Contributions to Determining Exact Ground-States of Ising Spin-Glasses and to their Physics

In the last decades, much research has focused on a better understanding of so-called spin glasses (e.g., the alloys CuMn and AuFe.) Spin glasses are not yet fully understood. In order to be able to test the different theories that have been proposed for the nature of spin glasses we have to analyze numerically generated data. We consider spin glasses in the Ising model, where the spins (magnetic dipoles) have exactly two possibilities for aligning themselves. We are interested in the low-temperature states of the system, in which the spins are `frozen' and disordered. Determining a state of minimum energy, a ground state, amounts to calculating a maximum cut in a graph. The max-cut problem is a prominent NP-hard problem from combinatorial optimization. Maximum cuts can be determined exactly with a branch-and-cut algorithm. This thesis consists of two parts. In the first part we introduce the max-cut problem and a branch-and-cut algorithm for solving reasonably sized problems. We present several approaches for improving its performance for Ising spin-glass instances. In the second part of this work, we study the physics of spin glasses. We first discuss what is known in the literature. Then we present results for Bethe spin glasses. Finally we study the nature of short-range three-dimensional spin glasses. Results of the former were obtained in collaboration with M. Palassini and A.K. Hartmann, results of the latter in cooperation with M. Palassini and A. Peter Young

A Fast Exact Algorithm for the Optimum Cooperation Problem

Given a graph G=(V,E) with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges having nodes in the same partition plus the number of resulting partitions. The problem is also known in the literature as the optimum attack problem in networks. It occurs as a subproblem in the separation of partition inequalities. Furthermore, a relevant physics application exists. Solution algorithms known in the literature require at least |V|-1 minimum cut computations in a corresponding network. In this work, we present a fast exact algorithm for the optimum cooperation problem. By graph-theoretic considerations and appropriately designed heuristics, we considerably reduce the number of minimum cut computations that are necessary in practice. We show the effectiveness of our method by comparing the performance of our algorithm with that of the fastest previously known method on instances coming from the physics application

Global Approaches for Facility Layout and VLSI Floorplanning

This paper summarizes recent advances in the global solution of several relevant facility layout problems

A simple MAX-CUT algorithm for planar graphs

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time can be bounded by O(|V|^(1.5)log|V|), similar to the fastest known methods. However, our transformation yields a much smaller associated graph than that of the known methods. Furthermore, it can be computed fast. As the practical running time strongly depends on the size of the associated graph, it can be expected that our algorithm is considerably faster than the methods known in the literature. More specifically, our program can determine maximum cuts in huge realistic and random planar graphs with up to 10^6 nodes

Preprocessing Maximum Flow Algorithms

Maximum-flow problems occur in a wide range of applications. Although already well-studied, they are still an area of active research. The fastest available implementations for determining maximum flows in graphs are either based on augmenting-path or on push-relabel algorithms. In this work, we present two ingredients that, appropriately used, can considerably speed up these methods. On the one hand, we present flow-conserving conditions under which subgraphs can be contracted to a single node. These rules are in the same spirit as presented by Padberg and Rinaldi (Math. Programming (47), 1990) for the minimum cut problem in graphs. On the other hand, we propose a two-step max-flow algorithm for solving the problem on instances coming from physics and computer vision. In the two-step algorithm flow is first sent along augmenting paths of restricted lengths only. Starting from this flow, the problem is then solved to optimality using some known max-flow methods. By extensive experiments on random instances and on instances coming from applications in theoretical physics and in computer vision, we show that a suitable combination of the proposed techniques speeds up traditionally used methods

A simple MAX-CUT algorithm for planar graphs

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time can be bounded by O(|V|^(1.5)log|V|), similar to the fastest known methods. However, our transformation yields a much smaller associated graph than that of the known methods. Furthermore, it can be computed fast. As the practical running time strongly depends on the size of the associated graph, it can be expected that our algorithm is considerably faster than the methods known in the literature. More specifically, our program can determine maximum cuts in huge realistic and random planar graphs with up to 10^6 nodes

Partitioning planar graphs: a fast combinatorial approach for max-cut

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. and that of Berman et al. The running time of the former can be bounded by O(|V|^(3/2)log|V|). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O(V{\ensuremath{|}}{\^{ }}(3/2)(log{\ensuremath{|}}V{\ensuremath{|}}){\^{ }}(3/2))alpha({\ensuremath{|}}V{\ensuremath{|}}), where alpha({\ensuremath{|}}V{\ensuremath{|}}) is the inverse Ackermann function, it can solve large instances in practice. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O({\ensuremath{|}}V{\ensuremath{|}}{\^{ }}(3/2)log{\ensuremath{|}}V{\ensuremath{|}}), similar to the bound achieved by Shih et al. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10{\^{ }}6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of Shih et al. and Berman et al. It turns out that our algorithm is considerably faster in practice than the one by Shih et al. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of Berman et al. However, whereas the procedure of generating the expanded graph in Berman et al. is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task

Global Approaches for Facility Layout and VLSI Floorplanning

This paper summarizes recent advances in the global solution of several relevant facility layout problems

A Fast Exact Algorithm for the Optimum Cooperation Problem

Given a graph G=(V,E) with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges having nodes in the same partition plus the number of resulting partitions. The problem is also known in the literature as the optimum attack problem in networks. It occurs as a subproblem in the separation of partition inequalities. Furthermore, a relevant physics application exists. Solution algorithms known in the literature require at least |V|-1 minimum cut computations in a corresponding network. In this work, we present a fast exact algorithm for the optimum cooperation problem. By graph-theoretic considerations and appropriately designed heuristics, we considerably reduce the number of minimum cut computations that are necessary in practice. We show the effectiveness of our method by comparing the performance of our algorithm with that of the fastest previously known method on instances coming from the physics application

A Fast Exact Algorithm for the Problem of Optimum Cooperation and Structure of Its Solutions

Given a graph with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges with nodes in the same class plus the number of the classes of the partition. The problem is also known in the literature as the optimum attack problem in networks. Furthermore, a relevant physics application exists. In this work, we present a fast exact algorithm for the optimum cooperation problem. Algorithms known in the literature require n-1 minimum cut computations in a corresponding network, where n is the number of nodes in the graph. By theoretical considerations and appropriately designed heuristics, we considerably reduce the numbers of minimum cut computations that are necessary in practice. We show the effectiveness of our method by presenting results on instances coming from the physics application. Furthermore, we analyze the structure of the optimal solutions