Markus Chimani

Improved Steiner tree algorithms for bounded treewidth

We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in O(Btw+2^[emailxa0protected][emailxa0protected]?|V|) time, where tw is the [emailxa0protected]?s treewidth and the Bell numberBk is the number of partitions of a k-element set. This is a linear-time algorithm for graphs with fixed treewidth and a polynomial algorithm for [emailxa0protected]?O(log|V|/loglog|V|). While being faster than the previously known algorithms, the coloring scheme used in our algorithm can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems with similar runtime bounds.

An SDP approach to multi-level crossing minimization

We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. We are given a layered graph (i.e., the graphs vertices are assigned to multiple parallel levels) and are asked for an ordering of the nodes on each level such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in what is probably the most widely used graph drawing scheme, the Sugiyama framework.n The problem has received a lot of attention in both the fields of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable, at least for small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances.n In this article, 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 by-product, we also obtain an SDP-based heuristic, which in practice always gives (near-)optimal solutions.n We conduct a large set of experiments, both on randomized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementation. 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 that the ILP solved that the SDP did not. Overall, our experiments 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.n Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this article, 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).

Fast alignment of fragmentation trees

Motivation: Mass spectrometry allows sensitive, automated and high-throughput analysis of small molecules such as metabolites. One major bottleneck in metabolomics is the identification of ‘unknown’ small molecules not in any database. Recently, fragmentation tree alignments have been introduced for the automated comparison of the fragmentation patterns of small molecules. Fragmentation pattern similarities are strongly correlated with the chemical similarity of the molecules, and allow us to cluster compounds based solely on their fragmentation patterns. Results: Aligning fragmentation trees is computationally hard. Nevertheless, we present three exact algorithms for the problem: a dynamic programming (DP) algorithm, a sparse variant of the DP, and an Integer Linear Program (ILP). Evaluation of our methods on three different datasets showed that thousands of alignments can be computed in a matter of minutes using DP, even for ‘challenging’ instances. Running times of the sparse DP were an order of magnitude better than for the classical DP. The ILP was clearly outperformed by both DP approaches. We also found that for both DP algorithms, computing the 1% slowest alignments required as much time as computing the 99% fastest. Contact: sebastian.boecker@uni-jena.de

Upward planarity testing via SAT

A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graphs triconnectivity structure which was able to solve sparse graphs. n nIn this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

Vertex insertion approximates the crossing number of apex graphs

An apex graph is a graph G from which only one vertex v has to be removed to make it planar. We show that the crossing number of such G can be approximated up to a factor of @D(G-v)@?d(v)/2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally chosen planar embedding of a planar graph. Since the latter problem can be solved in polynomial time, this establishes the first polynomial fixed-factor approximation algorithm for the crossing number problem of apex graphs with bounded degree. Furthermore, we extend this result by showing that the optimal solution for inserting multiple edges or vertices into a planar graph also approximates the crossing number of the resulting graph.

A tighter insertion-based approximation of the crossing number

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NP-hard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. n nIt is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph G + F, while computing the crossing number of G + F exactly is NP-hard already when |F| = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.

Exact Approaches to Multilevel Vertical Orderings

We present a semidefinite programming SDP approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering MLVO problem is a quadratic ordering problem and conceptually related to the well-studied problem of multilevel crossing minimization MLCM. In contrast to the latter, it can be formulated such that it does not merely consist of multiple sequentially linked bilevel quadratic ordering problems, but as a genuine multilevel problem with dense cost matrix. This allows us to describe the graphs structures more compactly and therefore obtain solutions for graphs too large for MLCM in practice. n nIn this paper we give motivation and mathematical models for MLVO. We formulate linear and quadratic programs, including some strengthening constraint classes, and an SDP relaxation for MLVO. We compare all approaches both theoretically and experimentally and show that MLVOs properties render linear and quadratic programming approaches inapplicable, even for small sparse graphs, while the SDP works surprisingly well in practice. This is in stark contrast to other ordering problems like MLCM, where such graphs are typically solved more efficiently with integer linear programs. Finally, we also compare our approach to related MLCM approaches.

Solving Two-Stage Stochastic Steiner Tree Problems by Two-Stage Branch-and-Cut

We consider the Steiner tree problem under a 2-stage stochastic model with recourse and finitely many scenarios (SSTP). Thereby, edges are purchased in the first stage when only probabilistic information on the set of terminals and the future edge costs is known. In the second stage, one of the given scenarios is realized and additional edges are purchased to interconnect the set of (now known) terminals. The goal is to choose an edge set to be purchased in the first stage while minimizing the overall expected cost of the solution.

Exact ILP solutions for phylogenetic minimum flip problems

In computational phylogenetics, the problem of constructing a consensus tree or supertree of a given set of rooted input trees can be formalized in different ways. We consider the Minimum Flip Consensus Tree and Minimum Flip Supertree problem, where input trees are transformed into a 0/1/?-matrix, such that each row represents a taxon, and each column represents a subtree membership. For the consensus tree problem, all input trees contain the same set of taxa, and no ?-entries occur. For the supertree problem, the input trees may contain different subsets of the taxa, and unrepresented taxa are coded with ?-entries. In both cases, the goal is to find a perfect phylogeny for the input matrix requiring a minimum number of 0/1-flips, i.e., matrix entry corrections. Both optimization problems are NP-hard.n We present the first efficient Integer Linear Programming (ILP) formulations for both problems, using three distinct characterizations of a perfect phylogeny. Although these three formulations seem to differ considerably at first glance, we show that they are in fact polytope-wise equivalent. Introducing a novel column generation scheme, it turns out that the simplest, purely combinatorial formulation is the most efficient one in practice. Using our framework, it is possible to find exact solutions for instances with ~100 taxa.

Shrinking the search space for clustered planarity

A clustered graph is a graph augmented with a hierarchical inclusion structure over its vertices, and arises very naturally in multiple application areas. While it is long known that planarity--i.e., drawability without edge crossings--of graphs can be tested in polynomial (linear) time, the complexity for the clustered case is still unknown. n nIn this paper, we present a new graph theoretic reduction which allows us to considerably shrink the combinatorial search space, which is of benefit for all enumeration-type algorithms. Based thereon, we give new classes of polynomially testable graphs and a practically efficient exact planarity test for general clustered graphs based on an integer linear program.