TY - JOUR

T1 - On the maximum weight minimal separator

AU - Hanaka, Tesshu

AU - Bodlaender, Hans L.

AU - van der Zanden, Tom C.

AU - Ono, Hirotaka

N1 - Funding Information:
This study is partially supported by NWO Gravity grant “Networks” (024.002.003) and JSPS KAKENHI Grant Numbers 17H01698, 17K19960, 18H06469.
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2019/12/3

Y1 - 2019/12/3

N2 - Given an undirected and connected graph G=(V,E) and two vertices s,t∈V, a vertex subset S that separates s and t is called an s-t separator, and an s-t separator is called minimal if no proper subset of S separates s and t. Moreover, we say that a set S is a minimal separator of G if S is a minimal s-t separator for some s and t. In this paper, we consider finding a minimal (s-t) separator with maximum weight on a vertex-weighted graph. We first prove that these problems are NP-hard. On the other hand, we give an O⁎(twO(tw))-time deterministic algorithm based on tree decompositions where O⁎ is the order notation omitting the polynomial factor of n. Moreover, we improve the algorithm by using the Rank-Based approach and the running time is O⁎(38⋅2ω)tw. Finally, we give an O⁎(9tw⋅W2)-time randomized algorithm to determine whether there exists a minimal (s-t) separator where W is its weight and tw is the treewidth of G.

AB - Given an undirected and connected graph G=(V,E) and two vertices s,t∈V, a vertex subset S that separates s and t is called an s-t separator, and an s-t separator is called minimal if no proper subset of S separates s and t. Moreover, we say that a set S is a minimal separator of G if S is a minimal s-t separator for some s and t. In this paper, we consider finding a minimal (s-t) separator with maximum weight on a vertex-weighted graph. We first prove that these problems are NP-hard. On the other hand, we give an O⁎(twO(tw))-time deterministic algorithm based on tree decompositions where O⁎ is the order notation omitting the polynomial factor of n. Moreover, we improve the algorithm by using the Rank-Based approach and the running time is O⁎(38⋅2ω)tw. Finally, we give an O⁎(9tw⋅W2)-time randomized algorithm to determine whether there exists a minimal (s-t) separator where W is its weight and tw is the treewidth of G.

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U2 - 10.1016/j.tcs.2019.09.025

DO - 10.1016/j.tcs.2019.09.025

M3 - Article

AN - SCOPUS:85073029754

SN - 0304-3975

VL - 796

SP - 294

EP - 308

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -