Gerold Jäger

Improved Approximation Algorithms for Maximum Graph Partitioning Problems

We consider the design of approximation algorithms for a number of maximum graph partitioning problems, among others MAX-k-CUT, MAX-k-DENSE-SUBGRAPH, and MAX-k-DIRECTED-UNCUT. We present a new version of the semidefnite relaxation scheme along with a better analysis, extending work of Halperin and Zwick (2002). This leads to an improvement over known approximation factors for such problems. The key to the improvement is the following new technique: It was already observed by Han et al. (2002) that a parameter-driven choice of the random hyperplane can lead to better approximation factors than obtained by Goemans and Williamson (1995). But it remained difficult to find a “good” set of parameters. In this paper, we analyze random hyperplanes depending on several new parameters. We prove that a sub-optimal choice of these parameters can be obtained by the solution of a linear program which leads to the desired improvement of the approximation factors. In this fashion a more systematic analysis of the semidefinite relaxation scheme is obtained.

Some basics on tolerances

In this paper we deal with sensitivity analysis of combinatorial optimization problems and its fundamental term, the tolerance. For three classes of objective functions (

The number of pessimistic guesses in Generalized Mastermind

\Sigma, \Pi, {\mbox{MAX}}

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

) we give some basic properties on upper and lower tolerances. We show that the upper tolerance of an element is well defined, how to compute the upper tolerance of an element, and give equivalent formulations when the upper tolerance is +∞ or > 0. Analogous results are given for the lower tolerance and some results on the relationship between lower and upper tolerances are given.

Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order

Mastermind is a famous two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. The code consists of 4 pegs, each of which is one of 6 colors. In Generalized Mastermind a general number p of pegs and a general number c of colors is considered. Let f(p,c) be the pessimistic number of questions for the generalization of Mastermind with an arbitrary number p of pegs and c of colors. By a computer program we compute ten new values of f(p,c). Combining this program with theoretical methods, we compute all values f(3,c) and a tight lower and upper bound for f(4,c). For f(p,2) we give an upper bound and a lower bound. Finally, combining results for fixed p and c, we give bounds for the general case f(p,c).

The number of pessimistic guesses in Generalized Black-peg Mastermind

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.

Tolerance based contract-or-patch heuristic for the asymmetric TSP

We introduce a new combinatorial optimization problem, which is a generalization of the Traveling Salesman Problem (TSP) and which we call Traveling Salesman Problem of Second Order (2-TSP). It is motivated by an application in bioinformatics, especially the Permuted Variable Length Markov model. We propose seven elementary heuristics and two exact algorithms for the 2-TSP, some of which are generalizations of similar algorithms for the Asymmetric Traveling Salesman Problem (ATSP), some of which are new ideas. Finally we experimentally compare the algorithms for random instances and real instances from bioinformatics. Our experiments show that for the real instances most heuristics lead to optimum or almost-optimum solutions, and for the random instances the exact algorithms need less time than for the real instances.

Exact and Heuristic Algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection

Mastermind is a famous two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible using information he receives from the codemaker after each guess. In Generalized Black-peg Mastermind for given arbitrary numbers p, c, the secret code consists of p pegs each having one of c colors, and the received information consists only of a number of black pegs, where this number equals the number of pegs matching in the corresponding question and the secret code. Let b(p,c) be the pessimistic number of questions for Generalized Black-peg Mastermind. By a computer program we compute several values b(p,c). By introducing some auxiliary games and combining this program with theoretical methods, for arbitrary c we obtain exact formulas for b(2,c), b(3,c) and b(4,c) and give upper and lower bounds for b(5,c) and a lower bound for b(6,c). Furthermore, for arbitrary p, we present upper bounds for b(p,2), b(p,3) and b(p,4). Finally, we give bounds for the general case b(p,c). In particular, we improve an upper bound recently proved by Goodrich.

How frugal is mother nature with haplotypes

In this paper we improve the quality of a recently suggested class of construction heuristics for the Asymmetric Traveling Salesman Problem (ATSP), namely the Contract-or-Patch heuristic. Our improvement is based on replacing the selection of each path to be contracted after deleting a heaviest arc from each short cycle in an Optimal Assignment Problem Solution (OAPS) by contracting a single arc from a short cycle in an OAPS with the largest upper tolerance with respect to one of the relaxed ATSP. The improved algorithm produces higher-quality tours than all previous COP versions and is clearly outperforming all other construction heuristics on robustness.

The worst case number of questions in Generalized AB game with and without white-peg answers

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.