Anja Fischer

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.

Exact algorithms and heuristics for the Quadratic Traveling Salesman Problem with an application in bioinformatics

In this paper we introduce an extension of the Traveling Salesman Problem (TSP), which is motivated by an important application in bioinformatics. In contrast to the TSP the costs do not only depend on each pair of two nodes traversed in succession in a cycle but on each triple of nodes traversed in succession. This problem can be formulated as optimizing a quadratic objective function over the traveling salesman polytope, so we call the combinatorial optimization problem quadratic TSP (QTSP). Besides its application in bioinformatics, the QTSP is a generalization of the Angular-Metric TSP and the TSP with reload costs. Apart from the TSP with quadratic cost structure we also consider the related Cycle Cover Problem with quadratic objective function (QCCP). In this work we present three exact solution approaches and several heuristics for the QTSP. The first exact approach is based on a polynomial transformation to a TSP, which is then solved by standard software. The second one is a branch-and-bound algorithm that relies on combinatorial bounds. The best exact algorithm is a branch-and-cut approach based on an integer programming formulation with problem-specific cutting planes. All heuristical approaches are extensions of classic heuristics for the TSP. Finally, we compare all algorithms on real-world instances from bioinformatics and on randomly generated instances. In these tests, the branch-and-cut approach turned out to be superior for solving the real-world instances from bioinformatics. Instances with up to 100 nodes could be solved to optimality in about ten minutes.

An Analysis of the Asymmetric Quadratic Traveling Salesman Polytope

The quadratic traveling salesman problem asks for a tour of minimal total costs where the costs are associated with each of two arcs that are traversed in succession. This structure arises, e.g., if the succession of two arcs represents loading processes in transport networks or a switch between different technologies in communication networks. Based on an integer program with quadratic objective function we present a linearized integer programming formulation and study the corresponding polyhedral structure of the asymmetric quadratic traveling salesman problem (AQTSP), where the costs may depend on the direction of traversal. The constructive approach that is used to establish the dimension of the underlying polytope allows us to prove the facetness of several classes of valid inequalities. Some of them are related to the Boolean quadric polytope. Two new classes are developed that exclude conflicting configurations. Among these the first one is separable in polynomial time, and the separation problem f...

Complete description for the spanning tree problem with one linearised quadratic term

Given an edge-weighted graph the minimum spanning tree problem (MSTP) asks for a spanning tree of minimal weight. The complete description of the associated polytope is well-known. Recently, Buchheim and Klein suggested studying the MSTP with one quadratic term in the objective function resp. the polytope arising after linearisation of that term, in order to better understand the MSTP with a general quadratic objective function. We prove a conjecture by Buchheim and Klein (2013) concerning the complete description of the associated polytope.

Simulation and optimization of robot driven production systems for peak-load reduction

One way to improve the energy efficiency in manufacturing is the use of energy-sensitive methods in production planning. So far, the energy consumption behavior of production facilities has not been investigated in great detail. Estimates are typically obtained by connected wattage values and concurrency factors. We present a new methodology to simulate and optimize complex robot driven production systems with special emphasis on energy aspects. In particular, we show how to translate the process descriptions and energy consumption profiles into a discrete-event-based simulation model and illustrate this with an example of a car body shop facility. In order to minimize the peak-load we set up an optimization model that is based on periodic time-expanded networks. A solution of this model corresponds to a process sequence for the robots that prescribes relative starting times via additional wait intervals. This sequence is then reinserted into the simulation model to validate the improvement.

An extended approach for lifting clique tree inequalities

We present a new lifting approach for strengthening arbitrary clique tree inequalities that are known to be facet defining for the symmetric traveling salesman problem in order to get stronger valid inequalities for the symmetric quadratic traveling salesman problem (SQTSP). Applying this new approach to the subtour elimination constraints (SEC) leads to two new classes of facet defining inequalities of SQTSP. For the special case of the SEC with two nodes we derive all known conflicting edges inequalities for SQTSP. Furthermore we extend the presented approach to the asymmetric quadratic traveling salesman problem (AQTSP).

Minimization and maximization versions of the quadratic travelling salesman problem

The travelling salesman problem (TSP) asks for a shortest tour through all vertices of a graph with respect to the weights of the edges. The symmetric quadratic travelling salesman problem (SQTSP) associates a weight with every three vertices traversed in succession. If these weights correspond to the turning angles of the tour, we speak of the angular-metric travelling salesman problem (Angle TSP). In this paper, we first consider the SQTSP from a computational point of view. In particular, we apply a rather basic algorithmic idea and perform the separation of the classical subtour elimination constraints on integral solutions only. Surprisingly, it turns out that this approach is faster than the standard fractional separation procedure known from the literature. We also test the combination with strengthened subtour elimination constraints for both variants, but these turn out to slow down the computation. Secondly, we provide a completely different, mathematically interesting MILP linearization for the Angle TSP that needs only a linear number of additional variables while the standard linearization requires a cubic one. For medium-sized instances of a variant of the Angle TSP, this formulation yields reduced running times. However, for larger instances or pure Angle TSP instances, the new formulation takes more time to solve than the known standard model. Finally, we introduce MaxSQTSP, the maximization version of the quadratic travelling salesman problem. Here, it turns out that using some of the stronger subtour elimination constraints helps. For the special case of the MaxAngle TSP, we can observe an interesting geometric property if the number of vertices is odd. We show that the sum of inner turning angles in an optimal solution always equals . This implies that the problem can be solved by the standard ILP model without producing any integral subtours. Moreover, we give a simple constructive polynomial time algorithm to find such an optimal solution. If the number of vertices is even, the optimal value lies between 0 and and these two bounds are tight, which can be shown by an analytic solution for a regular n-gon.

Computational recognition of RNA splice sites by exact algorithms for the quadratic traveling salesman problem

One fundamental problem of bioinformatics is the computational recognition of DNA and RNA binding sites. Given a set of short DNA or RNA sequences of equal length such as transcription factor binding sites or RNA splice sites, the task is to learn a pattern from this set that allows the recognition of similar sites in another set of DNA or RNA sequences. Permuted Markov (PM) models and permuted variable length Markov (PVLM) models are two powerful models for this task, but the problem of finding an optimal PM model or PVLM model is NP-hard. While the problem of finding an optimal PM model or PVLM model of order one is equivalent to the traveling salesman problem (TSP), the problem of finding an optimal PM model or PVLM model of order two is equivalent to the quadratic TSP (QTSP). Several exact algorithms exist for solving the QTSP, but it is unclear if these algorithms are capable of solving QTSP instances resulting from RNA splice sites of at least 150 base pairs in a reasonable time frame. Here, we investigate the performance of three exact algorithms for solving the QTSP for ten datasets of splice acceptor sites and splice donor sites of five different species and find that one of these algorithms is capable of solving QTSP instances of up to 200 base pairs with a running time of less than two days.

The traveling salesman problem on grids with forbidden neighborhoods

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases.

Solution Approaches for the Double-Row Equidistant Facility Layout Problem

We consider the Double-Row Equidistant Facility Layout Problem and show that the number of spaces needed to preserve at least one optimal solution is much smaller compared to the general double-row layout problem. We exploit this fact to tailor exact integer linear programming (ILP) and semidefinite programming (SDP) approaches that outperform other recent methods for this problem. We report computational results on a variety of benchmark instances showing that the ILP is preferable for small and medium instances whereas the SDP yields better results on large instances with up to 60 departments.