arXiv subjects

Franz Rendl

Publications and source records attributed to Franz Rendl.

10 recordsLinked to original sources

arXiv2024

Practical Experience with Stable Set and Coloring Relaxations

The stable set problem and the graph coloring problem are classes of NP-hard optimization problems on graphs. It is well known that even near-optimal solutions for these problems are difficult to find in polynomial time. The Lov\'asz theta function, introduced by Lov\'asz in the late 1970s, provides a powerful tool in the study of these problems. It can be expressed as the optimal value of a semidefinite program and serves as a relaxation for both problems. Over the years, considerable effort has been devoted to investigating additional cutting planes to strengthen these relaxations. In our work, we use these models and consider classes of cutting planes based on cliques, odd cycles, and odd antiholes contained in the underlying graph. We demonstrate that identifying such violated constraints can be done efficiently and that they often lead to significant improvements over previous bounds.

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arXiv2023

Stable-Set and Coloring bounds based on 0-1 quadratic optimization

We consider semidefinite relaxations of Stable-Set and Coloring, which are based on quadratic 0-1 optimization. Information about the stability number and the chromatic number is hidden in the objective function. This leads to simplified relaxations which depend mostly on the number of vertices of the graph. We also propose tightenings of the relaxations which are based on the maximal cliques of the underlying graph. Computational results on graphs from the literature show the strong potential of this new approach.

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arXiv2021

Lower Bounds for the Bandwidth Problem

The Bandwidth Problem seeks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions to bound the bandwidth of a graph. Our approach contains prior approaches for bounding the bandwidth as special cases. By varying sizes of partitions, we achieve a trade-off between quality of bounds and efficiency of computing them. To compute lower bounds, we derive a Semidefinite Programming relaxation. We evaluate the performance of our approach on several data sets, including real-world instances.

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arXiv2020

A Computational Study of Exact Subgraph Based SDP Bounds for Max-Cut, Stable Set and Coloring

The "exact subgraph" approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.

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arXiv2019

Performance of a Quantum Annealer for Ising Ground State Computations on Chimera Graphs

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue.

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arXiv2019

A Bundle Approach for SDPs with Exact Subgraph Constraints

The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.

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arXiv2018

Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.

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arXiv2018

Regularization of inverse problems via box constrained minimization

In the present paper we consider minimization based formulations of inverse problems $(x,Φ)\in\mbox{argmin}\{\mathcal{J}(x,Φ;y)\colon(x,Φ)\in M_{ad}(y) \}$ for the specific but highly relevant case that the admissible set $M_{ad}^δ(y^δ)$ is defined by pointwise bounds, which is the case, e.g., if $L^\infty$ constraints on the parameter are imposed in the sense of Ivanov regularization, and the $L^\infty$ noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems. Discretization of $(x,Φ)$ with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.

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arXiv2015

Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a non-convex cost function under a norm constraint, where non-convexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (non-convex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

math.NA↗

arXiv1999

Towards a characterisation of Pfaffian graphs

A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of $K_{3,3}$ such that $G - VH$ contains a 1-factor. However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs is currently not known. In this paper we describe a possible approach to the derivation of such a characterisation. We also extend the characterisation for bipartite graphs to a slightly more general class of graphs.

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