arXiv subjects

Florian Schager

Publications and source records attributed to Florian Schager.

6 recordsLinked to original sources

arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we consider the Santa Claus problem in the CONGEST model. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. Notably, each gift can have a different value. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we obtain the first results the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\hat \Theta(\sqrt n+D)$ rounds, where our $\widetilde\Omega(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.

cs.DS↗

arXiv2026

Fast Deterministic Distributed Degree Splitting

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)\Delta$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 \Delta \cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $\Delta = \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2\Delta - 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

cs.DS↗

arXiv2025

Towards Optimal Distributed Edge Coloring with Fewer Colors

There is a huge difference in techniques and runtimes of distributed algorithms for problems that can be solved by a sequential greedy algorithm and those that cannot. A prime example of this contrast appears in the edge coloring problem: while $(2Δ-1)$-edge coloring can be solved in $\mathcal{O}(\log^{\ast}(n))$ rounds on constant-degree graphs, the seemingly minor reduction to $(2Δ-2)$ colors leads to an $Ω(\log n)$ lower bound [Chang, He, Li, Pettie & Uitto, SODA'18]. Understanding this sharp divide between very local problems and inherently more global ones remains a central open question in distributed computing and it is a core focus of this paper. As our main contribution we design a deterministic distributed $\mathcal{O}(\log n)$-round reduction from the $(2Δ-2)$-edge coloring problem to the much easier $(2Δ-1)$-edge coloring problem. This reduction is optimal, as the $(2Δ-2)$-edge coloring problem admits an $Ω(\log n)$ lower bound, whereas the $2Δ-1$-edge coloring problem can be solved in $\mathcal{O}(\log^{\ast}n)$ rounds. By plugging in the $(2Δ-1)$-edge coloring algorithms from [Balliu, Brandt, Kuhn & Olivetti, PODC'22] running in $\mathcal{O}(\log^{12}Δ+ \log^{\ast} n)$ rounds, we obtain an optimal runtime of $\mathcal{O}(\log n)$ rounds as long as $Δ= 2^{\mathcal{O}(\log^{1/12} n)}$. Furthermore, on general graphs our reduction improves the runtime from $\widetilde{\mathcal{O}}(\log^3 n)$ to $\widetilde{\mathcal{O}}(\log^{5/3} n)$. In addition, we also obtain an optimal $\mathcal{O}(\log \log n)$-round randomized reduction of $(2Δ- 2)$-edge coloring to $(2Δ- 1)$-edge coloring. Lastly, we obtain an $\mathcal{O}(\log_Δn)$-round reduction from the $(2Δ-1)$-edge coloring, albeit to the somewhat harder maximal independent set (MIS) problem.

cs.DS↗

arXiv2025

On the Locality of Hall's Theorem

The last five years of research on distributed graph algorithms have seen huge leaps of progress, both regarding algorithmic improvements and impossibility results: new strong lower bounds have emerged for many central problems and exponential improvements over the state of the art have been achieved for the runtimes of many algorithms. Nevertheless, there are still large gaps between the best known upper and lower bounds for many important problems. The current lower bound techniques for deterministic algorithms are often tailored to obtaining a logarithmic bound and essentially cannot be used to prove lower bounds beyond $Ω(\log n)$. In contrast, the best deterministic upper bounds are often polylogarithmic, raising the fundamental question of how to resolve the gap between logarithmic lower and polylogarithmic upper bounds and finally obtain tight bounds. We develop a novel algorithm design technique aimed at closing this gap. In essence, each node finds a carefully chosen local solution in $O(\log n)$ rounds and we guarantee that this solution is consistent with the other nodes' solutions without coordination. The local solutions are based on a distributed version of Hall's theorem that may be of independent interest and motivates the title of this work. We showcase our framework by improving on the state of the art for the following fundamental problems: edge coloring, bipartite saturating matchings and hypergraph sinkless orientation. In particular, we obtain an asymptotically optimal $O(\log n)$-round algorithm for $3Δ/2$-edge coloring in bounded degree graphs. The previously best bound for the problem was $O(\log^4 n)$ rounds, obtained by plugging in the state-of-the-art maximal independent set algorithm from arXiv:2303.16043 into the $3Δ/2$-edge coloring algorithm from arXiv:1711.05469 .

cs.DS↗

arXiv2024

A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes

We present a novel bijection between stacked directed polyominoes and Motzkin paths with catastrophes. Further, we leverage this new bridge between these two worlds to obtain a better understanding of certain parameters of stacked directed animals. In particular, we obtain improved lower and upper bounds on the asymptotic width of stacked directed animals.

math.CO↗

arXiv2023

Fixed-Parameter Algorithms for Computing RAC Drawings of Graphs

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends (or determining that none exists) is fixed-parameter tractable parameterized by either the feedback edge number of $G$, or $b$ plus the vertex cover number of $G$.

cs.CG↗