TY - JOUR

T1 - Complexity of the multi-service center problem

AU - Ito, Takehiro

AU - Kakimura, Naonori

AU - Kobayashi, Yusuke

N1 - Funding Information:
Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan.Partially supported by JSPS KAKENHI Grant Numbers JP17K00028 and JP18H05291, Japan.Partially supported by JSPS KAKENHI Grant Numbers JP16K16010, JP17K19960, JP18H05291, and JP19H05485, Japan.
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/11/24

Y1 - 2020/11/24

N2 - The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars.

AB - The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars.

KW - Facility location

KW - Graph algorithm

KW - Multi-service location

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U2 - 10.1016/j.tcs.2020.07.021

DO - 10.1016/j.tcs.2020.07.021

M3 - Article

AN - SCOPUS:85089150490

VL - 842

SP - 18

EP - 27

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -