Role Assignments of Trees

1
Role Assignments of Trees
2
Graph Homomorphism

• Shows underlying structure of graph

3
Role Assignments

• Shows underlying structure of a network
• Additional condition
• Condition 1
• Condition 2

4
Example

• Used for social networks, to map agents to roles.

5
Applications

• Resource allocation on computer networks
• Role assignments on social networks
• Terrorist networks
• Somewhat small, many times hierarchical in
  structure
• Many activities are tree-like structures

6
Difficulty of the Problem

• Role assignments of graphs with two vertices
  Roberts, Sheng

Easy Bipartite Graphs
NP-complete
Easy Disconnected Graph
7
Difficulty of the Problem

• Theorem Fiala, Paulusma
• The question of R-assignability is NP-complete
  for every connected role graph R with more than
  one edge.

8
Assignments to Trees

• Theorem
• Any graph which is T-role assignable is
  bipartite.

Proof Role assignments are closed under
composition.
9
Assignments from Trees

• Theorem
• Any tree can only be assigned to a tree

10
Uniquely Mapped Vertices

• A vertex is uniquely mapped if
• Theorem
• If there is a uniquely mapped vertex, each level
  in a rooted tree is mapped to a disjoint set of
  roles (vertices) in Tr
• Theorem
• Each role assignment of a uniquely mapped tree
  can be accomplished by a series of folds of the
  tree

11
Algorithm for Uniquely Mapped Trees

• Preprocessing for TR
• Store all rooted subtrees as a list of child
  subtrees with multiplicities

12
Algorithm for Uniquely Mapped Trees

• Processing for TI
• Start at leaves and process upwards
• For each step i,
• Given children arrays c1,c2, ,ck construct the
  parent array
• For each subtree configuration sj found in the
  preprocessing step, determine if the subtree is
  possibly assignable to the nodes below.

13
Assignable Subtrees

• For a subtree si, scan to see that the
  intersection of cj and si is not empty, for each
  child j
• Solve the bipartite matching problem
• A vertex for every subtee of si
• A vertex for every child subtree array ci
• An edge between a child subtree and all subtrees
  in si for which it is assignable.

14
k-Assignable Trees

• Extend the algorithm to enumerate all mappings
  for a particular role
• Root tree at chosen vertices to provide similar
  distinct mappings.
• Important structures
• Chains between root nodes
• Subtrees extending from chains

15
Algorithm for k-Assignable Trees

• Modifications to accept bipartite graphs.
• Cannot be polynomial
• Current running time

16
References

• F.S. Roberts, L. Sheng, How hard is it to
  determine if a graph has a 2-role assignment?
  Networks, Volume 37, Issue 2, Pages 67-73
• J. Fiala, D. Paulusma, The Computational
  Complexity of the Role Assignment Problem
  Theoretical Computer Science, Volume 349, 2005,
  Pages 67-81
• L. Sheng, 2-role assignments on triangulated
  graphs Theoretical Computer Science, Volume 304,
  2003, Pages 201-214
• Everett, M.G., Borgatti, S.P. Role coloring a
  graph. Mathematical Social Sciences, Volume 26,
  1991, Pages 183188.