An Algorithmic Theory of Numbers, Graphs and Convexity - download pdf or read online

By Laszlo Lovasz

A research of the way complexity questions in computing engage with classical arithmetic within the numerical research of concerns in set of rules layout. Algorithmic designers involved in linear and nonlinear combinatorial optimization will locate this quantity specially helpful.

Two algorithms are studied intimately: the ellipsoid strategy and the simultaneous diophantine approximation procedure. even if either have been constructed to check, on a theoretical point, the feasibility of computing a few really expert difficulties in polynomial time, they seem to have useful purposes. The e-book first describes use of the simultaneous diophantine way to increase subtle rounding tactics. Then a version is defined to compute top and decrease bounds on a number of measures of convex our bodies. Use of the 2 algorithms is introduced jointly via the writer in a research of polyhedra with rational vertices. The booklet closes with a few functions of the implications to combinatorial optimization.

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If we only want that \\y — y\\ be 32 LASZL6 LOVASZ small then the best we can do is to round the entries of y independently of each other. But if we use simultaneous diophantine approximation then much more can be achieved: We can require that all linear inequalities which hold true for y and which have relatively simple coefficients remain valid for y ; even more, all such linear inequalities which "almost" hold for y should be "repaired" and hold true for y . We state this exactly in the following theorem.

How short is this 6 ? Suppose first that n > m . , am , and aTO+i = ... , 2~ 3/c3 a n ) belongs to £(A) . So in this case AN ALGORITHMIC THEORY 39 Here ja^l < 2fc by the hypothesis on the minimal polynomial of a and so Conversely, suppose that b  C,(A) is any vector such that We can write b = (g(a), 2~ 3fc 6 0 , . . , 2 3k bn}T , where g(x) = b0 + b\x + ... + bnxn is some polynomial with integral coefficients. Hence in particular and We claim that g(a) = 0 . e. the roots of its minimal polynomial.

Let £ be any lattice in R n and y 6 R n . Suppose that we want to find a lattice vector which is nearest to y . Let d(£, y) denote the minimum distance of y from the vectors in £ (it is obvious that this minimum is attained). To find d(£,y) is TVP-hard (van Emde-Boas (1981)). However, we describe an algorithm that determines this number up to a factor of (3/>/2)n in polynomial time, and also finds a vector 6 G £ such that ||6-y||<(3/>/2) n d(£,i/). Similarly, as in the shortest lattice vector case, we may observe that any such algorithm must provide an (implicit) proof of the fact that d(£, y) is not smaller than 2~ n ||fe — y|| .

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