Tight Bounds for Constant-Round Domination on Graphs of High Girth and Low Expansion

TL;DR

The work addresses the problem of achieving constant-round, constant-factor approximations for $k$-hop domination on graphs with high girth and bounded $k$-hop expansion $f(k)$. It develops time-optimal pruning-based algorithms in the port-numbering model and proves tight bounds: a $Θ(k f(k))$-approximation in $k$ rounds and, in $3k$ rounds, an $O(k+ f(k)^{k/(k+1)})$-approximation, while any constant-round deterministic algorithm cannot achieve $o(k+ f(k)^{k/(k+1)})$. The main techniques include distance-aware preprocessing, a Voronoi decomposition around a $k$-hop MDS, and careful analysis of leaf structures to bound the impact of $f(k)$. The results reveal that constant-time approximations can be sublinear in edge density for this graph class and highlight the role of high girth in enabling such tight bounds; lower bounds extend to the Local model, showing the necessity of the chosen rounds for achieving sublinear density-dependent guarantees.

Abstract

A long-standing open question is which graph class is the most general one permitting constant-time constant-factor approximations for dominating sets. The approximation ratio has been bounded by increasingly general parameters such as genus, arboricity, or expansion of the input graph. Amiri and Wiederhake considered $k$-hop domination in graphs of bounded $k$-hop expansion and girth at least $4k+3$; the $k$-hop expansion $f(k)$ of a graph family denotes the maximum ratio of edges to nodes that can be achieved by contracting disjoint subgraphs of radius $k$ and deleting nodes. In this setting, these authors to obtain a simple $O(k)$-round algorithm achieving approximation ratio $Θ(kf(k))$. In this work, we study the same setting but derive tight bounds: - A $Θ(kf(k))$-approximation is possible in $k$, but not $k-1$ rounds. - In $3k$ rounds an $O(k+f(k)^{k/(k+1)})$-approximation can be achieved. - No constant-round deterministic algorithm can achieve approximation ratio $o(k+f(k)^{k/(k+1)})$. Our upper bounds hold in the port numbering model with small messages, while the lower bounds apply to local algorithms, i.e., with arbitrary message size and unique identifiers. This means that the constant-time approximation ratio can be \emph{sublinear} in the edge density of the graph, in a graph class which does not allow a constant approximation. This begs the question whether this is an artefact of the restriction to high girth or can be extended to all graphs of $k$-hop expansion $f(k)$.

Figures (3)

• Figure 1: Example for the pruning procedure with k being 2. The deleted nodes are marked in red. Note that the set of deleted nodes induces a forest, i.e., nodes in a cycle are never deleted.
• Figure 2: Graphs used in the proof of \ref{['thm:kroundlower']}, for 2-hop domination and maximum degree 3. On the left, the tree consisting of two copies of T joined by an edge of color i is depicted. On the right, for each i a copy of T is attached to a root by an edge of color i. Matching port numbers to colors, nodes in the i-th copy of T cannot distinguish between the two graphs in fewer than k rounds.
• Figure 3: Lower bound graph of \ref{['thm:k-1lower']}, for 2-hop domination and maximum degree 3. On the left, the underlying ring and the sole cycle in the graph is depicted. On the right for one of the nodes the attached tree is depicted, with the root at the top. If the algorithm does not select the root and its children, it must cover each of the nodes marked in green by a different dominator.

Theorems & Definitions (49)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• definition 1: Pruned Graph
• proof
• lemma 1
• proof