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  # [Solved]: Minimum number of sets of unreachable vertices for directed acyclic graph (DAG)
  
  , ,
  Problem Detail:
  
  I have a DAG with vertices $V$ and edges $E$. If $v,w \in V$ are vertices such that $v$ is not reachable from $w$ and $w$ is not reachable from $v$, I will say that $\langle v,w \rangle$ is an unreachable pair.
  
  I want to implement an efficient algorithm which takes this DAG as input, and produces as output a list of sets of pvertices such that:
  
  1. For each unreachable pair of vertices $\langle v,w \rangle$, at least one of the output sets contains both $v$ and $w$,
  2. Every pair of vertices $v,w$ that are contained in the same output set must be an unreachable pair, and
  3. The number of output sets must be the minimum possible.
  
  Is there a polynomial-time algorithm for this problem?
  
  As an example, we have the following DAG with topologically ordered vertices A,B,C,D,E:
  
  A is connected to [D, E] B is connected to [D, E] C is connected to [E] D is connected to [] E is connected to [] 
  
  which means that we have the following unreachable vertex pairs:
  
  1. <A, B> 2. <A, C> 3. <B, C> 4. <C, D> 5. <D, E> 
  
  The expected output is composed of the following three vertex sets:
  
  1. {A, B, C} 2. {C, D} 3. {D, E} 
  
  The output was not {A,B} {A,C} {B,C} {C,D} {D,E} because the minimum number of possible output sets is 3 instead of 5 as given above. Otherwise it would satisfy the first two conditions, but not the third condition.
  
  As another example, consider the same DAG vertex set V but where the edge set E is empty. Since all vertex pairs would be unreachable, the expected output is the vertex set V itself.
  
  The problem is fixed-parameter tractable, where the parameter is the number of output sets.
  
  Define an undirected graph $G'$ with an edge $(v,w)$ for each unreachable pair of vertices in the original graph. Now you want to find a minimal edge clique cover of $G'$. The minimal edge clique cover problem is fixed-parameter tractable, where the parameter is the number of cliques.
  
  I don't know if there is a polynomial-time algorithm. The minimal edge clique cover problem is NP-complete for general graphs. I don't know if $G'$ has any special properties that makes the minimal edge clique cover problem easy on it. (It's not necessarily chordal, unfortunately.)
explanation: The text extract describes a problem related to directed acyclic graphs (DAGs) and asks for an efficient algorithm to find the minimum number of sets of unreachable vertices. The problem is detailed with examples and a proposed approach involving the transformation of the DAG into an undirected graph and solving the minimal edge clique cover problem. The extract contains a question about the existence of a polynomial-time algorithm for this problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: World's most popular travel blog for travel bloggers.
  
  # [Solved]: Minimum number of sets of unreachable vertices for directed acyclic graph (DAG)
  
  , ,
  Problem Detail:
  
  I have a DAG with vertices $V$ and edges $E$. If $v,w \in V$ are vertices such that $v$ is not reachable from $w$ and $w$ is not reachable from $v$, I will say that $\langle v,w \rangle$ is an unreachable pair.
  
  I want to implement an efficient algorithm which takes this DAG as input, and produces as output a list of sets of pvertices such that:
  
  1. For each unreachable pair of vertices $\langle v,w \rangle$, at least one of the output sets contains both $v$ and $w$,
  2. Every pair of vertices $v,w$ that are contained in the same output set must be an unreachable pair, and
  3. The number of output sets must be the minimum possible.
  
  Is there a polynomial-time algorithm for this problem?
  
  As an example, we have the following DAG with topologically ordered vertices A,B,C,D,E:
  
  A is connected to [D, E] B is connected to [D, E] C is connected to [E] D is connected to [] E is connected to [] 
  
  which means that we have the following unreachable vertex pairs:
  
  1. <A, B> 2. <A, C> 3. <B, C> 4. <C, D> 5. <D, E> 
  
  The expected output is composed of the following three vertex sets:
  
  1. {A, B, C} 2. {C, D} 3. {D, E} 
  
  The output was not {A,B} {A,C} {B,C} {C,D} {D,E} because the minimum number of possible output sets is 3 instead of 5 as given above. Otherwise it would satisfy the first two conditions, but not the third condition.
  
  As another example, consider the same DAG vertex set V but where the edge set E is empty. Since all vertex pairs would be unreachable, the expected output is the vertex set V itself.
  
  The problem is fixed-parameter tractable, where the parameter is the number of output sets.
  
  Define an undirected graph $G'$ with an edge $(v,w)$ for each unreachable pair of vertices in the original graph. Now you want to find a minimal edge clique cover of $G'$. The minimal edge clique cover problem is fixed-parameter tractable, where the parameter is the number of cliques.
  
  I don't know if there is a polynomial-time algorithm. The minimal edge clique cover problem is NP-complete for general graphs. I don't know if $G'$ has any special properties that makes the minimal edge clique cover problem easy on it. (It's not necessarily chordal, unfortunately.)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract describes a problem related to directed acyclic graphs (DAGs) and asks for an efficient algorithm to find the minimum number of sets of unreachable vertices. The problem is detailed with examples and a proposed approach involving the transformation of the DAG into an undirected graph and solving the minimal edge clique cover problem. The extract contains a question about the existence of a polynomial-time algorithm for this problem.
  
  Label: question