Read e-book online An Algorithmic Theory of Numbers, Graphs and Convexity PDF

By Laszlo Lovasz

ISBN-10: 0898712033

ISBN-13: 9780898712032

A learn of the way complexity questions in computing have interaction with classical arithmetic within the numerical research of concerns in set of rules layout. Algorithmic designers serious about linear and nonlinear combinatorial optimization will locate this quantity in particular worthy.

Two algorithms are studied intimately: the ellipsoid technique and the simultaneous diophantine approximation technique. even supposing either have been built to review, on a theoretical point, the feasibility of computing a few really expert difficulties in polynomial time, they seem to have sensible functions. The ebook first describes use of the simultaneous diophantine strategy to advance refined rounding strategies. Then a version is defined to compute top and reduce 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 ebook closes with a few functions of the consequences to combinatorial optimization.

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Additional info for An Algorithmic Theory of Numbers, Graphs and Convexity

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D^dk+i = 0 . We go on until we have chosen c i , . . , dn . Let c be the shortest vector among c i , . . , cn . Then 26 LASZL6 LOVASZ To complete the argument, it suffices to note that it is easy to construct a basis in £ whose Gram-Schmidt orthogonalization is just (d n /||d n || 2 ,... 15) with a(n] = b(n)2 . Remark. 6) is not too far from A(£): there exists a basis ( & i , . . , b n ) in any lattice £ such that Let 6 be a shortest non-zero vector in the lattice £ . We may not be able to prove in polynomial time that b is shortest, but we can prove in polynomial time that 6 is "almost shortest" in the sense that no non-zero lattice vector is shorter than ||6||/n .

2) VALIDITY PROBLEM. Given a halfspace {x : CTX < 7} = H (c £ Q , 7 6 Q) , does it contain K ? n In both cases, if the answer is "No", we may want to have a "certificate" of this. One way to "certify" our answer is formulated in the following two problems. 3) SEPARATION PROBLEM. Given a point y e Qn , decide if y e K , and if not, find a hyperplane separating y from K . 4) VIOLATION PROBLEM. Given a halfspace H , decide if K < H , and if not, find a point in H — K. Of course, the first two questions can be asked for any set in R n ; the reason for restricting our attention to closed convex sets is that in this case the MEMBERSHIP problem and the VALIDITY problem are "logically" equivalent: if we know the answer to either one of them for all inputs then we also know the answer to the other.

Let Y^i=i ^ + Y^i=i Pi — * and consider the vector 52 LASZLO LOVASZ By the convexity of K, z e K and hence also y^z + -^u G K . But -^z + Y^u = y , which is a contradiction. Also note that C contains the rotational cone with center y , axis y and half-angle arctg(n + 1) . So by the remark after the statement of the lemma, we are done. Case 2. g. vl £ K . Then we replace y by Vi and repeat the procedure. It remains to show that after a polynomial number of iterations in Case 2, Case 1 must occur; and also that the vertices of the cones do not "drift away" from y .

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An Algorithmic Theory of Numbers, Graphs and Convexity by Laszlo Lovasz

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