Florian Schager, Michael Wallner
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.
Tijn de Vos, Leo Wennmann, Malte Baumecker et al.
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 $\hat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.
Yannic Maus, Alexandre Nolin, Florian Schager
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 $(3 / 2 + \varepsilon)Î$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Î\cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $Î= \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2Î- 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].