A Polyhedral Approach to the Multi-Layer Crossing Minimization Problem

We study the multi-layer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multi-layer crossing minimization problem, we examine the 2-layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2- and 3-layer instances indicate, that the usage of the corresponding facet-defining inequalities in a branch-and-cut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted

An SDP approach to multi-level crossing minimization

We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization prob- lem. Thereby, we are given a layered graph (i.e., the graph´s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when draw- ing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the so- called Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this paper, we present a new SDP formulation for the general multi-level version that, for two-levels, is even stronger than the aforementioned specialized SDP. As a side- product, we also obtain an SDP-based heuristic which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on random- ized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementa- tion. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance the ILP solved, which the SDP did not. Overall, our experi- ments reveal that for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this paper we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum)

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis

Combinatorial Optimization

[no abstract available

Exact methods for nonlinear combinatorial optimization

We consider combinatorial optimization problems with nonlinear objective functions. Solution approaches for this class of problems proposed so far are either highly problem-specific or they apply generic algorithms for constrained nonlinear optimization, which often does not yield satisfactory results in practice. Our aim is to develop, implement and experimentally evaluate exact algorithms that address the nonlinearity of the objective function and at the same time exploit the underlying combinatorial structure of the problem. To this end we follow two approaches. The first combines good polyhedral descriptions of the objective function and the feasible set in a branch and cut-algorithm. The second approach is based on Lagrangean decomposition. By decomposing the original problem into an unconstrained nonlinear problem and a linear combinatorial problem, we are able to compute strong dual bounds for the optimal value. The computation of lower bounds is then embedded into a branch and bound-algorithm. For many applications there already exist efficient algorithms for the combinatorial subproblem, thus an important aspect of this thesis is the study of the corresponding unconstrained nonlinear subproblems. Both approaches have the advantage that they can easily be adapted to a wide range of nonlinear combinatorial problems.We devise both polyhedral and decomposition- based algorithms for submodular applications from wireless network design and portfolio optimization and evaluate their performance experimentally. Exploiting the equivalence between unconstrained binary quadratic optimization and the maximum cut problem gives rise to a branch and cut-algorithm for quadratic combinatorial problems which we use to compute optimal layouts of tanglegrams, an application from computational biology. Additionally we study the effect of quadratic reformulation of linear constraints, both theoretically and experimentally. The last class of nonlinear combinatorial problems we consider are two-scenario problems. Here we propose a new technique to compute lower bounds in the unconstrained subproblem of the decomposition. Our computational study of the two-scenario minimum spanning tree problem shows that the new Lagrangean decomposition-based algorithm is able to solve significantly larger instances than the standard linearization approach

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th