On algorithm complexity

This paper contains a review of the authors’ results in the theory of algorithm complexity. The results described concern methods for obtaining lower bounds (containing almost all exponential lower bounds on monotone complexity of monotone functions), synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.     

On the complexity of growth of the number of distinct fuzzy switching functions

Several attempts have been made to enumerate fuzzy switching (FSF's) and to develop upper and lower bounds for the number of FSF's of n variables in an effort to better understand the properties and the complexity of FSF's. Previous upper bounds are 2⁴ⁿ [9] and 2^{2–3n—2n—1} [7].It has also been shown that the exact numbers of FSF's of n variables for n = 0, 1, 2, 3, and 4 are 2, 6, 8, 84, 43 918 and 160 297 985 276 respectively.This paper will give a brief overview of previous approaches to the problem, study some of the properties of fuzzy switching functions and give improved upper and lower bounds for a general n.     

On the complexity of posets

The purpose of this paper is to discuss several invariants each of which provides a measure of the intuitive notion of complexity for a finite partially ordered set. For a poset X the invariants discussed include cardinality, width, length, breadth, dimension, weak dimension, interval dimension and semiorder dimension denoted respectively X, W(X), L(X), B(X), dim(X). Wdim(X), Idim(X) and Sdim(X). Among these invariants the following inequalities hold. $\text{B(}\text{X}\text{)?Idim(}\text{X}\text{)?Sdim(}\text{X}\text{)?Wdim(}\text{X}\text{)?dim(}\text{X}\text{)?W(}\text{X}\text{)}$. We prove that every poset X with three of more points contains a partly with $\text{Idim}\text{(}\text{X}\text{)}\text{Idim}\text{(}\text{X}\text{) {x,v})}$. If M denotes the set of maximal elements and A an arbitrary anticham of X we show that $\text{Idim}\text{(}\text{X}\text{)?W(}\text{X-M}\text{)}$ and $\text{Idim}\text{(}\text{X}\text{)?2W(}\text{X-A}\text{)}$. We also show that there exist functions f(n,t) and (gt) such that $\text{I(}\text{X}\text{)?n}$ and $\text{Idim}\text{(}\text{X}\text{)?t}$simply $\text{dim}\text{(}\text{X}\text{)?f(n,t and Sdim(}\text{X}\text{)?t}$ implies $\text{dim}\text{(}\text{X}\text{)?g(t)}$.     

On the relation between complexity and uncertainty

In practical problem situations data are usually inherently unreliable. A mathematical representation of uncertainty leads to stochastic optimization problems. In this paper the complexity of stochastic combinatorial optimization problems is discussed. Surprisingly, certain stochastic versions of NP-hard determinstic combinatorial problems appear to be solvable in polynomial time.     

On the complexity of constrained VC-classes

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n]={1,…,n} which have a margin less than or equal to N on all x∈[n] with h(x)=1, where the margin μ_h(x) of h at x∈[n] is defined as the largest non-negative integer a such that h is constant on the interval I_a(x)=[x-a,x+a]⊆[n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S⊆[n].     

On the computational complexity of cut-reduction

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.     

On the complexity of enumerating pseudo-intents

We investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P=NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P=NP.     

On strong embeddability and finite decomposition complexity

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.     

On the complexity and analysis of auxetic systems

A challenge in creating a model for an auxetic system based on a formalism that is fully computable, is the aim of this paper. Two major levels of complexity are discussed in a way of understanding the structure and processes that define an auxetic system. The auxeticity and structural complexity are interpreted in the light of Cosserat elasticity which admits degrees of freedom not present in classical elasticity, i.e. the rotation of points in the material, and a couple per unit area or the couple stress. The Young'modulus computing for a laminated periodic system made up of alternating aluminum and an auxetic material is an example of computing complexity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the complexity of polyhedral separability

It is NP-complete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NP-complete to recognize whether two sets of points in the plane can be separated withk lines. For every fixedk in any fixed dimension, it takes polynomial time to recognize whether two sets of points can be separated withk hyperplanes.     

On the complexity of diagram testing

In 1987, Neetil and Rödl [4] claimed to have proved that the problem of finding whether a given graphG can be oriented as the diagram of a partial order is NP-complete. A flaw was discovered in their proof by Thostrup [11]. Neetil and Rödl [5] have since corrected the proof, but the new version is rather complex. We give a simpler and more elementary proof, using a completely different approach.     

On the topological complexity of DC-sets

A DC-set is a set defined by means of convex constraints and one additional inverse convex constraint. Given an arbitrary closed subsetM of the Euclideann-space, we show that there exists a DC-set in (n+1)-space being homeomorphic to the given setM. Secondly, for any fixedn2, we construct a compactn-dimensional manifold with boundary, which is a DC-set and which has arbitrarily large Betti-numbersr _k fork n–2.     

On the complexity of stochastic integration

We study the complexity of approximating stochastic integrals with error for various classes of functions. For Ito integration, we show that the complexity is of order , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.

     

On the complexity of computing treelength

We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007) [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has a treelength at most k is NP-complete for every fixed k≥2, and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than . Additionally, we show that treelength can be computed in time O^∗(1.7549ⁿ) by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.