Partitioning into Isomorphic or Connected Subgraphs

This thesis deals mainly with the partitioning and connectedness of graphs. First, we show that the problem of partitioning the nodes of a graph into a specific number of subsets such that the induced subgraphs on these sets are isomorphic to one another is NP-complete. If the induced subgraphs have to be connected, the problem remains NP-complete. Then we inspect some special graph classes for which the problem is solvable in polynomial time. Afterwards, we deal with the problem of defining a polytope by incidence vectors of nodes, which induce a connected graph. We inspect some facet-defining inequalities and their general structure. For some graph classes we state the full description. We then proceed to the problem of partitioning the nodes of a graph into a given number of parts such that the induced graphs are connected. For the corresponding polytope we show the dimension and some facet defining inequalities. This theoretical inspection is advanced by the problem of partitioning a graph into different parts such that the parts induce a connected graph in order to maximize the induced cut. We introduce different ideas for solving this problem in SCIP and show the numerical results. This leads to interesting problems on MIPs in general. As the problem in literature generally deals with the feasible region, we focus on the objective function. To do that, we inspect the problem of finding MIPs for problems with nonlinear objective functions. We discuss properties and requirements showing the existence or non-existence of particular formulations. Lastly, we inspect the problem of partitioning the nodes of a graph such that all but one class are isomorphic. This problem becomes interesting if the part not inducing the isomorphism is minimized. For this purpose we also introduce a technique, which generates the parts by brute-force. Instead of partitioning the graph into isomorphic parts, we proceed to the problem of similar graphs. In this case we inspect different similarities and show algorithms which implement these

Partitioning into Isomorphic or Connected Subgraphs

This thesis deals mainly with the partitioning and connectedness of graphs. First, we show that the problem of partitioning the nodes of a graph into a specific number of subsets such that the induced subgraphs on these sets are isomorphic to one another is NP-complete. If the induced subgraphs have to be connected, the problem remains NP-complete. Then we inspect some special graph classes for which the problem is solvable in polynomial time. Afterwards, we deal with the problem of defining a polytope by incidence vectors of nodes, which induce a connected graph. We inspect some facet-defining inequalities and their general structure. For some graph classes we state the full description. We then proceed to the problem of partitioning the nodes of a graph into a given number of parts such that the induced graphs are connected. For the corresponding polytope we show the dimension and some facet defining inequalities. This theoretical inspection is advanced by the problem of partitioning a graph into different parts such that the parts induce a connected graph in order to maximize the induced cut. We introduce different ideas for solving this problem in SCIP and show the numerical results. This leads to interesting problems on MIPs in general. As the problem in literature generally deals with the feasible region, we focus on the objective function. To do that, we inspect the problem of finding MIPs for problems with nonlinear objective functions. We discuss properties and requirements showing the existence or non-existence of particular formulations. Lastly, we inspect the problem of partitioning the nodes of a graph such that all but one class are isomorphic. This problem becomes interesting if the part not inducing the isomorphism is minimized. For this purpose we also introduce a technique, which generates the parts by brute-force. Instead of partitioning the graph into isomorphic parts, we proceed to the problem of similar graphs. In this case we inspect different similarities and show algorithms which implement these

Sparsity of integer formulations for binary programs

\u3cp\u3eThis paper considers integer formulations of binary sets X of minimum sparsity, i.e., the maximal number of non-zeros for each row of the corresponding constraint matrix is minimized. Providing a constructive mechanism for computing the minimum sparsity, we derive sparsest integer formulations of several combinatorial problems, including the traveling salesman problem. We also show that sparsest formulations are NP-hard to separate, while (under mild assumptions) there exists a dense formulation of X separable in polynomial time.\u3c/p\u3

Mixed-Integer Programming Techniques for the Connected Max-k-Cut Problem

We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved

Finding the Best: Mathematical Optimization Based on Product and Process Requirements

The challenge of finding the best solution for a given problem plays a central role in many fields and disciplines. In mathematics, best solutions can be found by formulating and solving optimization problems. An optimization problem consists of an objective function, optimization variables, and optimization constraints, all of which define the solution space. Finding the optimal solution within this space means minimizing or maximizing the objective function by finding the optimal variables of the solution. Problems, such as geometry optimization of profiles (Hess and Ulbrich 2012), process control for stringer sheet forming (Bäcker et al. 2015) and optimization of the production sequence for branched sheet metal products (Günther and Martin 2006) are solved using mathematical optimization methods (Sects. 5.2 and 5.3). A variety of mathematical optimization methods is comprised within the field of engineering design optimization (EDO) (Roy et al. 2008)