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Complexity and Partitions
(2001)

Computational complexity theory usually investigates the complexity of sets, i.e., the complexity of partitions into two parts. But often it is more appropriate to represent natural problems by partitions into more than two parts. A particularly interesting class of such problems consists of classification problems for relations. For instance, a binary relation R typically defines a partitioning of the set of all pairs (x,y) into four parts, classifiable according to the cases where R(x,y) and R(y,x) hold, only R(x,y) or only R(y,x) holds or even neither R(x,y) nor R(y,x) is true. By means of concrete classification problems such as Graph Embedding or Entailment (for propositional logic), this thesis systematically develops tools, in shape of the boolean hierarchy of NP-partitions and its refinements, for the qualitative analysis of the complexity of partitions generated by NP-relations. The Boolean hierarchy of NP-partitions is introduced as a generalization of the well-known and well-studied Boolean hierarchy (of sets) over NP. Whereas the latter hierarchy has a very simple structure, the situation is much more complicated for the case of partitions into at least three parts. To get an idea of this hierarchy, alternative descriptions of the partition classes are given in terms of finite, labeled lattices. Based on these characterizations the Embedding Conjecture is established providing the complete information on the structure of the hierarchy. This conjecture is supported by several results. A natural extension of the Boolean hierarchy of NP-partitions emerges from the lattice-characterization of its classes by considering partition classes generated by finite, labeled posets. It turns out that all significant ideas translate from the case of lattices. The induced refined Boolean hierarchy of NP-partitions enables us more accuratly capturing the complexity of certain relations (such as Graph Embedding) and a description of projectively closed partition classes.

Starfree regular languages can be build up from alphabet letters by using only Boolean operations and concatenation. The complexity of these languages can be measured with the so-called dot-depth. This measure leads to concatenation hierarchies like the dot-depth hierarchy (DDH) and the closely related Straubing-Thérien hierarchy (STH). The question whether the single levels of these hierarchies are decidable is still open and is known as the dot-depth problem. In this thesis we prove/reprove the decidability of some lower levels of both hierarchies. More precisely, we characterize these levels in terms of patterns in finite automata (subgraphs in the transition graph) that are not allowed. Therefore, such characterizations are called forbidden-pattern characterizations. The main results of the thesis are as follows: forbidden-pattern characterization for level 3/2 of the DDH (this implies the decidability of this level) decidability of the Boolean hierarchy over level 1/2 of the DDH definition of decidable hierarchies having close relations to the DDH and STH Moreover, we prove/reprove the decidability of the levels 1/2 and 3/2 of both hierarchies in terms of forbidden-pattern characterizations. We show the decidability of the Boolean hierarchies over level 1/2 of the DDH and over level 1/2 of the STH. A technique which uses word extensions plays the central role in the proofs of these results. With this technique it is possible to treat the levels 1/2 and 3/2 of both hierarchies in a uniform way. Furthermore, it can be used to prove the decidability of the mentioned Boolean hierarchies. Among other things we provide a combinatorial tool that allows to partition words of arbitrary length into factors of bounded length such that every second factor u leads to a loop with label u in a given finite automaton.

Practical optimization problems often comprise several incomparable and conflicting objectives. When booking a trip using several means of transport, for instance, it should be fast and at the same time not too expensive. The first part of this thesis is concerned with the algorithmic solvability of such multiobjective optimization problems. Several solution notions are discussed and compared with respect to their difficulty. Interestingly, these solution notions are always equally difficulty for a single-objective problem and they differ considerably already for two objectives (unless P = NP). In this context, the difference between search and decision problems is also investigated in general. Furthermore, new and improved approximation algorithms for several variants of the traveling salesperson problem are presented. Using tools from discrepancy theory, a general technique is developed that helps to avoid an obstacle that is often hindering in multiobjective approximation: The problem of combining two solutions such that the new solution is balanced in all objectives and also mostly retains the structure of the original solutions. The second part of this thesis is dedicated to several aspects of systems of equations for (formal) languages. Firstly, conjunctive and Boolean grammars are studied, which are extensions of context-free grammars by explicit intersection and complementation operations, respectively. Among other results, it is shown that one can considerably restrict the union operation on conjunctive grammars without changing the generated language. Secondly, certain circuits are investigated whose gates do not compute Boolean values but sets of natural numbers. For these circuits, the equivalence problem is studied, i.\,e.\ the problem of deciding whether two given circuits compute the same set or not. It is shown that, depending on the allowed types of gates, this problem is complete for several different complexity classes and can thus be seen as a parametrized) representative for all those classes.

This thesis is devoted to the study of computational complexity theory, a branch of theoretical computer science. Computational complexity theory investigates the inherent difficulty in designing efficient algorithms for computational problems. By doing so, it analyses the scalability of computational problems and algorithms and places practical limits on what computers can actually accomplish. Computational problems are categorised into complexity classes. Among the most important complexity classes are the class NP and the subclass of NP-complete problems, which comprises many important optimisation problems in the field of operations research. Moreover, with the P-NP-problem, the class NP represents the most important unsolved question in computer science. The first part of this thesis is devoted to the study of NP-complete-, and more generally, NP-hard problems. It aims at improving our understanding of this important complexity class by systematically studying how altering NP-hard sets affects their NP-hardness. This research is related to longstanding open questions concerning the complexity of unions of disjoint NP-complete sets, and the existence of sparse NP-hard sets. The second part of the thesis is also dedicated to complexity classes but takes a different perspective: In a sense, after investigating the interior of complexity classes in the first part, the focus shifts to the description of complexity classes and thereby to the exterior in the second part. It deals with the description of complexity classes through leaf languages, a uniform framework which allows us to characterise a great variety of important complexity classes. The known concepts are complemented by a new leaf-language model. To a certain extent, this new approach combines the advantages of the known models. The presented results give evidence that the connection between the theory of formal languages and computational complexity theory might be closer than formerly known.

The thesis looks at the question asking for the computability of the dot-depth of star-free regular languages. Here one has to determine for a given star-free regular language the minimal number of alternations between concatenation on one hand, and intersection, union, complement on the other hand. This question was first raised in 1971 (Brzozowski/Cohen) and besides the extended star-heights problem usually refered to as one of the most difficult open questions on regular languages. The dot-depth problem can be captured formally by hierarchies of classes of star-free regular languages B(0), B(1/2), B(1), B(3/2),... and L(0), L(1/2), L(1), L(3/2),.... which are defined via alternating the closure under concatenation and Boolean operations, beginning with single alphabet letters. Now the question of dot-depth is the question whether these hierarchy classes have decidable membership problems. The thesis makes progress on this question using the so-called forbidden pattern approach: Classes of regular languages are characterized in terms of patterns in finite automata (subgraphs in the transition graph) that are not allowed. Such a characterization immediately implies the decidability of the respective class, since the absence of a certain pattern in a given automaton can be effectively verified. Before this work, the decidability of B(0), B(1/2), B(1) and L(0), L(1/2), L(1), L(3/2) were known. Here a detailed study of these classes with help of forbidden patterns is given which leads to new insights into their inner structure. Furthermore, the decidability of B(3/2) is proven. Based on these results a theory of pattern iteration is developed which leads to the introduction of two new hierarchies of star-free regular languages. These hierarchies are decidable on one hand, on the other hand they are in close connection to the classes B(n) and L(n). It remains an open question here whether they may in fact coincide. Some evidence is given in favour of this conjecture which opens a new way to attack the dot-depth problem. Moreover, it is shown that the class L(5/2) is decidable in the restricted case of a two-letter alphabet.