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This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revolutionizes communication. Traditional vertex coloring provides a conceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various extensions to cover technical and organizational side constraints. Among these ramifications are $T$-coloring and list coloring. To model all the subtleties, the techniques of integer programming have proven to be very useful. The ability to produce good frequency plans in practice is essential for the quality of mobile phone networks. The present algorithmic solution methods employ variants of some of the traditional coloring heuristics as well as more sophisticated machinery from mathematical programming. This paper will also address this issue. Finally, this paper discusses several practical frequency assignment problems in detail, states the associated mathematical models, and also points to public electronic libraries of frequency assignment problems from practice. The associated graphs have up to several thousand nodes and range from rather sparse to almost complete.
Mobile telecommunication systems establish a large number of communication links with a limited number of available frequencies; reuse of the same or adjacent frequencies on neighboring links causes interference. The task to find an assignment of frequencies to channels with minimal interference is the frequency assignment problem. The frequency assignment problem is usually treated as a graph coloring problem where the number of colors is minimized, but this approach does not model interference minimization correctly. We give in this paper a new integer programming formulation of the frequency assignment problem, the orientation model, and develop a heuristic two-stage method to solve it. The algorithm iteratively solves an outer and an inner optimization problem. The outer problem decides for each pair of communication links which link gets the higher frequency and leads to an acyclic subdigraph problem with additional longest path restrictions. The inner problem to find an optimal assignment respecting an orientation leads to a min-cost flow problem.
A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.
Combinatorial online optimization is an area with lots of applications and potential for significant progress, both in theory and practice. In this short note we sketch the ADACproblem, a typical large-scale online optimization problem, discuss some theoretical and pratical issues coming up, and explain, very briefly, how we approach this problem mathematically. Online problems are a battlefield of heuristics with many strong claims about their solution quality. We indicate that a stronger problem orientation and the use of a little more mathematics may yield.
A subset ${\cal C}$ of the power set of a finite set $E$ is called cardinality homogeneous if, whenever ${\cal C}$ contains some set $F$, ${\cal C}$ contains all subsets of $E$ of cardinality $|F|$. Examples of such set systems ${\cal C}$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of $E$. With each cardinality homogeneous set system ${\cal C}$, we associate the polytope $P({\cal C})$, the convex hull of the incidence vectors of all sets in ${\cal C}$, and provide a complete and nonredundant linear description of $P({\cal C})$. We show that a greedy algorithm optimizes any linear function over $P({\cal C})$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type $P({\cal C})$.
Optimization is the task of finding an optimum solution to a given problem. When the decision variables are discrete we speak of a combinatorial optimization problem. Such a problem is online when decisions have to be made before all data of the problem are known. And we speak of a real-time online problem when online decisions have to be computed within very tight time bounds. This paper surveys the are of combinatorial online and real-time optimization, it discusses, in particular, the concepts with which online and real-time algorithms can be analyzed.
This paper discusses online optimization of real-world transportation systems. We concentrate on transportation problems arising in production and manufacturing processes, in particular in company internal logistics. We describe basic techniques to design online optimization algorithms for such systems, but our main focus is decision support for the planner: which online algorithm is the most appropriate one in a particular setting? We show by means of several examples that traditional methods for the evaluation of online algorithms often do not suffice to judge the strengths and weaknesses of online algorithms. We present modifications of well-known evaluation techniques and some new methods, and we argue that the selection of an online algorithm to be employed in practice should be based on a sound combination of several theoretical and practical evaluation criteria, including simulation.
We present an approach to implement an auction of railway slots. Railway network, train driving characteristics, and safety requirements are described by a simplified, but still complex macroscopic model. In this environment, slots are modelled as combinations of scheduled track segments. The auction design builds on the iterative combinatorial auction. However, combinatorial bids are restricted to some types of slot bundles that realize positive synergies between slots. We present a bidding language that allows bidding for these slot bundles. An integer programming approach is proposed to solve the winner determination problem of our auction. Computational results for auction simulations in the Hannover-Fulda-Kassel area of the German railway network give evidence that auction approaches can induce a more efficient use of railway capacity.
Our main result is that every $n$-dimensional polytope can be described by at most $2n-1$ polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an $n$-dimensional pointed polyhedral cone we prove the bound $2n-2$ and for arbitrary polyhedra we get a constructible representation by $2n$ polynomial inequalities.
The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives. The transport company wishes to minimize operating costs, the passengers want to minimize travel times. We propose a n ew multi-commodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and that the lines are generated dynamically. We discuss properties of this model and investigate its complexity. Results with data for the city of Potsdam, Germany, are reported.