Collective Additive Tree Spanners of Bounded Tree-Breadth Graphs with Generalizations and Consequences

TL;DR

This paper explores how graphs with certain tree-decomposition properties admit small systems of collective additive or multiplicative tree spanners, providing polynomial algorithms to construct such systems with bounded stretch and size.

Contribution

It demonstrates converting multiplicative tree spanners into small sets of additive tree spanners and extends results to graphs with bounded tree-width, with polynomial-time algorithms.

Findings

Polynomial algorithms for constructing collective additive tree spanners.

Conversion of multiplicative to additive tree spanners with logarithmic size.

Extension to graphs with bounded tree-width and specific decomposition properties.

Abstract

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a system of $\mu$ collective additive tree $r$-spanners} (resp., {\em multiplicative tree $t$-spanners}) if there is a system $\cT(G)$ of at most $\mu$ spanning trees of $G$ such that for any two vertices $x,y$ of $G$ a spanning tree $T\in \cT(G)$ exists such that $d_T(x,y)\leq d_G(x,y)+r$ (resp., $d_T(x,y)\leq t\cdot d_G(x,y)$). When $\mu=1$ one gets the notion of {\em additive tree $r$-spanner} (resp., {\em multiplicative tree $t$-spanner}). It is known that if a graph $G$ has a multiplicative tree $t$-spanner, then $G$ admits a Robertson-Seymour's tree-decomposition with bags of radius at most $\lceil{t/2}\rceil$ in $G$. We use this to demonstrate that there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative tree $t$-spanner, constructs a system of at most $\log_2 n$ collective additive tree $O(t\log n)$-spanners of $G$. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph $G$ admits a multiplicative $t$-spanner with tree-width $k-1$, then $G$ admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most $k$ disks of $G$ of radius at most $\lceil{t/2}\rceil$ each. This is used to demonstrate that, for every fixed $k$, there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative $t$-spanner with tree-width $k-1$, constructs a system of at most $k(1+ \log_2 n)$ collective additive tree $O(t\log n)$-spanners of $G$.