Rüdiger Rudolf

Perspectives of Monge properties in optimization

Abstract An m × n matrix C is called Monge matrix if c ij + c rs ⩽ c is + c rj for all 1 ⩽ i r ⩽ m , 1 ⩽ j s ⩽ n . In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.

Three-dimensional axial assignment problems with decomposable cost coefficients

Abstract Given three n -element sequences a i , b i and c i of nonnegative real numbers, the aim is to find two permutations φ and Ψ such that the sum ∑ n i = 1 a i bφ ( i ) Cψ ( i ) is minimized (maximized, respectively). We show that the maximization version of this problem can be solved in polynomial time, whereas we present an NP-completeness proof for the minimization version. We identify several special cases of the minimization problem which can be solved in polynomial time, and suggest a local search heuristic for the general case.

Permuting matrices to avoid forbidden submatrices

This paper attaches a frame to a natural class of combinatorial problems and points out that this class includes many important special cases. A matrix M is said to avoid a set F of matrices if M does not contain any element of F as (ordered) submatrix. For F a fixed set of matrices, we consider the problem of deciding whether the rows and columns of a matrix can be permuted in such a way that the resulting matrix M avoids all matrices in F. We survey several known and new results on the algorithmic complexity of this problem, mostly dealing with (0,1)-matrices. Among others, we will prove that the problem is polynomial time solvable for many sets F containing a single, small matrix and we will exhibit some example sets F for which the problem is NP-complete.

The Cone of Monge Matrices: Extremal Rays and Applications

We present an additive characterization of Monge matrices based on the extremal rays of the cone of nonnegative Monge matrices. By using this characterization, a simple proof for an old result by Supnick (1957) on the traveling salesman problem on Monge matrices is derived.

A process scheduling problem arising from chemical production planning

In this paper we investigate scheduling problems which stem from real-world applications in the chemical process industry both from a theoretical and from a practical point of view. After providing a survey and a general mixed integer programming model, we present some results on the complexity of the process scheduling problem and investigate some important special cases. (We prove the NP-hardness in general and the polynomial solvability for specially structured cases.) Furthermore, we suggest a new heuristic approach and compare this to other heuristics known from the literature.

On the recognition of permuted bottleneck Monge matrices

An n × m matrix A is called bottleneck Monge matrix if max {aij, ars} <-max {ais, arj} for all 1<-i<r< -n, 1< -j <s< -m. The matrix A is termed permuted bottleneck Monge matrix, if there exist row and column permutations such that the permuted matrix becomes a bottleneck Monge matrix. We first show that the class of permuted 0–1 bottleneck Monge matrices can be recognized in O(nm) time. Then we present an O(nm(n+m)) time algorithm for the recognition of permuted bottleneck Monge matrices with arbitrary entries.

Monge matrices make maximization manageable

We continue the research on the effects of Monge structures in the area of combinatorial optimization. We show that three optimization problems become easy if the underlying cost matrix fulfills the Monge property: (A) The balanced max-cut problem, (B) the problem of computing minimum weight binary k-matchings and (C) the computation of longest paths in bipartite, edge-weighted graphs. In all three results, we first prove that the Monge structure imposes some special combinatorial property on the structure of the optimum solution, and then we exploit this combinatorial property to derive efficient algorithms.

On the recognition of permuted Supnick and incomplete Monge matrices

Incomplete Monge matrices are a generalization of standard Monge matrices: the values of some entries are not specified and the Monge property only must hold for all specified entries. We derive several combinatorial properties of incomplete Monge matrices and prove that the problem of recognizingpermuted incomplete Monge matrices is NP-complete. For the special case of permutedSupnick matrices, we derive a fast recognition algorithm and thereby identify a special case of then-vertex travelling salesman problem which can be solved inO(n2logn) time.

Sometimes Travelling is Easy: The Master Tour Problem

In 1975, Kalmanson proved that in case the distance matrix in the Travelling Salesman Problem (TSP) fulfills certain combinatorial conditions (nowadays called the Kalmanson conditions) then the TSP is solvable in polynomial time.

A general approach to avoiding two by two submatrices

A matrixC is said to avoid a set ℱ of matrices, if no matrix of ℱ can be obtained by deleting some rows and columns ofC. In this paper we consider the decision problem whether the rows and columns of a given matrixC can be permuted in such a way that the permuted matrix avoids all matrices of a given class ℱ. At first an algorithm is stated for deciding whetherC can be permuted such that it avoids a set ℱ of 2×2 matrices. This approach leads to a polynomial time recognition algorithm for algebraic Monge matrices fulfilling special properties. As main result of the paper it is shown that permuted Supnick matrices can be recognized in polynomial time. Moreover, we prove that the decision problem can be solved in polynomial time, if the set ℱ is sufficiently dense, and a sparse set of 2×2 matrices is exhibited for which the decision problem is NP-complete.ZusammenfassungEine MatrixC vermeidet eine Menge ℱ von Matrizen, wenn keine Matrix aus ℱ durch Streichen von Spalten und Zeilen vonC erhalten werden kann. In dieser Arbeit betrachten wir folgendes Entscheidungsproblem: Können die Zeilen bzw. Spalten einer MatrixC so vertauscht werden, daß die permutierte Matrix alle Matrizen aus einer gegebenen Menge ℱ vermeidet. Diese Arbeit enthält einen Algorithmus für den Fall, daß ℱ nur aus 2×2 Matrizen besteht. Dies führt zu einem polynomialen Erkennungsalgorithmus für spezielle algebraische Monge Matrizen. Als Hauptergebnis zeigen wir, daß permutierte Supnick Matrizen in polynomieller Zeit erkannt werden können. Zusätzlich wird bewiesen, daß im allgemeinen das Entscheidungsproblem NO-vollständig ist, es aber in polynomieller Zeit lösbar ist, wenn die Menge ℱ genügend dicht ist.