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)

Geometric aspects of linear programming : shadow paths, central paths, and a cutting plane method

Most everyday algorithms are well-understood; predictions made theoretically about them closely match what we observe in practice. This is not the case for all algorithms, and some algorithms are still poorly understood on a theoretical level. This is the case for many algorithms used for solving optimization problems from operations reserach. Solving such optimization problems is essential in many industries and is done every day. One important example of such optimization problems are Linear Programming problems. There are a couple of different algorithms that are popular in practice, among which is one which has been in use for almost 80 years. Nonetheless, our theoretical understanding of these algorithms is limited. This thesis makes progress towards a better understanding of these key algorithms for lineair programming, among which are the simplex method, interior point methods, and cutting plane methods

Exact linear programming circuits, curvature, and diameter

We study Linear Programming (LP) and present novel algorithms. In particular, we study LP in the context of circuits, which are support-minimal vectors of linear spaces. Our results will be stated in terms of the circuit imbalance (CI), which is the worst-case ratio of nonzero entries of circuits and whose properties we study in detail. We present following results with logarithmic dependency on CI. (i) A scaling-invariant Interior-Point Method, which solves LP in time that is polynomial in the dimensions, answering an open question by Monteiro-Tsuchiya in the affirmative. This closes a long line of work by Vavasis-Ye and Monteiro-Tsuchiya; (ii)We introduce a new polynomial-time path-following interior point method where the number of iterations admits a singly exponential upper bound. This complements recent results, that path-following method must take at least exponentially many iterations; (iii)We further provide similar upper bounds on a natural notion of curvature of the central path; (iv) A black-box algorithm that requires only quadratically many calls to an approximate LP solver to solve LP exactly. This significantly strengthens the framework by Tardos, which requires exact solvers and whose runtime is logarithmic in the maximum subdeterminant of the constraint matrix. The maximum subdeterminant is exponentially bigger than CI, already for fundamental combinatorial problems such as matchings; (v) Furthermore, we obtain a circuit diameter that is quadratic in the number of variables, giving the first polynomial bound for general LP where CI is exponential. Unlike in the simplex method, one does not have to augment around the edges of the polyhedron: Augmentations can be in any circuit direction; (vi) Lastly, we present an accelerated version of the Newton–Dinkelbach method, which extends the black-box framework to certain classes of fractional and parametric optimization problems. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain improved runtimes over the non-accelerated version

User hints for optimisation processes

Innovative improvements in the area of Human-Computer Interaction and User Interfaces have en-abled intuitive and effective applications for a variety of problems. On the other hand, there has also been the realization that several real-world optimization problems still cannot be totally auto-mated. Very often, user interaction is necessary for refining the optimization problem, managing the computational resources available, or validating or adjusting a computer-generated solution. This thesis investigates how humans can help optimization methods to solve such difficult prob-lems. It presents an interactive framework where users play a dynamic and important role by pro-viding hints. Hints are actions that help to insert domain knowledge, to escape from local minima, to reduce the space of solutions to be explored, or to avoid ambiguity when there is more than one optimal solution. Examples of user hints are adjustments of constraints and of an objective function, focusing automatic methods on a subproblem of higher importance, and manual changes of an ex-isting solution. User hints are given in an intuitive way through a graphical interface. Visualization tools are also included in order to inform about the state of the optimization process. We apply the User Hints framework to three combinatorial optimization problems: Graph Clus-tering, Graph Drawing and Map Labeling. Prototype systems are presented and evaluated for each problem. The results of the study indicate that optimization processes can benefit from human interaction. The main goal of this thesis is to list cases where human interaction is helpful, and provide an ar-chitecture for supporting interactive optimization. Our contributions include the general User Hints framework and particular implementations of it for each optimization problem. We also present a general process, with guidelines, for applying our framework to other optimization problems

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum