Efficient Algorithms for Node Disjoint Subgraph Homeomorphism Determination

Recently, great efforts have been dedicated to researches on the management of large scale graph based data such as WWW, social networks, biological networks. In the study of graph based data management, node disjoint subgraph homeomorphism relation between graphs is more suitable than (sub)graph isomorphism in many cases, especially in those cases that node skipping and node mismatching are allowed. However, no efficient node disjoint subgraph homeomorphism determination (ndSHD) algorithms have been available. In this paper, we propose two computationally efficient ndSHD algorithms based on state spaces searching with backtracking, which employ many heuristics to prune the search spaces. Experimental results on synthetic data sets show that the proposed algorithms are efficient, require relative little time in most of the testing cases, can scale to large or dense graphs, and can accommodate to more complex fuzzy matching cases.Comment: 15 pages, 11 figures, submitted to DASFAA 200

Irrelevant vertices for the planar Disjoint Paths Problem

The Disjoint Paths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),…,(sk,tk)(s1,t1),…,(sk,tk), whether there is a collection of k pairwise vertex-disjoint paths linking sisi and titi, for i=1,…,ki=1,…,k. In their f(k)⋅n3f(k)⋅n3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k)g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem , whose – very technical – proof gives a function g(k)g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k)=O(k3/2⋅2k)g(k)=O(k3/2⋅2k). Our bound is radically better than the bounds known for general graphs

Minor Embedding in Broken Chimera and Pegasus Graphs is NP-complete

The embedding is an essential step when calculating on the D-Wave machine. In this work we show the hardness of the embedding problem for both types of existing hardware, represented by the Chimera and the Pegasus graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the Pegasus graphs, the proof is then further extended to the Pegasus graphs

Minor Embedding in Broken Chimera and Pegasus Graphs is NP-complete

The embedding is an essential step when calculating on the D-Wave machine. In this work we show the hardness of the embedding problem for both types of existing hardware, represented by the Chimera and the Pegasus graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the Pegasus graphs, the proof is then further extended to the Pegasus graphs

A branch, price, and cut approach to solving the maximum weighted independent set problem

The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied NP-hard problems in the field of combinatorial optimization. In the first part of the dissertation, I explore efficient branch-and-price (B&P) approaches to solve MWISP exactly. B&P is a useful integer-programming tool for solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less challenging, on average, to solve. I use the B&P framework to solve MWISP on the original graph G using these specially constructed subproblems to generate columns. I demonstrate that vertex-disjoint partitioning scheme gives an effective approach for relatively sparse graphs. I also show that the edge-disjoint approach is less effective than the vertex-disjoint scheme because the associated DWD reformulation of the latter entails a slow rate of convergence. In the second part of the dissertation, I address convergence properties associated with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the edge-disjoint B&P scheme and show that these methods improve the rate of convergence. In the third part of the dissertation, I focus on identifying new cut-generation methods within the B&P framework. Such methods have not been explored in the literature. I present two new methodologies for generating generic cutting planes within the B&P framework. These techniques are not limited to MWISP and can be used in general applications of B&P. The first methodology generates cuts by identifying faces (facets) of subproblem polytopes and lifting associated inequalities; the second methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully demonstrate the feasibility of both approaches and present preliminary computational tests of each

Integer programming models for the branchwidth problem

We consider the problem of computing the branchwidth and an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced in 1991 by Robertson and Seymour and were used in the proof of Graph Minors Theorem (GMT), a well known conjecture (Wagner's conjecture) in graph theory. The notions of branchwidth and branch decompositions have been proved to be useful for solving many NP-hard problems that have applications in fields such as graph theory, network design, sensor networks and biology. Branch decompositions have been utilized for problems such as the traveling salesman problem by Cook and Seymour, general minor containment and the branchwidth problem by Hicks by means of the relevant branch decomposition-based algorithms. Branch decomposition-based algorithms are fixed parameter tractable algorithms obtained by combining dynamic programming techniques with branch decompositions. The running time and space of these algorithms strongly depend on the width of the utilized branch decomposition. Thus, finding optimal or close to optimal branch decompositions is very important for the efficiency of the branch decomposition-based algorithms. Motivated by the vastness of the fields of application, we aim to increase the efficiency of the branch decomposition-based algorithms by investigating effective techniques to find optimal branch decompositions. We present three integer programming models for the branchwidth problem. Two similar formulations are based on the relationship of branchwidth problem with a special case of the Steiner tree packing problem. The third formulation is based on the notion of laminar separations. We utilize upper and lower bounds obtained by heuristic algorithms, reduction techniques and cutting planes to increase the efficiency of our models. We use all three models for the branchwidth problem on hypergraphs as well. We compare the performance of three models both on graphs and hypergraphs. Furthermore we use the third model for rank-width problem and also offer a heuristic for finding good rank decompositions. We provide computational results for this problem, which can be a basis of comparison for future formulations

A branch, price, and cut approach to solving the maximum weighted independent set problem

The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied NP-hard problems in the field of combinatorial optimization. In the first part of the dissertation, I explore efficient branch-and-price (B&P) approaches to solve MWISP exactly. B&P is a useful integer-programming tool for solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less challenging, on average, to solve. I use the B&P framework to solve MWISP on the original graph G using these specially constructed subproblems to generate columns. I demonstrate that vertex-disjoint partitioning scheme gives an effective approach for relatively sparse graphs. I also show that the edge-disjoint approach is less effective than the vertex-disjoint scheme because the associated DWD reformulation of the latter entails a slow rate of convergence. In the second part of the dissertation, I address convergence properties associated with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the edge-disjoint B&P scheme and show that these methods improve the rate of convergence. In the third part of the dissertation, I focus on identifying new cut-generation methods within the B&P framework. Such methods have not been explored in the literature. I present two new methodologies for generating generic cutting planes within the B&P framework. These techniques are not limited to MWISP and can be used in general applications of B&P. The first methodology generates cuts by identifying faces (facets) of subproblem polytopes and lifting associated inequalities; the second methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully demonstrate the feasibility of both approaches and present preliminary computational tests of each

Novel Techniques for the Zero-Forcing and p-Median Graph Location Problems

This thesis presents new methods for solving two graph location problems, the p-Median problem and the zero-forcing problem. For the p-median problem, I present a branch decomposition based method that finds the best p-median solution that is limited to some input support graph. The algorithm can be used to either find an integral solution from a fractional linear programming solution, or it can be used to improve on the solutions given by a pool of heuristics. In either use, the algorithm compares favorably in running time or solution quality to state-of-the-art heuristics. For the zero-forcing problem, this thesis gives both theoretical and computational results. In the theoretical section, I show that the branchwidth of a graph is a lower bound on its zero-forcing number, and I introduce new bounds on the zero-forcing iteration index for cubic graphs. This thesis also introduces a special type of graph structure, a zero-forcing fort, that provides a powerful tool for the analysis and modeling of zero-forcing problems. In the computational section, I introduce multiple integer programming models for finding minimum zero-forcing sets and integer programming and combinatorial branch and bound methods for finding minimum connected zero-forcing sets. While the integer programming methods do not perform better than the best combinatorial method for the basic zero-forcing problem, they are easily adapted to the connected zero-forcing problem, and they are the best methods for the connected zero-forcing problem

Combinatorial Problems in Programming Quantum Annealers

Bevor auf einer Quanten-Annealing-Maschine, wie der der Firma D-Wave Systems Inc., Berechnungen durchgeführt werden können, sind zwei grundlegende Schritte notwendig, um das Originalproblem in ein Format zu übertragen, das von solchen Maschinen gelöst werden kann: Als Erstes muss ein mit dem Problem assoziierter Graph in den speziellen Hardwaregraphen eingebettet werden und als Zweites müssen die Parameter des eingebetteten Problems entsprechend weiterer Hardwarerestriktionen gewählt werden, sodass die Lösungen des eingebetteten Problems beweisbar äquivalent zu den ursprünglichen Lösungen sind. Diese Doktorarbeit adressiert graphentheoretische Fragestellungen und kombinatorische Optimierungsprobleme, die bei der genaueren Betrachtung beider Schritte auftreten. Im ersten Teil dieser Arbeit analysieren wir die Komplexität des Einbettungsproblems im Quanten- Annealing-Kontext, das heißt für Chimera- und Pegasus-Hardwaregraphen mit zum Teil nicht nutzbaren, "defekten" Qubits. Wir beweisen die Schwere des Hamiltonkreisproblems, einem Spezialfall des Einbettungsproblems, in solchen Graphen durch die Konstruktion defekter Chimera-Graphen aus speziellen Graphen, für welche die Schwere des Problems bereits bekannt ist. Da der Chimera- ein Subgraph des Pegasus-Graphen ist, können wir das Resultat auf letzteren übertragen. Ein weiterer Spezialfall ist die Einbettung eines vollständigen Graphen, welcher ein universelles Template für die Einbettung von beliebigen Graphen mit einer kleineren oder gleich großen Zahl an Knoten darstellt. Durch die Formulierung als Matchingproblem mit zusätzlichen linearen Nebenbedingungen können wir zeigen, dass das Problem eingeschränkt auf die sich natürlich ergebende Einbettungsstruktur "fixed-parameter tractable" ist, wenn wir die Zahl der defekten Qubits im Chimera-Graphen als Parameter betrachten. Wir vergleichen unser Verfahren mit vorherigen, heuristischen Ansätzen auf verschiedenen, zufällig generierten defekten Hardwaregraphen. Dabei können wir einen Vorteil unserer Methode gegenüber den anderen hinsichtlich der gefundenen Graphengrößen in der Praxis zeigen. Zusätzlich geben wir ein heuristisches Modell mit weniger Nebenbedingungen an, welches noch bessere Resultate liefert. Der zweite Teil beschäftigt sich mit derWahl der geeigneten Parameter, für welche wir hinreichende Bedingungen formulieren können. Durch die Betrachtung eines einzelnen Originalknotens und verschiedener, von der Hardware abgeleiteter Zielsetzungen können wir spezielle lineare Optimierungsprobleme extrahieren. Die Analyse eines entsprechenden Polyeders der zulässigen Lösungen zeigt, dass optimale Lösungen zu diesen Problemen in vielen Fällen in Linearzeit gefunden werden können. Für die verbleibenden Fälle konstruieren wir einen Algorithmus, der die Parameter in höchstens kubischer Laufzeit angibt. Aufgrund der Problemstruktur gelten diese Resultate sogar, wenn wir uns auf Ganzzahligkeit einschränken. --- Before being able to perform calculations on a quantum annealing device such as D-Wave's, two essential steps are required to transfer the original problem into a format which can be solved by these machines: First, a graph associated with the problem needs to be embedded into the specific hardware graph and, secondly, the parameters of the embedded problem need to be chosen, in accordance with further hardware restrictions, such that the solutions to the resulting problem are provably equivalent to those of the original problem. This thesis addresses graph theoretical questions and combinatorial optimization problems appearing in the closer examination of both steps. In the first part of this work, we analyze the complexity of the embedding problem in the quantum annealing context, this means when restricting to Chimera or Pegasus hardware graphs containing unavailable, "broken" qubits. We prove the hardness of the Hamiltonian cycle problem, a special case of the embedding problem, in such graphs by constructing broken Chimera graphs from certain graphs for which it is known that finding a Hamiltonian cycle is hard. As the Chimera graph is a subgraph of the Pegasus graph, we can easily extend the result to the latter. A further special case is the embedding of a complete graph, forming a universal template for the embedding of all arbitrary graphs with a smaller or equal number of vertices. By formulating this problem as a matching problem with additional linear constraints, we can prove that the problem restricted to the naturally arising embedding structure is fixed-parameter tractable in the number of broken vertices in the Chimera graph. By testing our model against previous, heuristic approaches on various random broken hardware graphs, we can further show that our method performs superior in practice. Additionally, we provide a heuristic model with less constraints, showing an even better performance. The second part is concerned with the problem of setting feasible parameters for the machine, for which we can formulate sufficient requirements. Considering a single original vertex and different objectives derived from the hardware restrictions, we extract certain linear optimization problems. By analyzing a corresponding polyhedron of feasible solutions, we can show that optimal solutions to these problems can be found in linear time for a lot of cases. For the remaining cases, we construct an algorithm providing the parameters in at most cubic time. Due to the problem structure, these results even hold if we restrict ourselves to integer problems