Thomas Niessen

Optimum positioning of base stations for cellular radio networks

Finding optimum base station locations for a cellular radio network is considered as a mathematical optimization problem. Dependent on the channel assignment policy, the minimization of interferences or the number of blocked channels, respectively, may be more favourable. In this paper, a variety of according analytical optimization problems are introduced. Each is formalized as an integer linear program, and in most cases optimum solutions can be given. Whenever by the complexity of the problem an exact solution is out of reach, simulated annealing is used as an approximate optimization technique. The performance of the different approaches is compared by extensive numerical tests.

The round-up property of the fractional chromatic number for proper circular arc graphs

A multigraph M with maximum degree Δ(M) is called critical, if the chromatic index χ2(M) > Δ(M) and χ2(M - e) = χ2(M) - 1 for each edge e of M. The weak critical graph conjecture [1, 7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c · Δ(M) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k e 5.

A Characterization of Graphs Having All (g, f)-Factors

LetGbe a graph with vertex setVand letg, f:V?Z+. We say thatGhas all (g, f)-factors ifGhas anh-factor for everyh:V?Z+such thatg(v)?h(v)?f(v) for everyv?Vand at least one suchhexists. In this note, we derive from Tuttesf-factor theorem a similar characterization for the property of having all (g, f)-factors. An analogous result for parity-factors is presented also.

How to find overfull subgraphs in graphs with large maximum degree

Let G be a simple graph with 3(G) > jVj .T heOverfull Graph Conjecture states that the chromatic index of G is equal to (G), if G does not contain an induced overfull subgraph H with (H )= (G), and otherwise it is equal to (G) + 1. We present an algorithm that determines these subgraphs in O(n 5=3 m) time, in general, and in O(n 3 ) time, if G is regular. Moreover, it is shown that G can have at most three of these subgraphs. If 2(G)jVj ,t henG contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time.

Regular factors of simple regular graphs and factor-spectra

Abstract Given integers n , r and λ, we determine all values of k for which every simple r -regular graph of order n and with edge-connectivity λ has a k -factor. Using this result we find for k ⩾ 2 the k -spectra Sp k ( n ) = { m : there exists a maximal set of m edge-disjoint k -factors of K n } which were introduced by Hoffman et al. (1993).

Frequency allocation and linear programming

The present paper deals with optimal fixed channel assignment for large real-world cellular radio networks. Examples are taken from data of the D2-network, operated by Mannesmann Mobilfunk (MMO) in Germany. Because of the huge size of the problems an exact optimal solution is presently out of reach. We present a heuristic iterative approach which performs extremely well, and significantly outperforms channel designs presently used by network operators. The basic ingredients of our approach are: (1) fast and well established simple heuristics as initial assignments; (2) splitting the whole problem into smaller subproblems which can be optimized efficiently by solving a binary linear program (BLP), and repeating this process iteratively; (3) past-processing the resulting near-optimal design to avoid undesirable properties. A lot of detailed problems must be solved, such as a powerful preprocessing of constraints for the BLPs, and a careful selection of the subproblems in (2). In summary, a very flexible tool is derived, also capable of taking into account external constraints from practical requirements.

Theory of maximum packing and related channel assignment strategies for cellular radio networks

Abstract. The maximum packing (MP) policy for dynamic channel assignment in cellular radio communication systems specifies that a new call attempt is admitted whenever there is some way of rearranging channels so that every call can be carried. Otherwise the call is blocked and removed from the system. We investigate the state space of MP and show that its description is in general more complicated than assumed to date. Furthermore, we prove that MP performs better than any fixed channel allocation and any hybrid policy under light traffic conditions.

Neighborhood unions and regular factors

We examine bounds on the size of the neighborhood union for two (independent) vertices of a graph that imply the existence of regular factors.

Minimum degree, independence number and regular factors

We investigate relations of the minimum degree and the independence number of a simple graph for the existence of regular factors.

Optimal channel allocation for several types of cellular radio networks

The channel allocation problem in cellular radio networks is formulated as an optimization problem on set-valued graph colorings. Thereby a common model is found for some optimization criteria that appeared formerly to be distinct. The optimization problem is then transformed to a weighted graph coloring problem. Several efficient algorithms for the weighted coloring on special classes of graphs are known. They are investigated, when they are applied to an instance resulting from a transformation of a channel allocation problem.