Abstract
AbstractGiven a combinatorial search problem, it may be highly useful to enumerate its (all) solutions besides just finding one solution, or showing that none exists. The same can be stated about optimal solutions if an objective function is provided. This work goes beyond the bare enumeration of optimal solutions and addresses the computational task of solution enumeration by optimality (SEO). This task is studied in the context of answer set programming (ASP) where (optimal) solutions of a problem are captured with the answer sets of a logic program encoding the problem. Existing answer set solvers already support the enumeration of all (optimal) answer sets. However, in this work, we generalize the enumeration of optimal answer sets beyond strictly optimal ones, giving rise to the idea of answer set enumeration in the order of optimality (ASEO). This approach is applicable up to the best k answer sets or in an unlimited setting, which amounts to a process of sorting answer sets based on the objective function. As the main contribution of this work, we present the first general algorithms for the aforementioned tasks of answer set enumeration. Moreover, we illustrate the potential use cases of ASEO. First, we study how efficiently access to the next-best solutions can be achieved in a number of optimization problems that have been formalized and solved in ASP. Second, we show that ASEO provides us with an effective sampling technique for Bayesian networks.
Highlights
In this paper, we address combinatorial problem solving where some solution components are combined to meet problem specific requirements
We address solution enumeration (SE) in the context of combinatorial optimization problems, the goal of which is to generate all optimal solutions to a given problem instance
We take sub-optimal solutions into consideration and propose algorithms that are able to recursively enumerate next-best solutions until (i) all solutions or, alternatively, (ii) the best k solutions have been enumerated. Such a procedure realizes our concept of solution enumeration by optimality (SEO)
Summary
The enumeration features of contemporary SAT/MaxSAT solvers are quite limited the algorithmic ideas are described by Gebser et al (2009) For this reason, we concentrate on ASP solvers that natively support SE and the enumeration of optimal solutions (SO) as discussed above, but not SEO in any systematic way. The second presumes a parameter k that gives the number of answer sets to be enumerated and, thereafter, the best k answer sets S are sought given f (S) The latter approach opens up new possibilities for organizing ASEO when k is small enough for storing intermediate solutions in memory.
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