Clemens Heuberger

Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results

Given a (combinatorial) optimization problem and a feasible solution to it, the corresponding inverse optimization problem is to find a minimal adjustment of the cost function such that the given solution becomes optimum.Several such problems have been studied in the last twelve years. After formalizing the notion of an inverse problem and its variants, we present various methods for solving them. Then we discuss the problems considered in the literature and the results that have been obtained. Finally, we formulate some open problems.

Minocycline for Huntington’s disease: An open label study

Despite advances in understanding the pathogenesis of Huntington’s disease (HD), there is no effective treatment. In the HD transgenic mouse, expression of a dominant-negative caspase-1 mutant extended survival and delayed the onset of symptoms, suggesting that caspase-1 is crucial in HD pathogenesis.1 In fact, the caspase inhibitor minocycline delayed disease progression and extended survival by 14% in the HD mouse model.2 Minocycline is a second-generation tetracycline commonly used for a prolonged period to treat acne. Minocycline crosses the blood–brain barrier and inhibits caspase-1 in acute stroke and decreases infarct size in a mouse stroke model.2 Minocycline protects neurons in mixed spinal cord cultures …

Cerebrospinal Fluid Tissue Transglutaminase as a Biochemical Marker for Alzheimer's Disease

Tissue transglutaminase (tTG) is an indicator of acute cell death in vitro. An increase in tTG protein level is found in postmortem Alzheimers disease (AD) brains as well as in Huntingtons disease. No study revealed tTG in vivo so far. We investigated the concentrations of tTG in the cerebrospinal fluid (CSF) obtained from 84 patients using ELISA assays. We compared 33 patients with probable AD to 18 patients with probable vascular dementia (VaD) and 33 control patients without neuropsychological deficit. Diagnosis was supported by CSF parameter and neuroimaging. We found a highly significant difference (P = 0.001) between the concentration of tTG in the AD groups (7.58 pg/ml) and controls (2.99 pg/ml). There was no statistical difference between controls and VaD (2.93 pg/ml). Interestingly, tTG did not show an association with tau protein, Abeta42, apoE4, neuropsychological items, or age. Males showed lower tTG values than females; however, this difference did not reach statistical significance. To our knowledge, this is the first demonstration that tTG is increased in AD in vivo. Our results suggest that tTG may be a powerful biochemical marker of the acute degenerating process in vivo. It may serve as completion of CSF analysis in the diagnosis of dementing disorders and may be a simple way of assessing the efficacy of possible new antiapoptotic drugs.

Chemical Trees Minimizing Energy and Hosoya Index

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.

Distribution results for low-weight binary representations for pairs of integers

We discuss an optimal method for the computation of linear combinations of elements of Abelian groups, which uses signed digit expansions. This has applications in elliptic curve cryptography. We compute the expected number of operations asymptotically (including a periodically oscillating second order term) and prove a central limit theorem. Apart from the usual right-to-left (i.e., least significant digit first) approach we also discuss a left-to-right computation of the expansions. This exhibits fractal structures that are studied in some detail.

Analysis of linear combination algorithms in cryptography

Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to this problem. Both methods are sped up using the sliding windows approach combined with precomputed lookup tables. We give explicit and asymptotic results for the number of group operations needed, assuming uniform distribution of the coefficients. Expected values, variances and a central limit theorem are proved using generating functions.Furthermore, we provide a new algorithm that calculates the digits of an optimal expansion of pairs of integers from left to right. This avoids storing the whole expansion, which is needed with the previously known right-to-left methods, and allows an online computation.

Minimality of the hamming weight of the τ-NAF for Koblitz curves and improved combination with point halving

In order to efficiently perform scalar multiplications on elliptic Koblitz curves, expansions of the scalar to a complex base associated with the Frobenius endomorphism are commonly used. One such expansion is the τ-adic NAF, introduced by Solinas. Some properties of this expansion, such as the average weight, are well known, but in the literature there is no proof of its optimality, i.e. that it always has minimal weight. In this paper we provide the first proof of this fact. Point halving, being faster than doubling, is also used to perform fast scalar multiplications on generic elliptic curves over binary fields. Since its computation is more expensive than that of the Frobenius, halving was thought to be uninteresting for Koblitz curves. At PKC 2004, Avanzi, Ciet, and Sica combined Frobenius operations with one point halving to compute scalar multiplications on Koblitz curves using on average 14% less group additions than with the usual τ-and-add method without increasing memory usage. The second result of this paper is an improvement over their expansion. The new representation, called the wide-double-NAF, is not only simpler to compute, but it is also optimal in a suitable sense. In fact, it has minimal Hamming weight among all τ-adic expansions with digits {0,±1} that allow one halving to be inserted in the corresponding scalar multiplication algorithm. The resulting scalar multiplication requires on average 25% less group operations than the Frobenius method, and is thus 12.5% faster than the previously known combination.

Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis

In this paper we prove the optimality and other properties of the τ-adic nonadjacent form: this expansion has been introduced in order to compute scalar multiplications on Koblitz curves efficiently. We also refine and extend results about double expansions of scalars introduced by Avanzi, Ciet and Sica in order to improve scalar multiplications further. Our double expansions are optimal and their properties are carefully analysed. In particular, we provide first- and second-order terms for the expected weight, determine the variance and prove a central limit theorem. Transducers for all the involved expansions are provided, as well as automata accepting all expansions of minimal weight.

On minimal expansions in redundant number systems: algorithms and quantitative analysis

Abstract We consider digit expansions in base q≥2 with arbitrary integer digits such that the length of the expansion plus the sum of the absolute values of the digits is minimal. Since this does not determine a unique minimal representation, we describe some reduced minimal expansions.We completely characterize its syntactical properties, give a simple algorithm to compute the reduced minimal expansion and a formula to compute a single digit without having to compute the others, and we calculate the average cost of such an expansion.

On the Number of Optimal Base 2 Representations of Integers

We study representations of integers n in binary expansions using the digits 0,±1. We analyze the average number of such representations of minimal “weight” (= number of non-zero digits). The asymptotic main term of this average involves a periodically oscillating function, which is analyzed in some detail. The main tool is the construction of a measure on [−1,1], which encodes the number of representations.