Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.5, Problem 1E
Program Plan Intro
To demonstrate that the sub-graph isomorphism problem is NP-complete.
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Check out a sample textbook solutionStudents have asked these similar questions
Recall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique.
Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP.
Q4.1
Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G).
Prove that for any subset of vertices…
The subgraph-isomorphism problem takes two graphs G1 and G2 and asks
whether G1 is isomorphic to a subgraph of G2. Show that
a) the subgraph-isomorphism problem is in NP;
b) it is NP-complete by giving a polynomial time reduction from SAT problem to it.
and
Note: Two graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there exists a
one-one and onto function f() from V1 to V2 such that for every two nodes u and
v in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 is
isomorphic to a subgraph with vertices {1,2,5,4} of G2 below.
G1
G2
1
1
2
4
5
4
3.
2.
3.
4. The subgraph-isomorphism problem takes two graphs Gl and G2 and asks whether
Gl is isomorphic to a subgraph of G2. Show that
a) the subgraph-isomorphism problem is in NP; and
b) it is NP-complete by giving a polynomial time reduction from SAT problem to it.
Note: Two graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there exists a one-
one and onto function f( ) from V1 to V2 such that for every two nodes u and v in V1,
(u,v) is in El if and only if (f(u), f(v)) is in E2. For examples, Gl is isomorphic to a
subgraph with vertices {1,2,5,4} of G2 below.
G1
G2
1
2
1
2
3
4
5
4
3
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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Assign to each vertex v in Va color c(v) such that 1< c(v)arrow_forwardThe third-clique problem is about deciding whether a given graph G = (V, E) has a clique of cardinality at least |V |/3.Show that this problem is NP-complete.arrow_forwardGraphs G and H are isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = { | G and H are isomorphic graphs}. Show that ISO is in the class NP.arrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forwardDetermine {G | G is a complete graph} is in P or NP-completearrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.arrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. You need to provide a clear expression using the qualifiersarrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. You need to provide a clear expression using the qualifiers.Please give clear explainationarrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)First, give an example yes-input and an example no-input, where in each case, F has at leastfive nodes and G has at least three nodes.Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.arrow_forwardA Vertex Cover of an undirected graph G is a subset of the nodes of G,such that every edge of G touches one of the selected nodes.The VERTEX-COVER problem is to decide if a graph G has a vertex cover of size k.VERTEX-COVER = { <G,k> | G is an undirected graph with a k-node vertex cover }The VC3 problem is a special case of the VERTEX-COVER problem where the value of k is fixed at 3.VERTEX-COVER 3 = { <G> | G is an undirected graph with a 3-node vertex cover }Use parts a-b below to show that Vertex-Cover 3 is in the class P.a. Give a high-level description of a decider for VC3.A high-level description describes an algorithmwithout giving details about how the machine manages its tape or head.b. Show that the decider in part a runs in deterministic polynomial time.arrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using quantifiersarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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