An application of simultaneous diophantine approximation in combinatorial optimization
 András Frank,
 Éva Tardos
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective functionw. Our preprocessing algorithm replacesw by an integral valuedw whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions asw.
As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial.
Further we apply the preprocessing technique to make H. W. Lenstra’s and R. Kannan’s Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used.
The method relies on simultaneous Diophantine approximation.
 R. E. Bixby, O. M.C. Marcotte andL. E. Trotter, Jr., Packing and covering with integral flows in integral supplydemand networks,Report No. 84327—OR, Institut für Ökonometrie und Operations Research, University Bonn, Bonn, West Germany.
 J. W. S. Cassels,An Introduction to the Theory of Numbers, Berlin, Heidelberg, New York, Springer, 1971.
 W. H. Cunningham, Testing membership in matroid polyhedra,Journal of Combinatorial Theory B,36 (1984), 161–188. CrossRef
 W. H. Cunningham andA. Frank, A primal dual algorithm for submodular flows,Mathematics of Operations Research.,10 (1985).
 E. A. Dinits, Algorithm for solution of a problem of maximum flow in a network with power estimation,Soviet Math. Dokl.,11 (1970), 1277–1280.
 J. Edmonds, Minimum partition of a matroid into independent subsets,Research of the Nat. Bureau of Standards 69 B (1965), 67–72.
 J. Edmonds, System of distinct representatives and linear algebra,J. Res. Nat. Bur. Standards,71 B (1967), 241–245.
 J. Edmonds andR. Giles, A minmax relation for submodular functions on graphs,Annals of Discrete Mathematics,1 (1977), 185–204. CrossRef
 J. Edmonds andR. M. Karp, Theoretical improvements in the algorithmic efficiency for network flow problems,J. ACM,19 (1972), 248–264. CrossRef
 S. Fujishige, A capacity rounding algorithm for the minimumcost circulation problem: a dual framework of the Tardos algorithm,Mathematical Programming,35 (1986), 298–309. CrossRef
 Z. Galil andÉ. Tardos, AnO(n ^{2}(m+n logn)·logn) minimum cost flow algorithm,in: Proc., 27th Annual Symposium on Foundations of Computer Science (1986), 1–9.
 M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization.Combinatorica,1 (1981), 169–197.
 M. Grötschel, L. Lovász andA. Schrijver,The ellipsoid method and combinatorial optimization, Springer Verlag,to appear.
 R. Kannan, Improved algorithms for integer programming and related lattice problems,in: Proc.,15 th Annual ACM Symposium on the Theory of Computing (1983), 193–206.Final version: Minkowski’s Convex Body Theorem and Integer Programming, CarnegieMellon University,Report No. 86–105.
 A. K. Lenstra, H. W. Lenstra, Jr. andL. Lovász, Factoring polynomials with rational coefficients,Math. Ann.,261 (1982), 515–534. CrossRef
 H. W. Lenstra, Jr., Integer programming with a fixed number of variables,Math. of Operations Research,8 (1983), 538–548. CrossRef
 É. Tardos, A strongly polynomial minimum cost circulation algorithm,Combinatorica,5 (1985), 247–255.
 É. Tardos, A strongly polynomial algorithm to solve combinatorial linear programs,Operations Research, (1986), No. 2
 U. Zimmermann, Minimization of submodular flows,Discrete Applied Math.,4 (1982), 303–323. CrossRef
 Title
 An application of simultaneous diophantine approximation in combinatorial optimization
 Journal

Combinatorica
Volume 7, Issue 1 , pp 4965
 Cover Date
 19870301
 DOI
 10.1007/BF02579200
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 68 E 10
 Industry Sectors
 Authors

 András Frank ^{(1)}
 Éva Tardos ^{(2)}
 Author Affiliations

 1. Institute of Mathematics, Eötvös University, P. O. B. 323, 1445, Budapest, Hungary
 2. Institute of Mathematics, Eötvös University, P. O. B. 323, 1445, Budapest, Hungary