Steffen Reith

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.

The complexity of base station positioning in cellular networks

We consider two optimization problems for cellular telephone networks, that arise in a recently discussed ITU proposal for a traffic load model. These problems address the positioning of base stations (on given possible locations) with the aim to maximize the number of supplied demand nodes and minimize the number of stations that have to be built. We show that these problems are hard to approximate, but their Euclidean versions allow a polynomial-time approximation scheme (PTAS). Furthermore, we consider other related optimization problems.

Bases for Boolean co-clones

The complexity of various problems in connection with Boolean constraints, like, for example, quantified Boolean constraint satisfaction, have been studied recently. Depending on what types of constraints may be used, the complexity of such problems varies. A very interesting observation of the recent past has been that the thus derived classification of constraints can be explained with the help of universal algebra. More precisely, the difficulty of such a constraint problem often depends on the co-clone the constraints are from. A co-clone is a set of Boolean relations that is closed under very natural closure operations. Nearly all these co-clones can be generated by said operators out of a finite set of relations, a so-called base. Knowing a, preferably simple, base for each co-clone can therefore be of great value when studying the complexity of Boolean constraint problems, since this knowledge reduces the infinitely many cases of equivalent problems to a single one--the constraint satisfaction problem for this base. In this paper we give a finite and simple base for every Boolean co-clone, where this is possible. We give evidence that the presented bases are as easy as possible.

The Complexity of Problems Defined by Boolean Circuits

We study the complexity of circuit-based combinatorial problems (e.g., the circuit value problem and the satisfiability problem) defined by boolean circuits w ith gates from an arbitrary finite base of boolean functions. Special cases have been investigated in the literature. We give a complete characterization of their complexity depending on the base . For example, for the satisfiability problem for boolean circuits with gates from we present a complete collection of (decidable) criteria which tell us for which this problem is in , is complete for , is complete for , is complete for , or is complete for . Our proofs make substantial use of the characterization of all closed classes of boolean functions given by E.L. POST already in the twenties.

Equivalence and Isomorphism for Boolean Constraint Satisfaction

A Boolean constraint satisfaction instance is a set of constraint applications where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a dichotomy theorem by showing that for all finite sets C of constraints, this problem is either polynomial-time solvable or coNP-complete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNP-hard if the corresponding equivalence problem is coNP-hard, and polynomial-time many-one reducible to the graph isomorphism problem in all other cases.

Optimal satisfiability for propositional calculi and constraint satisfaction problems

We consider the problems of finding the lexicographically minimal (or maximal) satisfying assignment of propositional formulas for different restricted classes of formulas. It turns out that for each class from our framework, these problems are either polynomial time solvable or complete for OptP. We also consider the problem of deciding if in the optimal assignment the largest variable gets value 1. We show that this problem is either in P or PNP complete.

On the Complexity of Some Equivalence Problems for Propositional Calculi

In the present paper we study the complexity of Boolean equivalence problems (i.e. have two given propositional formulas the same truthtable) and of Boolean isomorphism problems (i.e. does there exists a permutation of the variables of one propositional formula, such that the truthtable of this modified formula coincides with the truthtable of the second formula) of two given generalized propositional formulas and certain classes of Boolean circuits.

The Complexity of Boolean Constraint Isomorphism

We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time many-one equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.

On boolean lowness and boolean highness

The concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by Schoning [Sch85]. Informally, a set is low (high, resp.) for a relativizable class K of languages if it does not add (adds maximal, resp.) power to K when used as an oracle. In this paper we introduce the notions of boolean lowness and boolean highness. Informally, a set is boolean low (boolean high, resp.) for a class K of languages if it does not add (adds maximal, resp.) power to K when combined with K by boolean operations. We prove properties of boolean lowness and boolean highness which show a lot of similarities with the notions of lowness and highness. Using Kadin’s technique of hard strings (see [Kad88, Wag87, CK96, BCO93]) we show that the sets which are boolean low for the classes of the boolean hierarchy are low for the boolean closure of Σ2p. Furthermore, we prove a result on boolean lowness which has as a corollary the best known result (see [BCO93]; in fact even a bit better) on the connection of the collapses of the boolean hierarchy and the polynomial-time hierarchy: If BH = NP(k) then PH = Σ2p (k − 1) ⊕ NP(k).

The Complexity of Problems for Quantified Constraints

In this paper we are interested in quantified propositional formulas in conjunctive normal form with “clauses” of arbitrary shapes. i.e., consisting of applying arbitrary relations to variables. We study the complexity of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for such formulas, both with and without a bound on the number of quantifier alternations. For each of these computational goals we get full complexity classifications: We determine the complexity of each of these problems depending on the set of relations allowed in the input formulas. Thus, on the one hand we exhibit syntactic restrictions of the original problems that are still computationally hard, and on the other hand we identify non-trivial subcases that admit efficient algorithms.