Krzysztof Fleszar

Drawing Graphs on Few Lines and Few Planes

We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.

New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness

We study the classical \({\mathsf {NP}}\)-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is \({2^{\varOmega (\sqrt{\log n})}}\), assuming \({\mathsf {NP}\not \subseteq \mathsf {DTIME}(n^{\mathcal {O}(\log n)})}\). This constitutes a significant gap to the best known approximation upper bound of \({\mathcal {O}(\sqrt{n})}\) due to Chekuri et al. (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an \({\mathcal {O}(1)}\)-approximation when edges (or nodes) may be used by \({\mathcal {O}\left( \log n/\log \log n\right) }\) paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results: For MaxEDP, we give an \({\mathcal {O}(\sqrt{r} \log ({k}r))}\)-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio \({\mathcal {O}(\sqrt{n})}\) due to Chekuri et al., as \({r\le n}\). Further, we show how to route \({\varOmega ({\text {OPT}}^{*})}\) pairs with congestion bounded by \({\mathcal {O}(\log (kr)/\log \log (kr))}\), strengthening the bound obtained by the classic approach of Raghavan and Thompson. For MaxNDP, we give an algorithm that gives the optimal answer in time \({(k+r)^{\mathcal {O}(r)}\cdot n}\). This is a substantial improvement on the run time of \({2^kr^{\mathcal {O}(r)}\cdot n}\), which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is \({\mathsf {NP}}\)-hard even for \({r=1}\), and MaxNDP is \({\mathsf {W}[1]}\)-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless \({\mathsf {FPT}= \mathsf {W}[1]}\) and that our approximability results are relevant even for very small constant values of r.

Colored non-crossing Euclidean Steiner forest

Given a set of

Road segment selection with strokes and stability

k

Structural complexity of multiobjective NP search problems

-colored points in the plane, we consider the problem of finding

Minimum Rectilinear Polygons for Given Angle Sequences

k

Bi-factor approximation algorithms for hard capacitated k -median problems

trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For

The Complexity of Solving Multiobjective Optimization Problems and its Relation to Multivalued Functions.

k=1

The Complexity of Drawing Graphs on Few Lines and Few Planes

, this is the well-known Euclidean Steiner tree problem. For general

A Constant-Factor Approximation Algorithm for Uniform Hard Capacitated

k