, Volume 7, Issue 1, pp 49-65

An application of simultaneous diophantine approximation in combinatorial optimization

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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 functionw. Our preprocessing algorithm replacesw 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 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.

This research was done while the authors were visiting the Institute for Operations Research, University of Bonn, West Germany (1984–85), and while the second author was visiting MSRI, Berkeley. Her research was supported in part by NSF Grant 8120790.