In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials fe [X] into irreducible factors in 7L[X]. Here we call fe7L[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(n 12 +n 9 (logfI) 3). Here fE[X] is the polynomial to be factored, n = deg(f) is the degree of J and ZaiXi = for a polynomial Za 1 X 1 with real coefficients a 1. An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a p-adic irreducible factor h of J to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h 0 of f in ZL[X] that is divisible by h. The condition that h 0 is divisible by h means that h 0 belongs to a certain lattice, and the condition that h 0 divides f implies that the coefficients of h 0 are relatively small. It follows that we must look for a "small" element in that lattice, and this is done by means of a basis reduction algorithm. It turns out that this enables us to determine h 0. The algorithm is repeated until all irreducible factors of f have been found. The basis reduction algorithm that we employ is new, and it is described and analysed in Sect. 1. It improves the algorithm given in a preliminary version of [9, Sect. 3]. At the end of Sect. 1 we briefly mention two applications of the new algorithm to diophantine approximation. The connection between factors off and reduced bases of a lattice is treated in detail in Sect. 2. The theory presented here extends a result appearing in [8, Theorem 2]. It should be remarked that the latter result, which is simpler to prove, would in principle have sufficed for our purpose.

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PrefaceHistorically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well.Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics.The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization.The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation. Methods from the geometry of numbers were introduced in integer programming by H. W. Lenstra. An important element of his technique, called basis reduction, goes in fact back to Hermite. An efficient version of basis reduction yields a polynomial time algorithm useful not only in combinatorial optimization, but also in fields like number theory, algebra, and cryptography.A combination of these two methods results in a powerful tool for combinatorial optimization. It yields a theoretical framework in which the polynomial time solvability of a large number of combinatorial optimization problems can be shown quite easily. It establishes the algorithmic equivalence of problems which are "dual" in various senses.Being this general, this method cannot be expected to give running times comparable with special-purpose algorithms. Our policy in this book is, therefore, not to attempt to obtain the best possible running times; rather, it is to derive just the polynomial time solvability of the problems as quickly and painlessly as VI Preface possible. Thus, our results are best conceived as "almost pure" existence results for polynomial time algorithms for certain problems and classes of problems.Nevertheless, we could not get around quite a number of tedious technical details. We did try to outline the essential ideas in certain sections, which should give an outline of the underlying geometric an...

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