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We consider competitive location problems where two competing providers place their facilities sequentially and users can decide between the competitors. We assume that both competitors act non-cooperatively and aim at maximizing their own benefits. We investigate the complexity and approximability of such problems on graphs, in particular on simple graph classes such as trees and paths. We also develop fast algorithms for single competitive location problems where each provider places a single facilty. Voting location, in contrast, aims at identifying locations that meet social criteria. The provider wants to satisfy the users (customers) of the facility to be opened. In general, there is no location that is favored by all users. Therefore, a satisfactory compromise has to be found. To this end, criteria arising from voting theory are considered. The solution of the location problem is understood as the winner of a virtual election among the users of the facilities, in which the potential locations play the role of the candidates and the users represent the voters. Competitive and voting location problems turn out to be closely related.
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.
The first part of this thesis deals with the approximability of the traveling salesman problem. This problem is defined on a complete graph with edge weights, and the task is to find a Hamiltonian cycle of minimum weight that visits each vertex exactly once. We study the most important multiobjective variants of this problem. In the multiobjective case, the edge weights are vectors of natural numbers with one component for each objective, and since weight vectors are typically incomparable, the optimal Hamiltonian cycle does not exist. Instead we consider the Pareto set, which consists of those Hamiltonian cycles that are not dominated by some other, strictly better Hamiltonian cycles. The central goal in multiobjective optimization and in the first part of this thesis in particular is the approximation of such Pareto sets.
We first develop improved approximation algorithms for the two-objective metric traveling salesman problem on multigraphs and for related Hamiltonian path problems that are inspired by the single-objective Christofides' heuristic. We further show arguments indicating that our algorithms are difficult to improve. Furthermore we consider multiobjective maximization versions of the traveling salesman problem, where the task is to find Hamiltonian cycles with high weight in each objective. We generalize single-objective techniques to the multiobjective case, where we first compute a cycle cover with high weight and then remove an edge with low weight in each cycle. Since weight vectors are often incomparable, the choice of the edges of low weight is non-trivial. We develop a general lemma that solves this problem and enables us to generalize the single-objective maximization algorithms to the multiobjective case. We obtain improved, randomized approximation algorithms for the multiobjective maximization variants of the traveling salesman problem. We conclude the first part by developing deterministic algorithms for these problems.
The second part of this thesis deals with redundancy properties of complete sets. We call a set autoreducible if for every input instance x we can efficiently compute some y that is different from x but that has the same membership to the set. If the set can be split into two equivalent parts, then it is called weakly mitotic, and if the splitting is obtained by an efficiently decidable separator set, then it is called mitotic. For different reducibility notions and complexity classes, we analyze how redundant its complete sets are.
Previous research in this field concentrates on polynomial-time computable reducibility notions. The main contribution of this part of the thesis is a systematic study of the redundancy properties of complete sets for typical complexity classes and reducibility notions that are computable in logarithmic space. We use different techniques to show autoreducibility and mitoticity that depend on the size of the complexity class and the strength of the reducibility notion considered. For small complexity classes such as NL and P we use self-reducible, complete sets to show that all complete sets are autoreducible. For large complexity classes such as PSPACE and EXP we apply diagonalization methods to show that all complete sets are even mitotic. For intermediate complexity classes such as NP and the remaining levels of the polynomial-time hierarchy we establish autoreducibility of complete sets by locally checking computational transcripts. In many cases we can show autoreducibility of complete sets, while mitoticity is not known to hold. We conclude the second part by showing that in some cases, autoreducibility of complete sets at least implies weak mitoticity.
Given points in the plane, connect them using minimum ink. Though the task seems simple, it turns out to be very time consuming. In fact, scientists believe that computers cannot efficiently solve it. So, do we have to resign? This book examines such NP-hard network-design problems, from connectivity problems in graphs to polygonal drawing problems on the plane. First, we observe why it is so hard to optimally solve these problems. Then, we go over to attack them anyway. We develop fast algorithms that find approximate solutions that are very close to the optimal ones. Hence, connecting points with slightly more ink is not hard.