An efficient local search method for random 3-satisfiability

Seitz, Sakari; Orponen, Pekka

doi:10.1016/s1571-0653(04)00463-9

Cited by 11 publications

(14 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The theoretical analysis confirms a decreasing number of local maxima in the region of the phase transion for increasing values m of the number of clauses. We intend to analyse a variety of neighbourhood relations proposed in the literature [14,28,29], where it would be interesting to find out if the average number of local maxima can be related to the quality of the associated local search procedures. We intend to apply the Garnier/Kallel-method in a completely different context, namely to structure prediction problems in Computational Biology, such as RNA secondary structure prediction and protein folding simulation in various lattice models and for different types of the objective function.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of Local Search Landscapes for k-SAT Instances

2010

View full text Add to dashboard Cite

Abstract. Stochastic local search is a successful technique in diverse areas 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 preprocessing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parameter 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 relation. 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 provide 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 2 n−O( √ n/k) for the average number of local maxima, if m is in the region of 2 k ·n/k.Mathematics Subject Classification (2010). Primary 68R99; Secondary 90C27.

show abstract

Section: Discussionmentioning

confidence: 99%

“…For example, flipping values of truth assignments is determined by unsatisfied clauses only [29,28]. We consider a simple, unconstrained (i.e., features of clauses w.r.t.…”

Section: Basic Notationsmentioning

confidence: 99%

Analysis of Local Search Landscapes for k-SAT Instances

2010

View full text Add to dashboard Cite

show abstract

“…The expression Θ(2 k /k) is, of course, dependent on the specific neighbourhood underlying our calculations. We intend to analyse a variety of neighbourhood relations proposed in the literature [8], [20], [21], where it would be interesting to find out if the average number of local maxima (or how close the "breaking point" comes to the phase transition threshold) can be related to the quality of the associated local search procedure. Future research will also include a more comprehensive analysis of the Garnier/Kallel-approach for larger k-SAT instances and larger sets of parameter settings.…”

Section: Discussionmentioning

confidence: 99%

“…For example, flipping values of truth assignments is determined by unsatisfied clauses only, see also [21], [20]. We consider a simple, unconstrained (i.e., features of clauses w.r.t.…”

Section: Basic Notationsmentioning

confidence: 99%

See 1 more Smart Citation

Combinatorial Landscape Analysis for k-SAT Instances

Albrecht¹,

Lane²,

Steinhöfel³

2008

2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)

View full text Add to dashboard Cite

Abstract-Over the past ten years, methods from statistical physics have provided a deeper inside into the average complexity of hard combinatorial problems, culminating in a rigorous proof for the asymptotic behaviour of the k-SAT phase transition threshold by Achlioptas and Peres in 2004. On the other hand, when dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase might be helpful for an appropriate adjustment of local search-based procedures. In the present paper, we address both issues in the context of landscapes induced by k-SAT instances: Firstly, we utilize 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 relation. The objective function is given by the number of satisfied clauses. Secondly, we outline a method for obtaining upper bounds for the average number of local maxima in k-SAT instances which indicates some kind of phase transition for the neighbourhood-specific ratio m/n = Θ(2 k /k).

show abstract

From Spin Glasses to Hard Satisfiable Formulas

Jia

Moore

Selman

2005

Theory and Applications of Satisfiability Testing

View full text Add to dashboard Cite

We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably "glassy" behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true ground state, i.e., the unique satisfying assignment. We test the hardness of our formulas with two Davis-Putnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, WalkSAT and Record-to-Record Travel (RRT). We compare our formulas to random 3-XOR-SAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch's hgen2. For the complete solvers the running time of our formulas grows exponentially in √ n, and exceeds that of random 3-XOR-SAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For WalkSAT, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the parameter d is tuned to the height of the barriers between local minima, and we use this parameter to measure the barrier heights in random 3-XOR-SAT formulas as well.

show abstract

An efficient local search method for random 3-satisfiability

Cited by 11 publications

References 27 publications

Analysis of Local Search Landscapes for k-SAT Instances

Analysis of Local Search Landscapes for k-SAT Instances

Combinatorial Landscape Analysis for k-SAT Instances

From Spin Glasses to Hard Satisfiable Formulas

Contact Info

Product

Resources

About