3 research outputs found
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 sufficiently far below the critical
satisfiability threshold . 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 for this algorithm, such that its linear-time
regime extends as close to 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 -- 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
Palette-colouring: a belief-propagation approach
We consider a variation of the prototype combinatorial-optimisation problem
known as graph-colouring. Our optimisation goal is to colour the vertices of a
graph with a fixed number of colours, in a way to maximise the number of
different colours present in the set of nearest neighbours of each given
vertex. This problem, which we pictorially call "palette-colouring", has been
recently addressed as a basic example of problem arising in the context of
distributed data storage. Even though it has not been proved to be NP complete,
random search algorithms find the problem hard to solve. Heuristics based on a
naive belief propagation algorithm are observed to work quite well in certain
conditions. In this paper, we build upon the mentioned result, working out the
correct belief propagation algorithm, which needs to take into account the
many-body nature of the constraints present in this problem. This method
improves the naive belief propagation approach, at the cost of increased
computational effort. We also investigate the emergence of a satisfiable to
unsatisfiable "phase transition" as a function of the vertex mean degree, for
different ensembles of sparse random graphs in the large size ("thermodynamic")
limit.Comment: 22 pages, 7 figure