Approximation algorithms for flexible graph connectivity
We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26, 2020, Math Program 1–33, 2021). Let k ≥ 1 , p ≥ 1 and q ≥ 0 be integers. In an instance of the ( p , q )-Flexible Graph Connectivity problem,...
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| Published in: | Mathematical programming Vol. 204; no. 1-2; pp. 493 - 516 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2024
|
| Subjects: | |
| ISSN: | 0025-5610, 1436-4646 |
| Online Access: | Get full text |
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| Summary: | We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26, 2020, Math Program 1–33, 2021). Let
k
≥
1
,
p
≥
1
and
q
≥
0
be integers. In an instance of the (
p
,
q
)-Flexible Graph Connectivity problem, denoted
(
p
,
q
)
-FGC
, we have an undirected connected graph
G
=
(
V
,
E
)
, a partition of
E
into a set of safe edges
S
and a set of unsafe edges
U
, and nonnegative costs
c
:
E
→
R
≥
0
on the edges. A subset
F
⊆
E
of edges is feasible for the
(
p
,
q
)
-FGC
problem if for any set
F
′
⊆
U
with
|
F
′
|
≤
q
, the subgraph
(
V
,
F
\
F
′
)
is
p
-edge connected. The algorithmic goal is to find a feasible solution
F
that minimizes
c
(
F
)
=
∑
e
∈
F
c
e
. We present a simple 2-approximation algorithm for the
(
1
,
1
)
-FGC
problem via a reduction to the minimum-cost rooted 2-arborescence problem. This improves on the 2.527-approximation algorithm of Adjiashvili et al. Our 2-approximation algorithm for the
(
1
,
1
)
-FGC
problem extends to a
(
k
+
1
)
-approximation algorithm for the
(
1
,
k
)
-FGC
problem. We present a 4-approximation algorithm for the
(
k
,
1
)
-FGC
problem, and an
O
(
q
log
|
V
|
)
-approximation algorithm for the
(
p
,
q
)
-FGC
problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted
(
1
,
1
)
-FGC
problem by presenting a 16/11-approximation algorithm. The
(
p
,
q
)
-FGC
problem is related to the well-known
Capacitated
k
-
Connected Subgraph
problem (denoted
Cap-
k
-ECSS
) that arises in the area of Capacitated Network Design. We give a
min
(
k
,
2
u
max
)
-approximation algorithm for the
Cap-
k
-ECSS
problem, where
u
max
denotes the maximum capacity of an edge. |
|---|---|
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-023-01961-5 |