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**1 - 4**of**4**### Run-time Estimates for Protein Folding Simulation in the H-P Model

"... The hydrophobic-hydrophilic (H-P) model for protein folding was introduced by Dill et al. [6]. A problem instance consists of a sequence of amino acids, each labeled as either hydrophobic (H) or hydrophilic (P). The sequence must be placed on a 2D or 3D grid without overlapping, so that adjacent ..."

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The hydrophobic-hydrophilic (H-P) model for protein folding was introduced by Dill et al. [6]. A problem instance consists of a sequence of amino acids, each labeled as either hydrophobic (H) or hydrophilic (P). The sequence must be placed on a 2D or 3D grid without overlapping, so that adjacent amino acids in the sequence remain adjacent in the grid. The goal is to minimize the energy, which in the simplest variation corresponds to maximizing the number of adjacent hydrophobic pairs. Although the model is extremely simple, it captures the main features of the problem. The protein folding problem in the H-P model is NP-hard in both 2D and 3D. Recently, Fu and Wang [9] proved an exp(O(n ln n) algorithm for d-dimensional protein folding simulation in the HP-model. Our preliminary results on stochastic search applied to protein folding utilize complete move sets proposed by Lesh et al. [15] and Blazewicz et al. [3]. We obtain that after (n/# ) Markov chain transitions, the probability to be in a minimum energy conformation is at least 1 where n is the length of the instance, # is the maximum value of the minimum escape height from local minima of the underlying energy landscape, and c is a (small) constant.

### Analysis of Local Search Landscapes for k-SAT Instances

"... Abstract. Stochastic local search is a successful technique in diverse ar-eas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase i ..."

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Abstract. Stochastic local search is a successful technique in diverse ar-eas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parame-ter estimations in the context of landscapes induced by k-SAT instances: at first, we utilise a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood re-lation. The objective function is given by the number of satisfied clauses. The procedure provides good approximations of the actual number of local maxima, with a deviation typically around 10%. Secondly, we pro-vide a method for obtaining upper bounds for the average number of local maxima in k-SAT instances. The method allows us to obtain the upper bound 2n−O( n/k) for the average number of local maxima, if m is in the region of 2k ·n/k.