# An application of simultaneous diophantine approximation in combinatorial optimization

- 388 Downloads
- 74 Citations

## Abstract

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 function*w*. Our preprocessing algorithm replaces*w* by an integral valued*-w* whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions as*w*.

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.

## AMS subject classification (1980)

68 E 10## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R. E. Bixby, O. M.-C. Marcotte andL. E. Trotter, Jr., Packing and covering with integral flows in integral supply-demand networks,
*Report No.*84327—OR, Institut für Ökonometrie und Operations Research, University Bonn, Bonn, West Germany.Google Scholar - [2]J. W. S. Cassels,
*An Introduction to the Theory of Numbers*, Berlin, Heidelberg, New York, Springer, 1971.Google Scholar - [3]W. H. Cunningham, Testing membership in matroid polyhedra,
*Journal of Combinatorial Theory B*,**36**(1984), 161–188.zbMATHCrossRefMathSciNetGoogle Scholar - [4]W. H. Cunningham andA. Frank, A primal dual algorithm for submodular flows,
*Mathematics of Operations Research.*,**10**(1985).Google Scholar - [5]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.zbMATHGoogle Scholar - [6]J. Edmonds, Minimum partition of a matroid into independent subsets,
*Research of the Nat. Bureau of Standards***69 B**(1965), 67–72.MathSciNetGoogle Scholar - [7]J. Edmonds, System of distinct representatives and linear algebra,
*J. Res. Nat. Bur. Standards*,**71 B**(1967), 241–245.MathSciNetGoogle Scholar - [8]J. Edmonds andR. Giles, A min-max relation for submodular functions on graphs,
*Annals of Discrete Mathematics*,**1**(1977), 185–204.MathSciNetCrossRefGoogle Scholar - [9]J. Edmonds andR. M. Karp, Theoretical improvements in the algorithmic efficiency for network flow problems,
*J. ACM*,**19**(1972), 248–264.zbMATHCrossRefGoogle Scholar - [10]S. Fujishige, A capacity rounding algorithm for the minimum-cost circulation problem: a dual framework of the Tardos algorithm,
*Mathematical Programming*,**35**(1986), 298–309.zbMATHCrossRefMathSciNetGoogle Scholar - [11]Z. Galil andÉ. Tardos, An
*O*(*n*^{2}(*m*+*n*log*n*)·log*n*) minimum cost flow algorithm,*in: Proc., 27th Annual Symposium on Foundations of Computer Science*(1986), 1–9.Google Scholar - [12]M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization.
*Combinatorica*,**1**(1981), 169–197.zbMATHMathSciNetGoogle Scholar - [13]M. Grötschel, L. Lovász andA. Schrijver,
*The ellipsoid method and combinatorial optimization*, Springer Verlag,*to appear*.Google Scholar - [14]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, Carnegie-Mellon University,*Report*No. 86–105.Google Scholar - [15]A. K. Lenstra, H. W. Lenstra, Jr. andL. Lovász, Factoring polynomials with rational coefficients,
*Math. Ann.*,**261**(1982), 515–534.zbMATHCrossRefMathSciNetGoogle Scholar - [16]H. W. Lenstra, Jr., Integer programming with a fixed number of variables,
*Math. of Operations Research*,**8**(1983), 538–548.zbMATHMathSciNetCrossRefGoogle Scholar - [17]É. Tardos, A strongly polynomial minimum cost circulation algorithm,
*Combinatorica*,**5**(1985), 247–255.zbMATHMathSciNetGoogle Scholar - [18]É. Tardos, A strongly polynomial algorithm to solve combinatorial linear programs,
*Operations Research*, (1986), No. 2Google Scholar - [19]U. Zimmermann, Minimization of submodular flows,
*Discrete Applied Math.*,**4**(1982), 303–323.zbMATHCrossRefGoogle Scholar