On the Parameterized Complexity of the Expected Coverage Problem
The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k , the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is...
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| Vydáno v: | Theory of computing systems Ročník 66; číslo 2; s. 432 - 453 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.04.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 1432-4350, 1433-0490 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | The
Maximum Covering Location Problem
(MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer
k
, the MCLP seeks to find
k
potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge
e
of the network, we associate its survival probability 0 ≤
p
e
≤ 1, or equivalently, its failure probability 1 −
p
e
. The failure correlation in LRO is the following: If an edge
e
fails then every edge
e
′
with
p
e
′
≤
p
e
surely fails. The task is to identify the positions of
k
facilities that maximize the
expected
coverage. We refer to this problem as
Expected Coverage
problem. We study the
Expected Coverage
problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the
Expected Coverage
problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that
Expected Coverage
is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time
2
O
(
tw
log
Δ
)
n
O
(
1
)
, where
tw
is the treewidth,
Δ
is the maximum vertex degree, and
n
the number of vertices of the input graph. In particular, since
Δ
≤
n
, it means the problem is solvable in time
n
O
(
tw
)
, that is, is in XP parameterized by treewidth. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1432-4350 1433-0490 |
| DOI: | 10.1007/s00224-022-10073-0 |