Fabian Kuhn

Alkida Balliu, Sebastian Brandt, Fabian Kuhn et al.

One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes know the exact network size $n$, only a polynomial upper bound on $n$, or nothing at all. We ask whether these differences are merely technical or fundamentally affect the theory of Locally Checkable Labelings (LCLs), one of the most studied problem classes. LCLs are graph problems whose valid solutions can be characterized by a finite set of allowed constant-radius neighborhoods. Since their introduction by Naor and Stockmeyer [FOCS 1995], they have become central in distributed computing, and the last decade has seen major progress in understanding their complexity. For example, Chang, Kopelowitz, and Pettie [FOCS 2016] showed that the randomized complexity of any LCL on $n$-node graphs is at least its deterministic complexity on $\sqrt{\log n}$-node graphs. Later, Chang and Pettie [FOCS 2017] showed that any randomized $n^{o(1)}$-round algorithm for LCLs on bounded-degree trees can be turned into a deterministic $O(\log n)$-round algorithm. Then, Balliu et al. [STOC 2018] showed that such automatic speedups are impossible for general bounded-degree graphs. However, these results fundamentally rely on nodes knowing $n$. How much does this assumption affect the theory of LCLs? Our work shows that if nodes are oblivious to $n$, or know only a polynomial upper bound on it, then even on trees, the theory of LCLs changes significantly. While the fundamental classification of problems remains the same, we show the landscape becomes much more complex: for example, for LCLs, randomness helps in more cases; some problems have very unnatural complexities; and some have a lower bound that depends on which definition of $Ω$ we use!

Marc Fuchs, Fabian Kuhn

The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with $Δ+1$ colors, where $Δ$ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of $(Δ+1)$-coloring in the standard message passing model remains one of the most important open questions of the area. In this paper, we aim to improve our understanding of the deterministic complexity of $(Δ+1)$-coloring as a function of $Δ$ in a special family of graphs for which significantly faster algorithms are already known. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. In general, in graphs of neighborhood independence $θ=O(1)$ (e.g., line graphs), it is known that $(Δ+1)$-coloring can be solved in $2^{O(\sqrt{\logΔ})}+O(\log^* n)$ rounds. In the present paper, we significantly improve this result, and we show that in graphs of neighborhood independence $θ$, a $(Δ+1)$-coloring can be computed in $(θ\cdot\logΔ)^{O(\log\logΔ/ \log\log\logΔ)}+O(\log^* n)$ rounds and thus in quasipolylogarithmic time in $Δ$ as long as $θ$ is at most polylogarithmic in $Δ$. We also show that the known approach that leads to a polylogarithmic in $Δ$ algorithm for $(2Δ-1)$-edge coloring already fails for edge colorings of hypergraphs of rank at least $3$.

Marc Fuchs, Fabian Kuhn

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $β$, one can compute a coloring with $O(β^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdotβ$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrtΔ)+O(\log^* n)$-round algorithm for the $(Δ+1)$-coloring problem in the CONGEST model. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $θ$, one can compute a $d$-defective coloring with $O(θ\cdot Δ/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $θ$. In particular, if $θ=O(1)$, we get an $(\logΔ)^{O(\log\logΔ)}+O(\log^* n)$-round algorithm.