A rounding technique for the polymatroid membership problem
We present an efficient technique for finding a subset which maximizes ω( X) − ϱ( X) over all subsets of a set E, where ω and ϱ are real modular and polymatroid functions respectively, using as a subroutine an algorithm which finds such a set for functions ω , ϱ which are near ω, ϱ respectively. In...
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| Published in | Linear algebra and its applications Vol. 221; pp. 41 - 57 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.05.1995
|
| Online Access | Get full text |
| ISSN | 0024-3795 1873-1856 |
| DOI | 10.1016/0024-3795(93)00222-L |
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| Abstract | We present an efficient technique for finding a subset which maximizes
ω(
X) −
ϱ(
X) over all subsets of a set
E, where ω and ϱ are
real modular and polymatroid functions respectively, using as a subroutine an algorithm which finds such a set for functions
ω
,
ϱ
which are near ω, ϱ respectively. In particular we can choose
ω
,
ϱ
to be rational with denominators equal to 12|
E|
3 if we can assume, whenever
ϱ(
X) +
ϱ(
Y) >
ϱ(
X ∪
Y) +
ϱ(
X ∩
Y), that the difference between the two sides is at least one. By applying our technique, we construct an
O(|
E|
3
r
2) algorithm for the case where ϱ is a matroid rank function. |
|---|---|
| AbstractList | We present an efficient technique for finding a subset which maximizes
ω(
X) −
ϱ(
X) over all subsets of a set
E, where ω and ϱ are
real modular and polymatroid functions respectively, using as a subroutine an algorithm which finds such a set for functions
ω
,
ϱ
which are near ω, ϱ respectively. In particular we can choose
ω
,
ϱ
to be rational with denominators equal to 12|
E|
3 if we can assume, whenever
ϱ(
X) +
ϱ(
Y) >
ϱ(
X ∪
Y) +
ϱ(
X ∩
Y), that the difference between the two sides is at least one. By applying our technique, we construct an
O(|
E|
3
r
2) algorithm for the case where ϱ is a matroid rank function. |
| Author | Narayanan, H. |
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| Cites_doi | 10.1016/0095-8956(84)90023-6 10.1007/BF02579273 10.1287/mnsc.22.11.1268 10.1007/BF02579361 10.1287/moor.11.2.362 |
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| References | Picard (BIB5) 1976; 22 Tardos, Tovey, Trick (BIB4) 1986; 11 Welsh (BIB6) 1976 Cunningham (BIB2) 1985 Grotschel, Lovasz, Schrijver (BIB3) 1981; 1 Cunningham (BIB1) 1984; 36 Cunningham (10.1016/0024-3795(93)00222-L_BIB2) 1985 Tardos (10.1016/0024-3795(93)00222-L_BIB4) 1986; 11 Grotschel (10.1016/0024-3795(93)00222-L_BIB3) 1981; 1 Welsh (10.1016/0024-3795(93)00222-L_BIB6) 1976 Picard (10.1016/0024-3795(93)00222-L_BIB5) 1976; 22 Cunningham (10.1016/0024-3795(93)00222-L_BIB1) 1984; 36 |
| References_xml | – volume: 36 start-page: 161 year: 1984 end-page: 188 ident: BIB1 article-title: Testing membership in matroid polyhedra publication-title: J. Combin. Theory Ser. B – year: 1976 ident: BIB6 article-title: Matroid Theory – volume: 1 start-page: 169 year: 1981 end-page: 197 ident: BIB3 article-title: The ellipsoid method and its consequences in combinatorial optimisation publication-title: Combinatorica – volume: 11 start-page: 362 year: 1986 end-page: 370 ident: BIB4 article-title: Layered augmenting path algorithms publication-title: Math. Oper. Res. – start-page: 185 year: 1985 end-page: 190 ident: BIB2 article-title: On submodular function minimization publication-title: Combinatorica – volume: 22 start-page: 1268 year: 1976 end-page: 1277 ident: BIB5 article-title: Maximal closure of a graph and application to combinatorial problems publication-title: Management Sci. – volume: 36 start-page: 161 year: 1984 ident: 10.1016/0024-3795(93)00222-L_BIB1 article-title: Testing membership in matroid polyhedra publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(84)90023-6 – volume: 1 start-page: 169 issue: 2 year: 1981 ident: 10.1016/0024-3795(93)00222-L_BIB3 article-title: The ellipsoid method and its consequences in combinatorial optimisation publication-title: Combinatorica doi: 10.1007/BF02579273 – year: 1976 ident: 10.1016/0024-3795(93)00222-L_BIB6 – volume: 22 start-page: 1268 issue: 11 year: 1976 ident: 10.1016/0024-3795(93)00222-L_BIB5 article-title: Maximal closure of a graph and application to combinatorial problems publication-title: Management Sci. doi: 10.1287/mnsc.22.11.1268 – start-page: 185 year: 1985 ident: 10.1016/0024-3795(93)00222-L_BIB2 article-title: On submodular function minimization publication-title: Combinatorica doi: 10.1007/BF02579361 – volume: 11 start-page: 362 year: 1986 ident: 10.1016/0024-3795(93)00222-L_BIB4 article-title: Layered augmenting path algorithms publication-title: Math. Oper. Res. doi: 10.1287/moor.11.2.362 |
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| Snippet | We present an efficient technique for finding a subset which maximizes
ω(
X) −
ϱ(
X) over all subsets of a set
E, where ω and ϱ are
real modular and... |
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| StartPage | 41 |
| Title | A rounding technique for the polymatroid membership problem |
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