Lower bounds on collective additive spanners

TL;DR

This work establishes tight lower bounds for collective additive tree spanners and spanners of bounded treewidth across multiple graph classes. Using a gadget-based Master Theorem, the authors show that even powerful families of graphs (unit interval, chordal, weakly chordal, outerplanar, strongly chordal) require a non-constant number of spanning trees to achieve small additive surpluses, with precise bounds such as $\Omega((\log n)^{1/3})$ for unit interval graphs and nonexistence results for fixed surpluses in several classes. They extend these ideas to graphs of bounded treewidth by introducing $(k,h)$-snowflakes and proving that for any fixed $k\ge2$ and $c\ge1$, no spanning subgraph of treewidth $k-1$ can serve as a $c$-spanner of certain graphs of treewidth $k$. Collectively, the results complement known upper bounds and deepen understanding of the tradeoffs between the number of spanning trees, surplus, and structural graph properties in additive spanners.

Abstract

In this paper we present various lower bound results on collective tree spanners and on spanners of bounded treewidth. A graph $G$ is said to admit a system of $μ$ collective additive tree $c$-spanners if there is a system $\cal{T}$$(G)$ of at most $μ$ spanning trees of $G$ such that for any two vertices $u,v$ of $G$ a tree $T\in \cal{T}$$(G)$ exists such that the distance in $T$ between $u$ and $v$ is at most $c$ plus their distance in $G$. A graph $G$ is said to admit an additive $k$-treewidth $c$-spanner if there is a spanning subgraph $H$ of $G$ with treewidth $k$ such that for any pair of vertices $u$ and $v$ their distance in $H$ is at most $c$ plus their distance in $G$. Among other results, we show that: $\bullet$ Any system of collective additive tree $1$ -- spanners must have $Ω(\sqrt[3]{\log n})$ spanning trees for some unit interval graphs; $\bullet$ No system of a constant number of collective additive tree $2$-spanners can exist for strongly chordal graphs; $\bullet$ No system of a constant number of collective additive tree $3$-spanners can exist for chordal graphs; $\bullet$ No system of a constant number of collective additive tree $c$-spanners can exist for weakly chordal graphs as well as for outerplanar graphs for any constant $c\geq 0$; $\bullet$ For any constants $k \ge 2$ and $c \ge 1$ there are graphs of treewidth $k$ such that no spanning subgraph of treewidth $k-1$ can be an additive $c$-spanner of such a graph. All these lower bound results apply also to general graphs. Furthermore, they %results complement known upper bound results with tight lower bound results.

Figures (7)

• Figure 1: \ref{['fig:sub:house']} A house, \ref{['fig:sub:domino']} A domino.
• Figure 5: Idea of the gadget in the proof of Theorem\ref{['chordal3']}.
• Figure 6: Example of gadget for the case $k=3$. In addition to the edges depicted in the figure, all edges between the $b$ and the $d$ vertices exist. Terminal vertices are black and terminal edges are thick. The gray boxes around the $c$ vertices are to point out the adjacency of the corresponding $c$ vertex to the $b$ and $d$ vertices.
• Figure 7: Gadget for the case $k=4$.
• Figure 8: A $(3,1)$-snowflake (bold edges) is shown that is contained in a $(3,2)$-snowflake together with the construction tree of the $(3,1)$-snowflake (bold edges) contained in the construction tree of the $(3,2)$-snowflake. The corner vertices of the shaded area form a clique $K$ with the black vertex being the earliest vertex of $K$.

Theorems & Definitions (32)

• Theorem 1
• proof
• Claim 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• proof
• Lemma 6