Marc Fuchs

Marc Fuchs, Diana Ghinea, Zahra Parsaeian et al.

Approximate Agreement ($\mathcal{AA}$) is a fundamental primitive that, even in the presence of Byzantine faults, allows honest parties to obtain close (but not necessarily identical) outputs that lie within the range of their inputs. While the optimal round complexity of synchronous $\mathcal{AA}$ on real values is well understood, its extension to other input spaces has remained open, with fundamental questions regarding achievable resilience and round efficiency still unresolved. In this work, we investigate the optimal round complexity of synchronous $\mathcal{AA}$ on trees under Byzantine failures. In this setting, parties hold as inputs vertices of a publicly known labeled tree $T$ and must output $1$-close vertices lying in the convex hull of the honest inputs. We present a synchronous protocol with optimal resilience and round complexity $O\left(\frac{\log D(T)}{\log \log D(T)}\right)$, where $D(T)$ denotes the diameter of the input space tree. Complementing this result, we extend impossibility results for real-valued $\mathcal{AA}$ to any graph $G$ by proving a lower bound of $Ω\left(\frac{\log D(G)}{\log \log D(G) + \log \frac{n+t}{t}}\right)$ rounds, where $n$ is the number of parties and $t$ the number of Byzantine faults. Together, these results establish the asymptotic optimality of our protocol whenever $t \in Θ(n)$. We further extend our techniques to block graphs by leveraging their clique tree structure. This yields protocols for $\mathcal{AA}$ on block graphs with optimal resilience in both the synchronous and asynchronous models, and with optimal round complexity in the synchronous model.

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.