Homework Help is Here – Start Your Trial Now!
Engineering
Data Structures and Algorithms
Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U| = k, your algorithm should delete at most (c + 1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.

Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U| = k, your algorithm should delete at most (c + 1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.

icon
Related questions
Question
Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U| = k, your algorithm should delete at most (c + 1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.
Transcribed Image Text:Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U| = k, your algorithm should delete at most (c + 1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

For the answer you provided me with was rejected and commented as follows: You still insist on this approach, basically disregarding the feedback. I could almost literally repeat my earlier comments. "The approximation ratio c+1 is based on the observation ...", right, but then this observation is not used at all. There is no real analysis, you only claim the desired ratio, but it does not follow from anything that is said.

 

Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U] = k, your algorithm should delete at most (c +1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.
Transcribed Image Text:Exercise 1. A connected component in an undirected graph is a subgraph C with these two properties: C is connected, and no edges exist between nodes in C and nodes outside C. We consider the following problem of splitting a graph into small pieces by deleting some nodes: Given a graph G = (V, E) and an integer c, delete a subset UC V of nodes (and all their incident edges) from G such that, in the remaining graph, every connected component has at most c nodes, and the size |U| is as small as possible. The problem appears, e.g., in data analysis, where the nodes represent data items, an edge means similarity, and the data shall be partitioned into small clusters, thereby neglecting as few data as possible. Give a polynomial-time algorithm with approximation ratio no worse than c+ 1. That is, if there exists a solution U with |U] = k, your algorithm should delete at most (c +1)k nodes. The approximation ratio is generous, but make sure that you accurately prove it, and argue why you need only polynomial time. Advice: It is tempting to iteratively delete nodes with highest degrees in a greedy fashion, since this deletes many edges. However, this approach will fail, since the number of deleted edges is not quite related to the sizes of the remaining connected components. (This trap is not obvious, therefore we mention it here, to avoid frustration.) Instead, the following way is recommended: First study the special case c = 1 for a while, and then try to generalize your observations.
Solution
Bartleby Expert
SEE SOLUTION