arXiv subjects

Karl Däubel

Publications and source records attributed to Karl Däubel.

3 recordsLinked to original sources

arXiv2019

An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances.

cs.DM↗

arXiv2019

On orthogonal symmetric chain decompositions

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality.

math.CO↗

arXiv2019

Distance-preserving graph contractions

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding super-vertices remain at distance at least $φ(d)$ in the contracted graph, where $φ$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $φ(x)=x/α-β$, where $α\geq 1$ and $β\geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.

cs.DS↗