The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

Data Correcting Algorithms in Combinatorial Optimization

This paper describes data correcting algorithms. It provides the theory behind the algorithms and presents the implementation details and computational experience with these algorithms on the asymmetric traveling salesperson problem, the problem of maximizing submodular functions, and the simple plant location problem.

New solution approaches for the quadratic assignment problem

MSc., Faculty of Science, University of the Witwatersrand, 2011A vast array of important practical problems, in many di erent elds, can be modelled and solved as quadratic assignment problems (QAP). This includes problems such as university campus layout, forest management, assignment of runners in a relay team, parallel and distributed computing, etc. The QAP is a di cult combinatorial optimization problem and solving QAP instances of size greater than 22 within a reasonable amount of time is still challenging. In this dissertation, we propose two new solution approaches to the QAP, namely, a Branch-and-Bound method and a discrete dynamic convexized method. These two methods use the standard quadratic integer programming formulation of the QAP. We also present a lower bounding technique for the QAP based on an equivalent separable convex quadratic formulation of the QAP. We nally develop two di erent new techniques for nding initial strictly feasible points for the interior point method used in the Branch-and-Bound method. Numerical results are presented showing the robustness of both methods

Of keyboards and beyond - optimization in human-computer interaction

In this thesis, we present optimization frameworks in the area of Human-Computer Interaction. At first, we discuss keyboard layout problems with a special focus on a project we participated in, which aimed at designing the new French keyboard standard. The special nature of this national-scale project and its optimization ingredients are discussed in detail; we specifically highlight our algorithmic contribution to this project. Exploiting the special structure of this design problem, we propose an optimization framework that was efficiently computes keyboard layouts and provides very good optimality guarantees in form of tight lower bounds. The optimized layout that we showed to be nearly optimal was the basis of the new French keyboard standard recently published in the National Assembly in Paris. Moreover, we propose a relaxation for the quadratic assignment problem (a generalization of keyboard layouts) that is based on semidefinite programming. In a branch-and-bound framework, this relaxation achieves competitive results compared to commonly used linear programming relaxations for this problem. Finally, we introduce a modeling language for mixed integer programs that especially focuses on the challenges and features that appear in participatory optimization problems similar to the French keyboard design process.Diese Arbeit behandelt Ansätze zu Optimierungsproblemen im Bereich Human-Computer Interaction. Zuerst diskutieren wir Tastaturbelegungsprobleme mit einem besonderen Fokus auf einem Projekt, an dem wir teilgenommen haben: die Erstellung eines neuen Standards für die französische Tastatur. Wir gehen auf die besondere Struktur dieses Problems und unseren algorithmischen Beitrag ein: ein Algorithmus, der mit Optimierungsmethoden die Struktur dieses speziellen Problems ausnutzt. Mithilfe dieses Algorithmus konnten wir effizient Tastaturbelegungen berechnen und die Qualität dieser Belegungen effektiv (in Form von unteren Schranken) nachweisen. Das finale optimierte Layout, welches mit unserer Methode bewiesenermaßen nahezu optimal ist, diente als Grundlage für den kürzlich in der französischen Nationalversammlung veröffentlichten neuen französischen Tastaturstandard. Darüberhinaus beschreiben wir eine Relaxierung für das quadratische Zuweisungsproblem (eine Verallgemeinerung des Tastaturbelegungsproblems), die auf semidefinieter Programmierung basiert. Wir zeigen, dass unser Algorithmus im Vergleich zu üblich genutzten linearen Relaxierung gut abschneidet. Abschließend definieren und diskutieren wir eine Modellierungssprache für gemischt integrale Programme. Diese Sprache ist speziell auf die besonderen Herausforderungen abgestimmt, die bei interaktiven Optimierungsproblemen auftreten, welche einen ähnlichen Charakter haben wie der Prozess des Designs der französischen Tastatur

Well-solvable special cases of the TSP : a survey

The Traveling Salesman Problem belongs to the most important and most investigated problems in combinatorial optimization. Although it is an NP-hard problem, many of its special cases can be solved efficiently. We survey these special cases with emphasis on results obtained during the decade 1985-1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler and Shmoys. Keywords: Traveling Salesman Problem, Combinatorial optimization, Polynomial time algorithm, Computational complexity

The QAP-Polytope and the Star-Transformation

Polyhedral Combinatorics has been successfully applied to obtain considerable algorithmic progress towards the solution of many prominent hard combinatorial optimization problems. Until very recently, the quadratic assignment problem (QAP) was one of the few exceptions. Recent work of Rijal (1995) and Padberg and Rijal (1996) has on the one hand yielded some basic facts about the associated quadratic assignment polytope, but has on the other hand shown that investigations even of the very basic questions (like the dimension, the affine hull, and the trivial facets) soon become extremely complicated. In this paper, we propose an isomorphic transformation of the ''natural'' realization of the quadratic assignment polytope, which simplifies the polyhedral investigations enormously. We demonstrate this by giving short proofs of the basic results on the polytope that indicate that, exploiting the techniques developed in this paper, deeper polyhedral investigations of the QAP now become possible. Moreover, an 'ìnductive construction'' of the QAP-Polytope is derived that might be useful in branch-and-cut algorithms