TY - JOUR

T1 - Approximability of minimum certificate dispersal with tree structures

AU - Izumi, Taisuke

AU - Izumi, Tomoko

AU - Ono, Hirotaka

AU - Wada, Koichi

N1 - Funding Information:
This work is supported in part by KAKENHI no. 22700010 , 21500013 , 21680001 and 22700017 , and Foundation for the Fusion of Science and Technology (FOST).
Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/8/2

Y1 - 2015/8/2

N2 - 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 a 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; a tree structure is frequently adopted to construct an efficient communication network. 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(n1+ε|R|), where ε is an arbitrarily small positive constant number.

AB - 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 a 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; a tree structure is frequently adopted to construct an efficient communication network. 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(n1+ε|R|), where ε is an arbitrarily small positive constant number.

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U2 - 10.1016/j.tcs.2015.01.007

DO - 10.1016/j.tcs.2015.01.007

M3 - Article

AN - SCOPUS:84943615489

SN - 0304-3975

VL - 591

SP - 5

EP - 14

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -