arXiv subjects

Markus Chimani

Publications and source records attributed to Markus Chimani.

At least 37 records · Page 2Linked to original sources

arXiv2021

Spanner Approximations in Practice

A multiplicative $α$-spanner $H$ is a subgraph of $G=(V,E)$ with the same vertices and fewer edges that preserves distances up to the factor $α$, i.e., $d_H(u,v)\leqα\cdot d_G(u,v)$ for all vertices $u$, $v$. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch.

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arXiv2021

Approximating Multistage Matching Problems

In multistage perfect matching problems we are given a sequence of graphs on the same vertex set and asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios. We propose new approximation algorithms and present methods to transfer results between different problem variants without loosing approximation guarantees.

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arXiv2020

On the Bond Polytope

Given a graph $G=(V,E)$, the maximum bond problem searches for a maximum cut $δ(S) \subseteq E$ with $S \subseteq V$ such that $G[S]$ and $G[V\setminus S]$ are connected. This problem is closely related to the well-known maximum cut problem and known under a variety of names such as largest bond, maximum minimal cut and maximum connected (sides) cut. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While cut polytopes have been intensively studied, there are no results on bond polytopes. We start a structural study of the latter. We investigate the relation between cut- and bond polytopes and study the effect of graph modifications on bond polytopes and their facets. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and $3$-connected planar $(K_5-e)$-minor free graphs. Moreover we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on $3$-connected graphs. This yields a linear time algorithm for maximum bond on $(K_5-e)$-minor free graphs.

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arXiv2020

An Experimental Study of ILP Formulations for the Longest Induced Path Problem

Given a graph $G=(V,E)$, the longest induced path problem asks for a maximum cardinality node subset $W\subseteq V$ such that the graph induced by $W$ is a path. It is a long established problem with applications, e.g., in network analysis. We propose novel integer linear programming (ILP) formulations for the problem and discuss efficient implementations thereof. Comparing them with known formulations from literature, we prove that they are beneficial in theory, yielding stronger relaxations. Moreover, our experiments show their practical superiority.

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arXiv2020

Crossing Numbers and Stress of Random Graphs

Consider a random geometric graph over a random point process in $\mathbb{R}^d$. Two points are connected by an edge if and only if their distance is bounded by a prescribed distance parameter. We show that projecting the graph onto a two dimensional plane is expected to yield a constant-factor crossing number (and rectilinear crossing number) approximation. We also show that the crossing number is positively correlated to the stress of the graph's projection.

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arXiv2020

Maximum Cut Parameterized by Crossing Number

Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.

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arXiv2019

Crossing Numbers of Beyond-Planar Graphs

We study the 1-planar, quasi-planar, and fan-planar crossing number in comparison to the (unrestricted) crossing number of graphs. We prove that there are $n$-vertex 1-planar (quasi-planar, fan-planar) graphs such that any 1-planar (quasi-planar, fan-planar) drawing has $Ω(n)$ crossings, while $O(1)$ crossings suffice in a crossing-minimal drawing without restrictions on local edge crossing patterns.

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arXiv2019

Computing Stable Demers Cartograms

Cartograms are popular for visualizing numerical data for map regions. Maintaining correct adjacencies is a primary quality criterion for cartograms. When there are multiple data values per region (over time or different datasets) shown as animated or juxtaposed cartograms, preserving the viewer's mental-map in terms of stability between cartograms is another important criterion. We present a method to compute stable Demers cartograms, where each region is shown as a square and similar data yield similar cartograms. We enforce orthogonal separation constraints with linear programming, and measure quality in terms of keeping adjacent regions close (cartogram quality) and using similar positions for a region between the different data values (stability). Our method guarantees ability to connect most lost adjacencies with minimal leaders. Experiments show our method yields good quality and stability.

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arXiv2019

Toroidal grid minors and stretch in embedded graphs

We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded graph G, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar crossing number of G. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree.

math.CO↗

arXiv2019

Cut Polytopes of Minor-free Graphs

The cut polytope of a graph $G$ is the convex hull of the indicator vectors of all cuts in $G$ and is closely related to the MaxCut problem. We give the facet-description of cut polytopes of $K_{3,3}$-minor-free graphs and introduce an algorithm solving MaxCut on those graphs, which only requires the running time of planar MaxCut. Moreover, starting a systematic geometric study of cut polytopes, we classify graphs admitting a simple or simplicial cut polytope.

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arXiv2018

A Simple Primal-Dual Approximation Algorithm for 2-Edge-Connected Spanning Subgraphs

We propose a simple and natural approximation algorithm for the problem of finding a 2-edge-connected spanning subgraph of minimum total edge cost in a graph. The algorithm maintains a spanning forest starting with an empty edge set. In each iteration, a new edge incident to a leaf is selected in a natural greedy manner and added to the forest. If this produces a cycle, this cycle is contracted. This growing phase ends when the graph has been contracted into a single node and a subsequent cleanup step removes redundant edges in reverse order. We analyze the algorithm using the primal-dual method showing that its solution value is at most 3 times the optimum. Although this only matches the ratio of existing primal-dual algorithms, we require only a single growing phase, thereby addressing a question by Williamson. Also, we consider our algorithm to be not only conceptually simpler than the known approximation algorithms but also easier to implement in its entirety. For n and m being the number of nodes and edges, respectively, it runs in O(min{nm, m + n^2 log n}) time and O(m) space without data structures more sophisticated than binary heaps and graphs, and without graph algorithms beyond depth-first search.

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arXiv2018

Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast

The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph $H$ of a given graph $G$ such that $H$ has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes, together with a lifting of the polyhedron, to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in $G$. This paper discusses both the theoretical as well as the practical sides of this strengthening.

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arXiv2018

Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments

Given a graph $G$, the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of $G$ with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski's famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance.

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arXiv2017

Crossing Number for Graphs With Bounded Pathwidth

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth~3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an $O(n)\times O(n)$-grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a $4w^3$-approximation for maximal graphs of pathwidth $w$. This is a constant approximation for bounded pathwidth graphs.

cs.CG↗

arXiv2017

Planar L-Drawings of Directed Graphs

We study planar drawings of directed graphs in the L-drawing standard. We provide necessary conditions for the existence of these drawings and show that testing for the existence of a planar L-drawing is an NP-complete problem. Motivated by this result, we focus on upward-planar L-drawings. We show that directed st-graphs admitting an upward- (resp. upward-rightward-) planar L-drawing are exactly those admitting a bitonic (resp. monotonically increasing) st-ordering. We give a linear-time algorithm that computes a bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or reports that there exists none.

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arXiv2017

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.

cs.CG↗

arXiv2016

Placing Arrows in Directed Graph Drawings

We consider the problem of placing arrow heads in directed graph drawings without them overlapping other drawn objects. This gives drawings where edge directions can be deduced unambiguously. We show hardness of the problem, present exact and heuristic algorithms, and report on a practical study.

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arXiv2016

A Note on the Practicality of Maximal Planar Subgraph Algorithms

Given a graph $G$, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of $G$ with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but---to the best of our knowledge---they have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, focusing on practical runtime, solution quality, and implementation complexity. Surprisingly, a seemingly only theoretically strong approximation forms the building block of the strongest choice.

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