Christian Glaßer

A polynomial-time approximation scheme for base station positioning in UMTS networks

We consider the following optimization problem for UMTS networks: For a specified teletraffic demand and possible base station locations, choose positions for base stations such thatthe construction costs are below a given limit, as much teletraffic as possible is supplied, the ongoing costs are minimal, and the intra-cell interference in the range of each base station is low. We prove that for a particular specification of teletraffic (the so called demand node concept), this problem has a polynomial-time approximation scheme, but cannot have a fully polynomial-time approximation scheme unless P = NP.

Error-bounded probabilistic computations between MA and AM

We introduce the probabilistic class SBP. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show that SBP is in the polynomial-time hierarchy, between MA and AM on the one hand and between BPP and BPPpath on the other hand. We provide evidence that SBP does not coincide with these and other known complexity classes. In particular, in a suitable relativized world SBP is not contained in Σ2P. In the same world, BPPpath is not contained in Σ2P, which solves an open question raised by Han, Hemaspaandra, and Thierauf. We study the question of whether SBP has many-one complete sets. We relate this question to the existence of uniform enumerations and construct an oracle relative to which SBP and AM do not have many-one complete sets. We introduce the operator SB. and prove that, for any class C with certain properties, BP ċ ∃ ċ C contains every class defined by applying an operator sequence over {Uċ, ∃ċ, BPċ, SBċ} to C.

Languages of Dot-Depth 3/2

We prove an effective characterization of languages having dot-depth 3/2. Let B3/2 denote this class, i.e., languages that can be written as finite unions of languages of the form u0L1u1L2u2 ... Lnun, where ui ∈ A* and Li are languages of dot-depth one. Let F be a deterministic finite automaton accepting some language L. Resulting from a detailed study of the structure of B3/2, we identify a pattern P (cf. Fig. 2) such that L belongs to B3/2 if and only if F does not have pattern P in its transition graph. This yields an NL-algorithm for the membership problem for B3/2. Due to known relations between the dot-depth hierarchy and symbolic logic, the decidability of the class of languages definable by Σ2-formulas of the logic FO[<, min,max, S, P] follows. We give an algebraic interpretation of our result.

Survey of disjoint NP-pairs and relations to propositional proof systems

We survey recent results on disjoint NP-pairs. In particular, we survey the relationship of disjoint NP-pairs to the theory of proof systems for propositional calculus.

Approximability and hardness in multi-objective optimization

We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are: - We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions. - These solution notions induce corresponding NP-hardness notions for which we prove implication and separation results. - We define approximative solution notions and investigate in which cases polynomial-time solvability translates from one to another notion. Moreover, for problems where all objectives have to be minimized, approximability results translate from single-objective to multi-objective optimization such that the relative error degrades only by a constant factor. Such translations are not possible for problems where all objectives have to be maximized (unless P = NP). As a consequence we see that in contrast to single-objective problems (where the solution notions coincide), the situation is more subtle for multiple objectives. So it is important to exactly specify the NP-hardness notion when discussing the complexity of multi-objective problems.

Splitting NP-Complete Sets

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long-standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaser et al., complete sets for all of the following complexity classes are m-mitotic:

Redundancy in complete sets

\mathrm{NP}

Autoreducibility of complete sets for log-space and polynomial-time reductions

,

A Protocol for Serializing Unique Strategies

\mathrm{coNP}

Equivalence problems for circuits over sets of natural numbers

,