Part I1 Lattices and linear diophantine equations Lattices and linear diophantine equations can be described, to a large extent, parallel to linear spaces and linear equations (Part I) and to polyhedra and linear inequalities (Part 111). In this part we first discuss in Chapter 4 the theoretical side of lattices and linear diophantine equations, basically due to Hermite and Minkowski. Fundamental is the Hermite normalform of'a matrix. Next, in Chapters 5 and 6, we go into the algorithmic side. In Chapter 5 we discuss first the classical Euclidean algorithm for finding the g.c.d., which also can be used for solving one linear diophantine equation in polynomial time. Next an extension to solving systems of linear diophantine equations in polynomial time is described, based on finding the Hermite normal form of a matrix. Chapter 6 starts with a discussion of the classical continuedfraction method for approxi- mating a real number by rational numbers of low denominator (diophantine approxi- mation). The continued fraction method is a variant of the Euclidean algorithm. Next we describe Lovasz's important basis reduction method for lattices. This method applies to simultaneous diophantine approximation, to solving systems of linear diophantine equations, to linear programming (see Section 14.I), to integer linear programming (see Section 18.4), to factoring polynomials over Q in polynomial time, and to several other problems. Part I1 concludes with historical and further notes on lattices and linear diophantine equations. Copyright © 1998. Wiley. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost)43 - printed on 1/15/2015 10:15 AM via TEXAS A&M UNIV - COLLEGE STATION AN: 17885 ; Schrijver, Alexander.; Theory of Linear and Integer Programming Account: s8516548 This page intentionally left blank Copyright © 1998. Wiley. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 1/15/2015 10:15 AM via TEXAS A&M UNIV - COLLEGE STATION AN: 17885 ; Schrijver, Alexander.; Theory of Linear and Integer Programming Account: s8516548 4 Theory of lattices and linear diophantine equations In this chapter we describe some results from elementary number theory concerning lattices and linear diophantine equations. In Section 4.1 we show that each rational matrix of full row rank can be brought into the so-called Hermite normalform. We derive a characterization for the feasibility of a system of linear diophantine equations and derive the existence of a linearly independent basis for any rational lattice. In Section 4.2 we show uniqueness of the Hermite normal form. In Section 4.3 we study unirnodular matrices, and in Section 4.4 we go into some further theoretical aspects. 4.1. THE HERMITE NORMAL FORM A matrix of full row rank is said to be in Hermite normal form if it has the form [B 01, where B is a nonsingular, lower triangular, nonnegative matrix, in which each row has a unique maximum entry, which is located on the main diagonal of B. The following operations on a matrix are called elementary (unirnodular) column operations: (1) (i) exchanging two columns; (ii) multiplying a column by - 1; (iii) adding an integral multiple of one column to another column. Theorem 4.1 (Hermite normal form theorem). Each rational matrix of full row rank can be brought into Hermite normal form by a series of elementary column operations. Proof. Let A be a rational matrix of full row rank. Without loss of generality, A is integral. Suppose we have transformed A, by elementary column operations, to the form where B is lower triangular and with positive diagonal. Copyright © 1998. Wiley. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. [ i] EBSCO Publishing : eBook Collection (EBSCOhost)45 - printed on 1/15/2015 10:15 AM via TEXAS A&M UNIV - COLLEGE STATION AN: 17885 ; Schrijver, Alexander.; Theory of Linear and Integer Programming Account: s8516548 46 4 Theory of lattices and linear diophantine equations Now with elementary column operations we can modify D so that its first row (dl1,...,dlk) is nonnegative, and so that the sum 6,, +"'+61k is as small as possible. We may assume that 6 , 3 6 ,,3 ' 3 6lk. Then 6, > 0, as A has full row rank. Moreover, if S,, > 0, by subtracting the second column of D from the first column of D,the first row will have smaller sum, contradicting our assumption. Hence 6, ,= *. * = 61, =0, and we have obtained a larger lower triangular matrix (this last procedure corresponds to the Euclidean algorithm- cf. Section 5.1). By repeating this procedure, the matrix A finally will be transformed into [B 01 with B=(pij) lower triangular with positive diagonal. Next do the following: (2) for i = 2,. ,n (:= order of B), do the following: for j = 1,...,i - 1, add an integer multiple of the ith column of B to thejth column of B so that the (i,j)th entry of B will be nonnegative and less than pii. (So the procedure is applied in the order: (i,j) = (2, l), (3, I), (3,2), (4, l), (4,2), (4,3),....) It is easy to see that after these elementary column operations the matrix is in Hermite normal form. 0 In Theorem 4.2 below we shall see that any rational matrix of full row rank has a unique Hermite normal form. So we can speak of the Hermite normal form of a matrix. A first corollary gives necessary and suficient conditions for the feasibility of systems of linear diophantine equations. This result is a corollary of Kronecker's approximation theorem [1884b1 (cf. Koksma [1936: p. 831 and the historical notes at the end of Part IT). Corollary 4.la. Let A be a rational matrix and let b be a rational column vector. Then the system Ax = b has an integral solution x, ifand only ifyb is an integer .for each rational row vector y.for which yA is inteyrul. Proof. Necessity of the condition is trivial: if x and yA are integral vectors and Ax = b, then yb =yAx is an integer. To see sufficiency, suppose yb is an integer whenever yA is integral. Then Ax = b has a (possibly fractional) solution, since otherwise yA =0 and yb =4 for some rational vector y (by linear algebra-see Corollary 3.1 b). So we may assume that the rows of A are linearly independent. Now both sides of the equivalence to be proved are invariant under elementary column operations (1). So by Theorem 4.1 we may assume that A is in Hermite normal form [B 01. Since B-'[B O] = [I O] is an integral matrix, it follows from our assumption that also B-'b is an integral vector. Since (3) [B O]( 'i'") = b the vector x:= (';lb) is an integral solution for Ax = b. 0 Copyright © 1998. Wiley. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 1/15/2015 10:15 AM via TEXAS A&M UNIV - COLLEGE STATION AN: 17885 ; Schrijver, Alexander.; Theory of Linear and Integer Programming Account: s8516548 4.1 The Hermite normal form 47 Corollary 4.la can be seen as the integer analogue of Corollary 3.lb and of Corollary 7.ld (Farkas' lemma). It can also be interpreted in terms of groups and lattices. A subset A of R" is called an (additive) group if: (4) (i) OEA (ii) if x,y~Athen x + YEAand - XEA. The group is said to be generated by a,, . ,a, if (5) A = {Alal+ - * - + AmamI AI,. ..,&,E Z). The group is called a lattice if it can be generated by linearly independent vectors. These vectors then are called a basis for the lattice. Note that if matrix B comes from matrix A by elementary column operations, then the columns of B generate the same group as the columns of A. Theorem 4.1 has also the following consequence. Corollary 4.lb. If a,, . ,a, are rational vectors, then the group generated by a,, ... ,a, is a lattice, i.e. is generated by linearly independent vectors. Proof. We may assume that a,, ...,a, span all space. (Otherwise we could apply a linear transformation to a lower dimensional space.) Let A be the matrix with columns a,, . ,a,,, (so A has full row rank). Let [B 01 be the Hermite normal form of A. Then the columns of B are linearly independent vectors generating the same group as a,, . .,a,. U So if a,, . .,a, are rational vectors we can speak of the lattice generated by a,,. ..,U,,,, Given a rational matrix A, Corollary 4.la gives necessary and sufficient conditions for being an element of the lattice A generated by the columns of A. The proof of Corollary 4.la implies that if A has full row rank, with Hermite normal form [B 0) (B lower triangular), then b belongs to A if and only if B- 'b is integral. Another consequence of Theorem 4.1 is: Corollary 4.1~.Let A be an integral m x n-matrix of full row rank. Then the following are equivalent: (i) the g.c.d. of the subdeterminants of A of order m is 1; (ii) the system Ax = b has an integral solution x,for each integral vector b; (iii) for each vector y, ifyA is integral, then y is integral.