14 research outputs found
FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
We show the existence of a fully polynomial-time approximation scheme (FPTAS)
for the problem of maximizing a non-negative polynomial over mixed-integer sets
in convex polytopes, when the number of variables is fixed. Moreover, using a
weaker notion of approximation, we show the existence of a fully
polynomial-time approximation scheme for the problem of maximizing or
minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes,
when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin
Ranks of twists of elliptic curves and Hilbert's Tenth Problem
In this paper we investigate the 2-Selmer rank in families of quadratic
twists of elliptic curves over arbitrary number fields. We give sufficient
conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer
rank, and we give lower bounds for the number of twists (with bounded
conductor) that have a given 2-Selmer rank. As a consequence, under appropriate
hypotheses we can find many twists with trivial Mordell-Weil group, and
(assuming the Shafarevich-Tate conjecture) many others with infinite cyclic
Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our
results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth
Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
Comparative similarity, tree automata, and diophantine equations
The notion of comparative similarity âX is more similar or closer to Y than to Zâ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similarity-based reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the âpropositionalâ logic with the binary operator âcloser to a set Ï 1 than to a set Ï 2â and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTime-complete for the classes of all finite symmetric and all finite (possibly non-symmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our âcloserâ operator has the same expressive power as the standard operator > of conditional logic, these results may have interesting implications for conditional logic as well
On Intersection Problems for Polynomially Generated Sets
Some classes of sets of vectors of natural numbers are introduced as generalizations of the semi-linear sets, among them the `simple semi-polynomial sets.' Motivated by verification problems that involve arithmetical constraints, we show results on the intersection of such generalized sets with semi-linear sets, singling out cases where the nonemptiness of intersection is decidable. Starting from these initial results, we list some problems on solvability of arithmetical constraints beyond the semi-linear ones
The Isomorphism Problem for Ï-Automatic Trees
The main result of this paper is that the isomorphism problem for Ï-automatic trees of finite height is at least as hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, MontalbĂĄn, and Nies [9] showing that the isomorphism problem for Ï-automatic structures is not â_2^1. Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for Ï-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for Ï-automatic trees of every finite height: (i) It is decidable (â_1^0-complete, resp.) for height 1 (2, resp.), (ii) (â_1^1-hard and in (â_2^1 for height 3, and (iii) (â_(n-3)^1- and (â_(n-3)^1-hard and in (â_(2n-4)^1(assuming CH) for all nââ„â4. All proofs are elementary and do not rely on theorems from set theory. Complete proofs can be found in [18]