35,723 research outputs found
The complexity of solution-free sets of integers for general linear equations
Given a linear equationL, a setAof integers isL-free ifAdoes not contain anynon-trivial solutions toL. Meeks and Treglown [6] showed that for certain kindsof linear equations, it isNP-complete to decide if a given set of integers containsa solution-free subset of a given size. Also, for equations involving three variables,they showed that the problem of determining the size of the largest solution-freesubset isAPX-hard, and that for two such equations (representing sum-free andprogression-free sets), the problem of deciding if there is a solution-free subset withat least a specified proportion of the elements is alsoNP-complete.We answer a number of questions posed by Meeks and Treglown, by extendingthe results above to all linear equations, and showing that the problems remain hardfor sets of integers whose elements are polynomially bounded in the size of the set.For most of these results, the integers can all be positive as long as the coefficientsdo not all have the same sign.We also consider the problem of counting the number of solution-free subsets ofa given set, and show that this problem is #P-complete for any linear equation inat least three variables
On the complexity of finding and counting solution-free sets of integers
Given a linear equation , a set of integers is
-free if does not contain any `non-trivial' solutions to
. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers has an -free subset
of a given size, and counting all such -free subsets. We also
raise a number of open problems.Comment: 27 page
On the Complexity of Hilbert Refutations for Partition
Given a set of integers W, the Partition problem determines whether W can be
divided into two disjoint subsets with equal sums. We model the Partition
problem as a system of polynomial equations, and then investigate the
complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a
given set of integers is not partitionable. We provide an explicit construction
of a minimum-degree certificate, and then demonstrate that the Partition
problem is equivalent to the determinant of a carefully constructed matrix
called the partition matrix. In particular, we show that the determinant of the
partition matrix is a polynomial that factors into an iteration over all
possible partitions of W.Comment: Final versio
More Than 1700 Years of Word Equations
Geometry and Diophantine equations have been ever-present in mathematics.
Diophantus of Alexandria was born in the 3rd century (as far as we know), but a
systematic mathematical study of word equations began only in the 20th century.
So, the title of the present article does not seem to be justified at all.
However, a linear Diophantine equation can be viewed as a special case of a
system of word equations over a unary alphabet, and, more importantly, a word
equation can be viewed as a special case of a Diophantine equation. Hence, the
problem WordEquations: "Is a given word equation solvable?" is intimately
related to Hilbert's 10th problem on the solvability of Diophantine equations.
This became clear to the Russian school of mathematics at the latest in the mid
1960s, after which a systematic study of that relation began.
Here, we review some recent developments which led to an amazingly simple
decision procedure for WordEquations, and to the description of the set of all
solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings
of CAI 2015, Stuttgart, Germany, September 1 - 4, 201
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
Bounds on the Automata Size for Presburger Arithmetic
Automata provide a decision procedure for Presburger arithmetic. However,
until now only crude lower and upper bounds were known on the sizes of the
automata produced by this approach. In this paper, we prove an upper bound on
the the number of states of the minimal deterministic automaton for a
Presburger arithmetic formula. This bound depends on the length of the formula
and the quantifiers occurring in the formula. The upper bound is established by
comparing the automata for Presburger arithmetic formulas with the formulas
produced by a quantifier elimination method. We also show that our bound is
tight, even for nondeterministic automata. Moreover, we provide optimal
automata constructions for linear equations and inequations
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
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