A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands
Given an edge-weighted directed graph G = ( V , E ) on n vertices and a set T = { t 1 , t 2 , … , t p } of p terminals, the objective of the Strongly Connected Steiner Subgraph ( p -SCSS) problem is to find an edge set H ⊆ E of minimum weight such that G [ H ] contains an t i → t j path for each 1 ≤...
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| Vydáno v: | Algorithmica Ročník 77; číslo 4; s. 1216 - 1239 |
|---|---|
| Hlavní autoři: | , , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.04.2017
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Given an edge-weighted directed graph
G
=
(
V
,
E
)
on
n
vertices and a set
T
=
{
t
1
,
t
2
,
…
,
t
p
}
of
p
terminals, the objective of the
Strongly Connected Steiner Subgraph
(
p
-SCSS) problem is to find an edge set
H
⊆
E
of minimum weight such that
G
[
H
] contains an
t
i
→
t
j
path for each
1
≤
i
≠
j
≤
p
. The
p
-SCSS problem is NP-hard, but Feldman and Ruhl [FOCS ’99; SICOMP ’06] gave a novel
n
O
(
p
)
time algorithm. In this paper, we investigate the computational complexity of a variant of 2-SCSS where we have demands for the number of paths between each terminal pair. Formally, the
2
-SCSS-
(
k
1
,
k
2
)
problem is defined as follows: given an edge-weighted directed graph
G
=
(
V
,
E
)
with weight function
ω
:
E
→
R
≥
0
, two terminal vertices
s
,
t
, and integers
k
1
,
k
2
; the objective is to find a set of
k
1
paths
F
1
,
F
2
,
…
,
F
k
1
from
s
⇝
t
and
k
2
paths
B
1
,
B
2
,
…
,
B
k
2
from
t
⇝
s
such that
∑
e
∈
E
ω
(
e
)
·
ϕ
(
e
)
is minimized, where
ϕ
(
e
)
=
max
{
|
{
i
∈
[
k
1
]
:
e
∈
F
i
}
|
,
|
{
j
∈
[
k
2
]
:
e
∈
B
j
}
|
}
. For each
k
≥
1
, we show the following:
The
2
-SCSS-
(
k
,
1
)
problem can be solved in time
n
O
(
k
)
.
A matching lower bound for our algorithm: the
2
-SCSS-
(
k
,
1
)
problem does not have an
f
(
k
)
·
n
o
(
k
)
time algorithm for any computable function
f
, unless the Exponential Time Hypothesis fails.
Our algorithm for
2
-SCSS-
(
k
,
1
)
relies on a structural result regarding an optimal solution followed by using the idea of a “token game” similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the
2
-SCSS-
(
k
1
,
k
2
)
problem if
min
{
k
1
,
k
2
}
≥
2
. Therefore
2
-SCSS-
(
k
,
1
)
is the most general problem one can attempt to solve with our techniques. To obtain the lower bound matching the algorithm, we reduce from a special variant of the
Grid Tiling
problem introduced by Marx [FOCS ’07; ICALP ’12]. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-016-0145-8 |