Maximum agreement subtree problem

The maximum agreement subtree problem is any of several closely related problems in graph theory and computer science. In all of these problems one is given a collection of trees

      T
      
        1
      
    
    ,
    …
    ,
    
      T
      
        m
      
    
  

{\displaystyle T_{1},\ldots ,T_{m}}

each containing

    n
  

{\displaystyle n}

leaves. The leaves of these trees are given labels from some set

    L
  

{\displaystyle L}

with

      |
    
    L
    
      |
    
    =
    n
  

{\displaystyle |L|=n}

so that no pair of leaves in the same tree sharing the same label, within the same tree the labelling for each leaf is distinct. In this problem one would like to find the largest subset

      L
      ′
    
    ⊂
    L
  

{\displaystyle L'\subset L}

such that the minimal spanning subtrees containing the leaves in

      L
      ′
    
  

{\displaystyle L'}

, of

      T
      
        1
      
    
    ∣
    S
    ,
    …
    ,
    
      T
      
        m
      
    
    ∣
    S
  

{\displaystyle T_{1}\mid S,\ldots ,T_{m}\mid S}

are the "same" while preserving the labelling.

Source:

This version requires that the subtrees

      T
      
        1
      
    
    ∣
    S
    ,
    …
    ,
    
      T
      
        m
      
    
    ∣
    S
  

{\displaystyle T_{1}\mid S,\ldots ,T_{m}\mid S}

are homeomorphic to one another.

This version is the same as the maximum homeomorphic agreement subtree, but we further assume that

      T
      
        1
      
    
    ,
    …
    ,
    
      T
      
        m
      
    
  

{\displaystyle T_{1},\ldots ,T_{m}}

are rooted and that the subtrees

      T
      
        1
      
    
    ∣
    S
    ,
    …
    ,
    
      T
      
        m
      
    
    ∣
    S
  

{\displaystyle T_{1}\mid S,\ldots ,T_{m}\mid S}

contain the root node. This version of the maximum agreement subtree problem is used for the study of phylogenetic trees. Because of its close ties with phylogeny this formulation is often what is mean when one refers to the "maximum agreement subtree" problem.

There exits other formulations for example the (rooted) maximum isomorphic agreement subtree where we require the subtrees to be isomorphic to one another.

• Frequent subtree mining

• Kao, Ming-Yang; Lam, Tak-Wah; Sung, Wing-Kin; Ting, Hing-Fung (August 2001). "An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings". Journal of Algorithms. 40 (2): 212–233. arXiv:cs/0101010. doi:10.1006/jagm.2001.1163.