Enhancing set constraint solvers with lexicographic bounds

Since their beginning in constraint programming, set solvers have been applied to a wide range of combinatorial search problems, such as bin-packing, set partitioning, circuit and combinatorial design. In this paper we present and evaluate a new means towards improving the practical reasoning power of Finite Set (FS) constraint solvers to better address such set problems with a particular attention to the challenging symmetrical set problems often cast as Combinatorial Design Problems (CDPs). While CDPs find a natural formulation in the language of sets, in constraint programming literature, alternative models are often used due to a lack of efficiency of traditional FS solvers. We first identify the main structural components of CDPs that render their modeling suitable to set languages but their solving a technical challenge. Our new prototype solver extends the traditional subset variable domain with lexicographic bounds that better approximate a set domain by satisfying the cardinality restrictions applied to the variable, and allow for active symmetry breaking using ordering constraints. Our contribution includes the formal and practical framework of the new solver implemented on top of a traditional set solver, and an empirical evaluation on benchmark CDPs.     

Existence of finite test-sets for k-power-freeness of uniform morphisms

A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. When k?3, we provide such an algorithm for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite test-sets for k-power-freeness.     

On the decidability of equational theories of varieties of rings

A result illustrating the complexity of describing the varieties of rings with undecidable equational theory is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 873–881, June, 1998.The author wishes to thank Yu. M. Vazhenin for his supervision of this research.     

Recognizing splicing languages: Syntactic monoids and simultaneous pumping

We use syntactic monoid methods, together with an enhanced pumping lemma, to investigate the structure of splicing languages. We obtain an algorithm for deciding whether a regular language is a reflexive splicing language, but the general question remains open.     

Bidding under price uncertainty in multi-unit pay-as-bid procurement auctions for power systems reserve

In this paper a methodology for profit maximized bidding under price uncertainty in a day-ahead, multi-unit and pay-as-bid procurement auction for power systems reserve is proposed. Within this novel methodology a bidder is considered to follow a Bayes-strategy. Thereby, one bidder is assumed to behave strategically and the behavior of the remaining is summarized in a probability distribution of the market price and a reaction function to price dumping by the strategic bidder. With this approach two problems arise: First, as a pay-as-bid auction is considered, no uniform price and therefore no single probability distribution of the market price is readily available. Second, if historic bidding data of all participants are used to estimate such a distribution and market power is a relevant factor, the bid of the strategically behaving bidder is likely to influence the distribution. Within this paper for both of the problems solutions are presented. It is shown that by estimating a probability of acceptance the optimal bidding price with respect to a given capacity can be calculated by maximizing a stochastic non-linear objective function of expected profit. Taking the characteristics of recently established markets in Germany into account, the methodology is applied using exemplary data. It is shown that the methodology helps to manage existing price uncertainties and hence supports the trading decisions of a bidder. It is inferred that the developed methodology may also be used for bidding on other auction markets with a similar market design.     

Integer-Programming Software Systems

Recent developments in integer-programming software systems have tremendously improved our ability to solve large-scale instances. We review the major algorithmic components of state-of-the-art solvers and discuss the options available to users for adjusting the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer-programming software systems and a perspective on features we may expect to see in these systems in the near future.     

Hierarchical sets in mathematical programming modeling languages

The mathematical notation commonly applied for the formulation of mathematical programming models is extended to include hierarchical structures. The proposed notation is related to hierarchical set concepts in the languages UIMP, AMPL, and LPL. With the proposed notation it is possible to aggregate and disaggregate over hierarchical structures. In addition, views are introduced to permit the use of hierarchical substructures and to create new hierarchies out of existing ones. The proposed notation for hierarchical sets and views is illustrated by applying it to the representation and estimation of social accounting matrices (SAMs).     

Piecewise testable languages via combinatorics on words

A regular language L over an alphabet A is called piecewise testable if it is a finite Boolean combination of languages of the form A^∗a₁A^∗a₂A^∗…A^∗a_?A^∗, where a₁,…,a_?∈A, ?≥0. An effective characterization of piecewise testable languages was given in 1972 by Simon who proved that a language L is piecewise testable if and only if its syntactic monoid is J-trivial. Nowadays, there exist several proofs of this result based on various methods from algebraic theory of regular languages. Our contribution adds a new purely combinatorial proof.     

Analytical Formulation of the Classical Friction Model for Motion Analysis and Simulation

Starting from a precise definition of friction torque when velocity vanishes that distinguishes the case of instantaneous zero crossing from that where the velocity is zero over a time interval, this paper proposes a compact analytical formulation of the classical discontinuous friction model that is useful for motion analysis. A finite state machine that allows a numerically robust computation of motion equations when velocity vanishes or motion restarts is then defined. Simulation results show that the discontinuous model can be seen as an asymptotic approximation, infinitely fast, of a recently proposed continuous, dynamic friction model.     

Regular expression searching on compressed text

We present a solution to the problem of regular expression searching on compressed text. The format we choose is the Ziv–Lempel family, specifically the LZ78 and LZW variants. Given a text of length u compressed into length n, and a pattern of length m, we report all the R occurrences of the pattern in the text in O(2^m+mn+Rmlogm) worst case time. On average this drops to O(m²+(n+Rm)logm) or O(m²+n+Ru/n) for most regular expressions. This is the first nontrivial result for this problem. The experimental results show that our compressed search algorithm needs half the time necessary for decompression plus searching, which is currently the only alternative.