: If someone discovers an efficient algorithm to one

of these difficult problems, then does that mean that there

are efficient algorithms for other problems? If so, how do we

obtain such an algorithm.

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Computational Intractability NP-Complete Problems: This is a large class of problems

such that all problems in this class are equivalent in thefollowing sense:

A polynomial time algorithm for any one problem in this class implies

the existence of polynomial time algorithm for all of them.

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Computational Intractability NP-Complete Problems: This is a large class of problems

such that all problems in this class are equivalent in thefollowing sense:

A polynomial time algorithm for any one problem in this class implies

the existence of polynomial time algorithm for all of them.

Polynomial time reduction: Consider two problemsand .

Suppose there is a black boxthat solves arbitrary instances of

problem.

Suppose any arbitrary instance of problem can be solved usinga polynomial number of standard computational steps and a

polynomial number of calls to the black box that solves instance

of problemcorrectly.

We say that is polynomial time reducible toor .

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Computational Intractability Polynomial time reduction:

Consider two problemsand . Suppose there is a black boxthat solves arbitrary instances of

problem.

Suppose any arbitrary instance of problem can be solved using

a polynomial number of standard computational steps and apolynomial number of calls to the black box that solves instance



Is ?

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Computational Intractability Polynomial time reduction:

Consider two problemsand . Suppose there is a black boxthat solves arbitrary instances of

problem.

Suppose any arbitrary instance of problem can be solved using

a polynomial number of standard computational steps and apolynomial number of calls to the black box that solves instance



Claim 1: Suppose . Ifcan be solved in polynomialtime, then can be solved in polynomial time.

Claim 2: Suppose . If cannot be solved in

polynomial time, thencannot be solved in polynomial time.

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Computational Intractability

Polynomial-time reductions

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Computational Intractability: Reduction Problem(Independent set): Given a graph = (, )and

an integer , check if there is an independent set of size at leastin .

Independent set: A subset of vertices is called an independent set

if there is no edge between any pair of vertices in .

Problem(Maximum independent set): Given a graph = (, ), output the size of the independent setofofmaximum size.

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Problem(Maximum independent set): Given a graph = (, ), output the size of the independent set ofofmaximum size.

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Problem(Maximum independent set): Given a graph

= (, ), output the size of the independent set ofof

maximum size.

Claim 1: Maximum-independent-set Independent-set

Claim 2: Independent-set Maximum-independent-set

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Computational Intractability: Reduction Problem(Vertex cover): Given a graph = (, )and an

integer , check if there is a vertex cover of size at most in . Vertex cover: A subset of vertices is called a vertex cover of if

for any edge (,)in the graph is in or is in .

Problem(Minimum vertex cover): Given a graph =(,)

, output the size of thevertex cover

of

of minimumsize.

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Computational Intractability: Reduction Problem(Vertex cover): Given a graph = (, )and an

integer , check if there is a vertex cover of size at most in . Vertex cover: a subset of vertices is called a vertex cover of if

for any edge (,)in the graph is in or is in .

Problem(Minimum vertex cover): Given a graph = (, ),output the size of the vertex cover of of minimum size.

Claim 1: Minimum-vertex-cover

Vertex-cover.

Claim 2: Vertex-cover Minimum-vertex-cover

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Computational Intractability: Reduction Claim: Independent-set Vertex-cover.

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Proof:

Claim 1: Let be an independent set of , then is a

vertex cover of .

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Proof:


vertex cover of .

Claim 2: Let be a vertex cover of , then is an

independent set of .

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Proof:


vertex cover of .



Claim 3: has an independent set of size at least if and only if

has a vertex cover of size at most ( ).

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Proof:


vertex cover of .



Claim 3: has an independent set of size at least if and only if

has a vertex cover of size at most ( ).

Given an instance of the independent set problem (,)create

an instance of the vertex cover problem (, ), make aquery to the black box solving the vertex cover problem and

return the answer that is returned by the black box.

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Computational Intractability: Reduction Claim: Minimum-vertex-cover Maximum-independent-

set.

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Computational Intractability: Reduction Claim: Minimum-vertex-cover p Maximum-independent-

set. Proof 1: has an independent set of size at least if and

only if has a vertex cover of size at most ( ).

Proof 2:

Minimum-vertex-cover Vertex-cover.

Vertex-cover Independent-set.

Independent-set Maximum-independent-set.

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Computational Intractability: Reduction Claim: Minimum-vertex-cover p Maximum-independent-

set. Proof 1: has an independent set of size at least if and

only if has a vertex cover of size at most ( ).

Proof 2:

Minimum-vertex-cover Vertex-cover.

Vertex-cover Independent-set.

Independent-set Maximum-independent-set.

Theorem: If and , then .

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Computational Intractability: Reduction Problem(Deg-3-Independent set): Given a graph

= (, )of bounded degree3and an integer , check ifthere is an independent set of size at least in .

Graph with bounded degree 3: A graph is said to have bounded

degree 3if the degrees of all vertices in the graph is at most 3.

Claim: Independent-set Deg-3-Independent-set.

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Computational Intractability: Reduction

Claim: Independent-set

Deg-3-Independent-set.

Claim: has an independent set of size at least if and only

if has an independent set of size at least ( + 1).

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End

lec-1 (11)

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