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 unsatisfied clauses 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 α sufficiently far below the critical satisfiability threshold α c ≈ 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 η for this algorithm, such that its linear-time regime extends as close to α 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 α > 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.

We study the performance of stochastic local search algorithms for random instances of the K-satisfiability (K-SAT) problem. We present a stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT is a focused algorithm in the sense that it focuses on variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large K-SAT instances almost surely in linear time, up to high clause-to-variable ratios ␣; for example, for K ‫؍‬ 4 we observe linear-time performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms.geometry of solutions ͉ local search ͉ performance ͉ random K-SAT

Controlled crosses of seven Scots pine (Pinus sylvestris L.) trees produced 49 families that included both reciprocals and selfings. Embryogenic cultures were initiated from immature megagametophytes and after 6 months in maintenance culture, mature somatic embryos were produced from the surviving 166 lines. The effect of parent genotypes on the cultures was evaluated at initiation of the tissue culture period, after 6 months in maintenance culture and at embryo maturation. The effect of the maternal parent was most pronounced at culture initiation. After 6 months in tissue culture, the maternal effect had decreased and the effects of both parents were significant. By the somatic embryo maturation stage, the maternal effect was still considerable but the paternal effect was no longer detectable. There was little correlation between the ranking of mothers and fathers, indicating that the maternal effect was caused by factors other than the paternal effect. No mother x father interaction was found, indicating that mothers successful at initiation and after 6 months in tissue culture, pollinated by any of the successful fathers, produced somatic lines and mature somatic embryos.

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