Phase Transition in the Number Partitioning Problem

Number partitioning is an NP-complete problem of combinatorial optimization. A statistical mechanics analysis reveals the existence of a phase transition that separates the easy from the hard to solve instances and that reflects the pseudo-polynomiality of number partitioning. The phase diagram and the value of the typical ground state energy are calculated.Comment: minor changes (references, typos and discussion of results

Random Costs in Combinatorial Optimization

The random cost problem is the problem of finding the minimum in an exponentially long list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. This explains the bad performance of heuristic approaches to the number partitioning problem and allows us to calculate the probability distributions of the optimum and sub-optimum costs.Comment: 4 pages, Revtex, 2 figures (eps), submitted to PR

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.Comment: 4 pages, 3 figure

Symmetry breaking in numeric constraint problems

Symmetry-breaking constraints in the form of inequalities between variables have been proposed for a few kind of solution symmetries in numeric CSPs. We show that, for the variable symmetries among those, the proposed inequalities are but a specific case of a relaxation of the well-known LEX constraints extensively used for discrete CSPs. We discuss the merits of this relaxation and present experimental evidences of its practical interest.Postprint (author’s final draft

Observational constraints on the origin of the elements. V. Non-LTE abundance ratios of [Ni/Fe] in Galactic stars and enrichment by sub-Chandrasekhar mass SNe

We constrain the role of different SN Ia channels in the chemical enrichment of the Galaxy by studying the abundances of nickel in Galactic stars. We investigate four different SN Ia sub-classes, including the classical single-degenerate near-Chandrasekhar mass SN Ia, the fainter SN Iax systems associated with He accretion from the companion, as well as two sub-Ch mass SN Ia channels. The latter include the double-detonation of a white dwarf accreting helium-rich matter and violent white dwarf mergers. NLTE models of Fe and Ni are used in the abundance analysis. In the GCE models, we include new delay time distributions arising from the different SN Ia channels, as well as recent yields for core-collapse supernovae and AGB stars. The data-model comparison is performed using a Markov chain Monte Carlo framework that allows us to explore the entire parameter space allowed by the diversity of explosion mechanisms and the Galactic SN Ia rate, taking into account the uncertainties of the observed data. We show that NLTE effects have a non-negligible impact on the observed [Ni/Fe] ratios in the Galactic stars. The NLTE corrections to Ni abundances are not large, but strictly positive, lifting the [Ni/Fe] ratios by ~+0.15 dex at [Fe/H] =-2. We find that that the distributions of [Ni/Fe] in LTE and in NLTE are very tight, with a scatter of < 0.1 dex at all metallicities, supporting earlier work. In LTE, most stars have scaled-solar Ni abundances, [Ni/Fe] = 0, with a slight tendency for sub-solar [Ni/Fe] ratios at lower [Fe/H]. In NLTE, however, we find a mild anti-correlation between [Ni/Fe] and metallicity, and a slightly elevated [Ni/Fe] ratios at [Fe/H] < -1.0. The NLTE data can be explained by the GCE models calculated with a substantial, ~ 75%, fraction of sub-Ch SN Ia.Comment: accepted for publication in Astronomy & Astrophysics, abridged version of the abstrac

Theory Learning with Symmetry Breaking

This paper investigates the use of a Prolog coded SMT solver in tackling a well known constraints problem, namely packing a given set of consecutive squares into a given rectangle, and details the developments in the solver that this motivates. The packing problem has a natural model in the theory of quantifier-free integer difference logic, a theory supported by many SMT solvers. The solver used in this work exploits a data structure consisting of an incremental Floyd-Warshall matrix paired with a watch matrix that monitors the entailment status of integer difference constraints. It is shown how this structure can be used to build unsatisfiable theory cores on the fly, which in turn allows theory learning to be incorporated into the solver. Further, it is shown that a problem-specific and non-standard approach to learning can be taken where symmetry breaking is incorporated into the learning stage, magnifying the effect of learning. It is argued that the declarative framework allows the solver to be used in this white box manner and is a strength of the solver. The approach is experimentally evaluated

Епоха "пост": людина в перспективі "нової духовності"

Розглянуто основні проблеми сучасного суспільства, пов’язані із антропологічною кризою, кардинальними змінами у темпоральності, способі буття людини. Проаналізовано ситуацію заміни попередніх гуманістичних ціннісних систем на більш “технізовані”, що призводить до виникнення нового типу людини – “постлюдини”. Посилюється вплив на людей “світу віртуальної реальності”. Нове штучне середовище “сканує”, приймає лише інформаційний аспект людини, вводячи її як цілісну істоту в стан кризи. Доведено актуальність відповідей на смислові запитання епохи, пов’язані із світоглядом, духовністю та цінностями.The main problems of modern society, related to the anthropological crisis, fundamental changes in temporality, ways of human being are considered. There is an analysis of situation of replacing the previous humanistic value systems on a more “technicized”, which causes a new type of man – “posthuman”. The impact on “the world of virtual reality” people is growing. New artifi cial environment “scans”, takes only informational aspect of human, introducing it as complete being in a state of crisis. The urgency of responses to semantic question of the epoch, related to the outlook, spirituality and values is justifi ed

Phase Transition in Multiprocessor Scheduling

The problem of distributing the workload on a parallel computer to minimize the overall runtime is known as Multiprocessor Scheduling Problem. It is NP-hard, but like many other NP-hard problems, the average hardness of random instances displays an ``easy-hard'' phase transition. The transition in Multiprocessor Scheduling can be analyzed using elementary notions from crystallography (Bravais lattices) and statistical mechanics (Potts vectors). The analysis reveals the control parameter of the transition and its critical value including finite size corrections. The transition is identified in the performance of practical scheduling algorithms.Comment: 6 pages, revtex