Minimum certificate dispersal with tree structures

  • Authors:

  • Taisuke Izumi;Tomoko Izumi;Hirotaka Ono;Koichi Wada

  • Affiliations:

  • Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Japan;College of Information Science and Engineering, Ritsumeikan University, Kusatsu, Japan;Faculty of Economics, Kyushu University, Fukuoka, Japan;Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Japan

  • Venue:
  • TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

Abstract

Given an n -vertex graph G =(V ,E ) and a set R ⊆{{x ,y }|x ,y ∈V } of requests, we consider to assign a set of edges to each vertex in G so that for every request {u , v } in R the union of the edge sets assigned to u and v contains a path from u to v . The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex, which is originally motivated by the design of secure communications in distributed computing. This problem has been shown to be LOGAPX-hard for general directed topologies of G and R . In this paper, we consider the complexity of MCD for more practical topologies of G and R , that is, when G or R forms an (undirected) tree; tree structures are frequently adopted to construct efficient communication networks. We first show that MCD is still APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree Δ of the tree: MCD for tree request set with constant Δ is solvable in polynomial time, while that with Δ=Ω(n ) is 2.78-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that if G is a tree, the problem can be solved in O (n 1+ε |R |), where ε is an arbitrarily small positive constant number.