Focused Local Search for Random 3-Satisfiability

A local search algorithm solving an NP-complete optimisation problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satisfiability problem, the heuristic of focusing the local moves on the presently unsatisfiedclauses is known to be very effective: the time to solution has been observed to grow only linearly in the number of variables, for a given clauses-to-variables ratio $\alpha$ sufficiently far below the critical satisfiability threshold $\alpha_c \approx 4.27$. We present numerical results on the behaviour of three focused local search algorithms for this problem, considering in particular the characteristics of a focused variant of the simple Metropolis dynamics. We estimate the optimal value for the ``temperature'' parameter $\eta$ for this algorithm, such that its linear-time regime extends as close to $\alpha_c$ as possible. Similar parameter optimisation is performed also for the well-known WalkSAT algorithm and for the less studied, but very well performing Focused Record-to-Record Travel method. We observe that with an appropriate choice of parameters, the linear time regime for each of these algorithms seems to extend well into ratios $\alpha > 4.2$ -- much further than has so far been generally assumed. We discuss the statistics of solution times for the algorithms, relate their performance to the process of ``whitening'', and present some conjectures on the shape of their computational phase diagrams.Comment: 20 pages, lots of figure

Optimal adult growth of Daphnia in a seasonal environment

1. The cladoceran Daphnia serves as an example of an iteroparous organism, with overlapping generations, that is capable of substantial adult growth. The life history of Daphnia was modelled as the consequence of a series of decisions about allocation of energetic resources to growth and reproduction. 2. We used numerical methods to find resource allocation patterns that maximized fitness of Daphnia in a temporally variable environment. Temporal variation was modelled as alternating active and dormant seasons; length of the active season was uniformly distributed. Fitness was measured by the geometric mean of resting eggs produced at the end of the active season. We examined effects of mean and range of the active season on the optimal life history; we also examined effects of increasing (invertebrate predation), constant (non-selective) and decreasing (fish) size-specific survival rates. For comparison, we found resource allocation patterns that maximized fitness in a constant environment, where fitness was measured by the intrinsic rate of increase r. 3. Life histories optimized for seasonal environments generally showed earlier maturity and greater adult growth than those optimized for constant environments. Adult growth occurred with non-selective predation, and even with fish predation, conditions under which it does not occur in the optimal life histories for constant environments. 4. Greatest size at maturity and adult growth occurred in life histories optimized to invertebrate predation in seasonal environments. Smallest size at maturity and least adult growth occurred in life histories optimized to fish predation. 5. In the optimal life histories, size at maturity generally increased with mean length of the active season. Adult growth reached a maximum for mean seasons of length equal to about one-half to one life span of Daphnia. 6. Increasing the variation in season length decreased adult growth in the optimal life history, but had little effect on size at maturity. 7. We expect that life histories are adapted to the long-term average of season length and its variation. If the animals can detect the type of predator, selection could favour phenotypic variation in resource allocation

Downward Nominal Rigidity in US Wage Data from the PSID - An Application of the Kernel-Location Approach

Earlier studies of US wage data from the PSID with a variety of methods have led to mixed results with respect to the existence and extent of downward nominal wage rigidity. Here the kernellocation approach to the analysis of downward nominal wage rigidity in micro data is applied to that data for the first time, in order to non-parametrically estimate counterfactual and factual distributions of annual nominal wage changes, the rigidity function and the average degree of downward nominal wage rigidity. Avoiding several problems of earlier studies by the use of the kernellocation approach, a substantial degree of downward nominal wage rigidity is found, and earlier evidence in favor of the hypothesis of downwardly rigid nominal wages is corroborated, weakening the institutionalist view of downward nominal wage rigidity. Frühere Untersuchungen zu US-amerikanischen Lohndaten der PSID unter Verwendung unterschiedlicher Methoden haben recht verschiedene Ergebnisse bzgl. der Existenz und des Ausmaßes von Abwärtsnominallohnstarrheit gezeitigt. In diesem Beitrag wird zum ersten Mal der Kernel-Location-Approach zur Analyse der Abwärtsnominallohnstarrheit in Mikrodaten verwandt. Mit ihm wird eine nicht-parametrische Schätzung der kontrafaktischen und der faktischen Verteilungen, der Starrheitsfunktion und des durchschnittlichen Grades der Abwärtsnominallohnstarrheit durchgeführt. Die Vermeidung vieler Probleme früherer Studien durch die Anwendung des Kernel-Location-Approach ermöglicht den Nachweis eines erheblichen Grades an Abwärtsnominallohnstarrheit. Dadurch können frühere, die Hypothese abwärtsstarrer Nominallöhne stützende, Ergebnisse untermauert werden, wobei gleichzeitig die institutionalistische Position zur Abwärtsnominallohnstarrheit geschwächt wird