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Towards Ambitious Approximation Algorithms in Stubborn Set Optimization

Varpaaniemi K.

Fundamenta Informaticae

2003

Vol. 54, nr 2,3

279-294

This article continues research on the stubborn set method that constructs on-the-fly a reduced LTS that is CFFD-equivalent to the parallel composition of given LTSs. In particular, minimization of the number of successor states of a given state is reconsidered. The earlier suggested and/or-graph approach requires solving #P-complete counting problems in order to get the weights for the vertices of the and/or-graph. The ``branch-and-bound'' decision problem corresponding to the minimization of the sum of the computed weights is ``only'' NP-complete. Unfortunately, #P-complete counting does not seem easily avoidable in the general case because it is PP-complete to check whether a given stubborn set produces at most as many successor states as another given stubborn set. Instead of solving each of the subproblems, one could think of computing approximate solutions in such a way that the total effect of the approximations is a useful approximation itself.

Minimizing the Number of Successor States in the Stubborn Set Method

Varpaaniemi K.

Fundamenta Informaticae

2002

Vol. 51, nr 1,2

215-234

Combinatorial explosion which occurs in parallel compositions of LTSs can be alleviated by letting the stubborn set method construct on-the-fly a reduced LTS that is CFFD- or CSP-equivalent to the actual parallel composition. This article considers the problem of minimizing the number of successor states of a given state in the reduced LTS. The problem can be solved by constructing an and/or-graph with weighted vertices and by finding a set of vertices that satisfies a certain constraint such that no set of vertices satisfying the constraint has a smaller sum of weights. Without weights, the and/or-graph can be constructed in low-degree polynomial time w.r.t. the length of the input of the problem. However, since actions can be nondeterministic and transitions can share target states, it is not known whether the weights are generally computable in polynomial time. Consequently, it is an open problem whether minimizing the number of successor states is as ``easy'' as minimizing the number of successor transitions.