Many combinatorial optimization problems entail a number of hierarchically dependent optimization problems. An often used solution is to associate a suitably large cost with each individual optimization problem, such that the solution of the resulting aggregated optimization problem solves the original set of hierarchically dependent optimization problems. This paper starts by studying the package upgradeability problem in software distributions. Straightforward solutions based on Maximum Satisfiability (MaxSAT) and pseudo-Boolean (PB) optimization are shown to be ineffective, and unlikely to scale for large problem instances. Afterwards, the package upgradeability problem is related to multilevel optimization. The paper then develops new algorithms for Boolean Multilevel Optimization (BMO) and highlights a large number of potential applications. The experimental results indicate that the proposed algorithms for BMO allow solving optimization problems that existing MaxSAT and PB solvers would otherwise be unable to solve.
Deep Dive into On Solving Boolean Multilevel Optimization Problems.
Many combinatorial optimization problems entail a number of hierarchically dependent optimization problems. An often used solution is to associate a suitably large cost with each individual optimization problem, such that the solution of the resulting aggregated optimization problem solves the original set of hierarchically dependent optimization problems. This paper starts by studying the package upgradeability problem in software distributions. Straightforward solutions based on Maximum Satisfiability (MaxSAT) and pseudo-Boolean (PB) optimization are shown to be ineffective, and unlikely to scale for large problem instances. Afterwards, the package upgradeability problem is related to multilevel optimization. The paper then develops new algorithms for Boolean Multilevel Optimization (BMO) and highlights a large number of potential applications. The experimental results indicate that the proposed algorithms for BMO allow solving optimization problems that existing MaxSAT and PB solver
Many real problems require an optimal solution rather than any solution. Whereas decision problems require a yes/no answer, optimization problems require the best solution, thus differentiating the possible solutions. In practice, there must be a classification scheme to determine how one solution compares with the others. Such classification may be seen as a way of establishing preferences that express cost or satisfaction.
A special case of combinatorial optimization problems may require a set of optimization criteria to be observed, for which is possible to define a hierarchy of importance. Suppose that instead of requiring a balance between price, horsepower and fuel consumption for choosing a new car, you have made a clear hierarchy in your mind: you have a strict limit on how much you can afford, then you will not consider a car with less than 150 horsepower and after that the less the fuel consumption the better. Not only you establish a hierarchy in your preferences, but also the preferences are defined in such a way that the set of potential solutions gets subsequently reduced. Such kind of problems are present not only in your daily life but also in many real applications.
Clearly, the kind of problems we target can be encoded as a constraint optimization problem, making use of the available technology for dealing with preferences. Preference handling is one of the current hot topics in AI with active research lines in constraint satisfaction and optimization [24]. Broadly, preferences over constraints may be expressed quantitatively or qualitatively. For example, one may wish to fly in the afternoon or simply choose the less expensive flight of that day. Soft constraints model quantitative preferences by associating a level of satisfaction with each of the solutions [23], whereas CP-nets model qualitative preferences by expressing preferential dependencies with pairwise comparisons [7]. Furthermore, preference-based search algorithms can be generalized to handle multi-criteria optimization [16].
A straightforward approach to solve a special case of a constraint optimization problem, for which there is a total ranking of the criteria, would be to establish a lexicographic ordering over variables and domains, such that optimal solutions would come first in the search tree [13]. But this has the potential disadvantage of producing a thrashing behavior whenever assignments that are not supported by any solution are considered, as a result of decisions made at the first nodes of the search tree [16].
Maximum satisfiability (MaxSAT) naturally encodes a constraint optimization problem over Boolean variables where constraints are encoded as clauses. A solution to the MaxSAT problem maximizes the number of satisfied clauses. Weights may also be associated with clauses, in which case the sum of the weights of the satisfied clauses is to be maximized. The use of the weighted MaxSAT formalism allows to solve a set of hierarchically dependent optimization problems. Pseudo-Boolean (PB) optimization may also be used to solve this kind of problems, given that weighted MaxSAT problem instances can be translated to PB. Each clause is extended with a relaxation variable that is then included in the cost function, jointly with the respective weight.
Boolean satisfiability (SAT) and PB have been extended in the past to handle preferences. For example, SAT-based planning has been extended to include conflicting preferences [14], for each of which weights are associated, thus requiring the use of an objective function involving the preferences and their weights. The proposed solution modifies a SAT backtracking algorithm to search first for optimal plans by branching according to the partial order induced by the preferences. In addition, algorithms for dealing with multi-objective PB problems have been developed [19], in contrast to traditional algorithms that optimize a single linear function.
This paper is organized as follows. The next section describes the problem of package upgradeability in software systems. This problem comes from a real application and has been the drive for the algorithms being developed. Section 3 introduces multilevel optimization and relates it with a variety of problems. Afterwards, specific multilevel optimization algorithms are proposed, being based in MaxSAT and PB. Experimental results show the effectiveness of the new algorithms. Finally, the paper concludes.
We have all been through a situation where the installation of a new piece of software turns out to be a nightmare. Not only you do not get the new computer program installed, but also some other programs may eventually stop working properly. And this may also happen when you simply want to upgrade to a more recent version of a program that you have been using for some time. Although this seems to be a software engineering problem, behind the nightmare is a hard computational problem, and therefore an intelligent solution
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