Completeness Results for Parameterized Space Classes

The parameterized complexity of a problem is considered "settled" once it has been shown to lie in FPT or to be complete for a class in the W-hierarchy or a similar parameterized hierarchy. Several natural parameterized problems have, however, resisted such a classification. At least in some cases, the reason is that upper and lower bounds for their parameterized space complexity have recently been obtained that rule out completeness results for parameterized time classes. In this paper, we make progress in this direction by proving that the associative generability problem and the longest common subsequence problem are complete for parameterized space classes. These classes are defined in terms of different forms of bounded nondeterminism and in terms of simultaneous time--space bounds. As a technical tool we introduce a "union operation" that translates between problems complete for classical complexity classes and for W-classes.Comment: IPEC 201

On measuring nondeterminism in regular languages

AbstractIt is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language. This paper studies situations intermediate between forbidding nondeterminism and allowing it. The amount of nondeterminism used by a finite automaton is quantified, so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied. These intermediate situations are shown always to lie between two extremes:(1) there are no savings as the amount of nondeterminism increases incrementally, so that savings occur only when the amount of nondeterminism becomes unlimited;(2) each increment of nondeterminism results in additional savings, the number s of states decreasing approximately as s1i, until exponential savings have been achieved after about i = logslog log s increments