Local Approximability of Max-Min and Min-Max Linear Programs
In a max-min LP , the objective is to maximise ω subject to A x ≤ 1 , C x ≥ ω 1 , and x ≥ 0 . In a min-max LP , the objective is to minimise ρ subject to A x ≤ ρ 1 , C x ≥ 1 , and x ≥ 0 . The matrices A and C are nonnegative and sparse: each row a i of A has at most Δ I positive elements, and each r...
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| Published in: | Theory of computing systems Vol. 49; no. 4; pp. 672 - 697 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer-Verlag
01.11.2011
Springer Nature B.V |
| Subjects: | |
| ISSN: | 1432-4350, 1433-0490 |
| Online Access: | Get full text |
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| Summary: | In a
max-min LP
, the objective is to maximise
ω
subject to
A
x
≤
1
,
C
x
≥
ω
1
, and
x
≥
0
. In a
min-max LP
, the objective is to minimise
ρ
subject to
A
x
≤
ρ
1
,
C
x
≥
1
, and
x
≥
0
. The matrices
A
and
C
are nonnegative and sparse: each row
a
i
of
A
has at most Δ
I
positive elements, and each row
c
k
of
C
has at most Δ
K
positive elements.
We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on
local algorithms
(constant-time distributed algorithms). We show that for any Δ
I
≥2, Δ
K
≥2, and
ε
>0 there exists a local algorithm that achieves the approximation ratio Δ
I
(1−1/Δ
K
)+
ε
. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio Δ
I
(1−1/Δ
K
) for any Δ
I
≥2 and Δ
K
≥2. |
|---|---|
| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
| ISSN: | 1432-4350 1433-0490 |
| DOI: | 10.1007/s00224-010-9303-6 |