CSE202: Algorithm Design and Analysiscseweb.ucsd.edu/~rajaiswal/cse202/Notes/Week-10/lec-1.pdf · 2015. 11. 24. · An algorithm is said to be e cient if it runs in time polynomial

CSE202: Algorithm Design and Analysis

Ragesh Jaiswal, CSE, UCSD

Ragesh Jaiswal, CSE, UCSD CSE202: Algorithm Design and Analysis

Network Flow: Applications of Network Flow


Network FlowImage Segmentation

You are given an image as a 2-D matrix of pixels.We want to determine the foreground and the backgroundpixels.Each pixel i , has an integer a(i) associated with it denotinghow likely it is to be a foreground pixel.Similarly, each pixel i , has an integer b(i) associated with itdenoting how likely it is to be a foreground pixel.For neighboring pixels, i and j , there is an associated penaltyp(i , j) with putting i and j in different sets.

Problem

Find a partition of the pixels into F and B such that:∑i∈F

a(i) +∑i∈B

b(i)−∑

i and j are neighbors but in different sets

p(i , j)

is maximized.Ragesh Jaiswal, CSE, UCSD CSE202: Algorithm Design and Analysis


Problem


a(i) +∑i∈B

b(i)−∑


p(i , j)

is maximized.



Problem


a(i) +∑i∈B

b(i)−∑


p(i , j)

is maximized.

Consider the network below:

Figure: Idea: The s-t min-cut in the above network gives the optimalpartition.



Let C =∑

i a(i) +∑

i b(i).Claim 1: Consider a partition (F ,B) of the set of pixels. Let S = F ∪ {s},T = B ∪{t}. Then the capacity of the s-t cut (S ,T ) in the network is given by

C (S ,T ) = C−

∑i∈F

a(i) +∑i∈B

b(j)−∑


p(i , j)



Let C =∑

i a(i) +∑


C (S ,T ) = C−

∑i∈F

a(i) +∑i∈B

b(j)−∑


p(i , j)

Claim 2: Consider an s-t cut (S ,T ) in the network. Let F = A \ {s},B = T \ {t}. Then

C (S ,T ) = C−

∑i∈F

a(i) +∑i∈B

b(j)−∑


p(i , j)



Let C =∑

i a(i) +∑


C (S ,T ) = C−

∑i∈F

a(i) +∑i∈B

b(j)−∑


p(i , j)

Claim 2: Consider an s-t cut (S ,T ) in the network. Let F = A \ {s},B = T \ {t}. Then

C (S ,T ) = C−

∑i∈F

a(i) +∑i∈B

b(j)−∑


p(i , j)

Form Claims 1 and 2, we get that if (S ,T ) is a s-t min-cut in the network,then F = S \ {s},B = T \ {t} is an optimal solution to the ImageSegmentation problem


Network FlowEdge-disjoint paths

Definition (Edge-disjoint path)

Two paths P1 and P2 between from vertex s to vertex t are callededge-disjoint if P1 and P2 do not share any edges.

Problem

Given an unweighted directed graph G , find the maximum number ofedge-disjoint paths between s and t in G .



Problem


How many paths from s to t are present in this graph?How many edge-disjoint paths from s to t are present in thisgraph?



Problem


Consider the network G ′ below:



Problem


Claim 1: If there are k edge-disjoint paths in G , then here is ans-t flow in the network with value at least k .



Problem


Claim 1: If there are k edge-disjoint paths in G , then here is ans-t flow in the network with value at least k .Claim 2: If there is an s-t flow in G ′ of value k , then there are atleast k edge-disjoint paths in G .

Idea: Use induction on the number of edges with non-zero flowvalue.


Course Overview

Basic graph algorithms

Algorithm Design Techniques:

Greedy AlgorithmsDivide and ConquerDynamic ProgrammingNetwork Flow

Computational Intractability


Computational Intractability


Computational IntractabilityIntroduction

Definition (Efficient Algorithms)

An algorithm is said to be efficient if it runs in time polynomial inthe input size. Such algorithms are also called polynomial-timealgorithms.





Question 1: Given a problem, does there exist an efficientalgorithm to solve the problem?






There are lots of problems arising in various fields for whichthis question is unresolved.

Question 2: Are these problems related in some manner?






There are lots of problems arising in various fields for whichthis question is unresolved.

Question 2: Are these problems related in some manner?

Question 3: If someone discovers an efficient algorithm to oneof these difficult problems, then does that mean that there areefficient algorithms for other problems? If so, how do weobtain such an algorithm.


Computational IntractabilityPolynomial-time reduction

NP-complete problems: This is a large class of problems suchthat all problems in this class are equivalent in the followingsense:

The existence of a polynomial-time algorithm forany one problem in this class implies the existence ofpolynomial-time algorithm for all of them.



NP-complete problems: This is a large class of problems suchthat all problems in this class are equivalent in the following sense:

The existence of a polynomial-time algorithm for anyone problem in this class implies the existence ofpolynomial-time algorithm for all of them.

Polynomial-time reduction:

Consider two problems X and Y .Suppose there is a black box that solves arbitrary instances ofproblem X .Suppose any arbitrary instance of problem Y can be solved usinga polynomial number of standard computational steps and apolynomial number of calls to the black box that solves instanceof problem X .If the previous statement is true, then we say that Y ispolynomial-time reducible to X . A short notation for this isY ≤p X .





Claim 1: BIPARTITE-MATCHING ≤p MAX-FLOW.





Claim 2: Suppose Y ≤p X . If X can be solved in polynomialtime, then Y can be solved in polynomial time.





Claim 2: Suppose Y ≤p X . If X can be solved in polynomialtime, then Y can be solved in polynomial time.Claim 3: Suppose Y ≤p X . If Y cannot be solved inpolynomial time, then X cannot be solved in polynomial time.



Definition (Independent Set)

Given a graph G = (V ,E ), a subset I ⊆ V of vertices is called anindependent set of G if there are no edges between any pair ofvertices in I .

Problem

INDEPENDENT-SET: Given a graph G = (V ,E ) and an integer k ,check if there is an independent set of size at least k in G .

Problem

MAXIMUM-INDEPENDENT-SET: Given a graph G = (V ,E ),output the size of independent set of G of maximum cardinality.




Given a graph G = (V ,E ), a subset I ⊆ V of vertices is called anindependent set of G if there are no edges between any pair of verticesin I .

Problem


Problem

MAXIMUM-INDEPENDENT-SET: Given a graph G = (V ,E ), outputthe size of independent set of G of maximum cardinality.

Claim 1: MAXIMUM-INDEPENDENT-SET ≤p INDEPENDENT-SET.




Given a graph G = (V ,E ), a subset I ⊆ V of vertices is called anindependent set of G if there are no edges between any pair of verticesin I .

Problem


Problem

MAXIMUM-INDEPENDENT-SET: Given a graph G = (V ,E ), outputthe size of independent set of G of maximum cardinality.

Claim 1: MAXIMUM-INDEPENDENT-SET ≤p INDEPENDENT-SET.Claim 2: INDEPENDENT-SET ≤p MAXIMUM-INDEPENDENT-SET.



Definition (Vertex Cover)

Given a graph G = (V ,E ), a subset S ⊆ V of vertices is called avertex cover of G if for any edge (u, v) in the graph at least one ofu, v is in S .

Problem

VERTEX-COVER: Given a graph G = (V ,E ) and an integer k,check if there is a vertex cover of size at most k in G .

Problem

MINIMUM-VERTEX-COVER: Given a graph G = (V ,E ), outputthe size of vertex cover of G of minimum cardinality.





Problem

VERTEX-COVER: Given a graph G = (V ,E ) and an integer k, checkif there is a vertex cover of size at most k in G .

Problem

MINIMUM-VERTEX-COVER: Given a graph G = (V ,E ), output thesize of vertex cover of G of minimum cardinality.

Claim 3: MINIMUM-VERTEX-COVER ≤p VERTEX-COVER.





Problem

VERTEX-COVER: Given a graph G = (V ,E ) and an integer k, checkif there is a vertex cover of size at most k in G .

Problem

MINIMUM-VERTEX-COVER: Given a graph G = (V ,E ), output thesize of vertex cover of G of minimum cardinality.

Claim 3: MINIMUM-VERTEX-COVER ≤p VERTEX-COVER.Claim 4: VERTEX-COVER ≤p MINIMUM-VERTEX-COVER.



Claim 5: INDEPENDENT-SET ≤p VERTEX-COVER.

Proof of Claim 5

Claim 5.1: Let I be an independent set of G , then V − I is avertex cover of G .




Proof of Claim 5

Claim 5.1: Let I be an independent set of G , then V − I is avertex cover of G .Claim 5.2: Let S be a vertex cover of G , then V − S is anindependent set of G .




Proof of Claim 5

Claim 5.1: Let I be an independent set of G , then V − I is avertex cover of G .Claim 5.2: Let S be a vertex cover of G , then V − S is anindependent set of G .Claim 5.3: G has an independent set of size at least k if and onlyif G has a vertex cover of size at most n − k.




Proof of Claim 5

Claim 5.1: Let I be an independent set of G , then V − I is avertex cover of G .Claim 5.2: Let S be a vertex cover of G , then V − S is anindependent set of G .Claim 5.3: G has an independent set of size at least k if and onlyif G has a vertex cover of size at most n − k.Given an instance (G , k) of the independent set problem, createan instance (G , n − k) of the vertex cover problem, make a singlequery to the block box for solving the vertex cover problem andreturn the answer that is returned by the black box.



Claim 6: MINIMUM-VERTEX-COVER ≤pMAXIMUM-INDEPENDENT-SET.




Proof of Claim 6

Claim 6.1: G has an independent set of size k if and only if Ghas a vertex cover of size n − k.Make a single call to the black box for the maximum independentproblem with input G . If the black box returns k , then returnn − k .




Proof of Claim 6

Claim 6.1: G has an independent set of size k if and only if Ghas a vertex cover of size n − k.Make a single call to the black box for the maximum independentproblem with input G . If the black box returns k , then returnn − k .

Another proof of Claim 6

MINIMUM-VERTEX-COVER ≤p VERTEX-COVERVERTEX-COVER ≤p INDEPENDENT-SETINDEPENDENT-SET ≤p MAXIMUM-INDEPENDENT-SET



Theorem

If X ≤p Y and Y ≤p Z , then X ≤p Z .



Problem

DEG-3-INDEPENDENT-SET: Given a graph G = (V ,E ) of boundeddegree 3 (i.e., all vertices have degree ≤ 3) and an integer k , check ifthere is an independent set of size at least k in G .

Claim 1: INDEPENDENT-SET ≤p DEG-3-INDEPENDENT-SET



Problem

DEG-3-INDEPENDENT-SET: Given a graph G = (V ,E ) of boundeddegree 3 (i.e., all vertices have degree ≤ 3) and an integer k , check ifthere is an independent set of size at least k in G .

Claim 1: INDEPENDENT-SET ≤p DEG-3-INDEPENDENT-SETIdea: “Split” all vertices.



Claim 1: INDEPENDENT-SET ≤p DEG-3-INDEPENDENT-SET

Proof of Claim 1

Consider graph G ′ constructed by “splitting” a vertex of G .Claim 1.1: G has an independent set of size at least k if and onlyif G ′ has an independent set of size at least (k + 1).



Problem

SET-COVER: Given a set U of n elements, a collection S1, ...,Smof subsets of U, and an integer k, determine if there exist acollection of at most k of these sets whose union is equal to U.



Problem

SET-COVER: Given a set U of n elements, a collection S1, ...,Smof subsets of U, and an integer k, determine if there exist acollection of at most k of these sets whose union is equal to U.

Claim 1: VERTEX-COVER ≤p SET-COVER.


End


Documents

CSE202: Algorithm Design and Analysiscseweb.ucsd.edu/~rajaiswal/cse202/Notes/Week-10/lec-1.pdf · 2015. 11. 24. · An algorithm is said to be e cient if it runs in time polynomial