[Paper Review] On distance r-dominating and 2r-independent sets in sparse graphs

Zdenĕk Dvořák|arXiv (Cornell University)|Oct 27, 2017

Advanced Graph Theory Research7 references3 citations

TL;DR

This paper establishes tighter bounds between distance $r$-dominating sets and $2r$-independent sets in sparse graphs using LP duality and graph augmentation techniques. It proves that the ratio $\gamma_r(G)/\alpha_{2r}(G)$ is bounded by $\text{wcol}_r^2(G)$, improving prior results and enabling constant-factor approximation algorithms in graphs with bounded expansion or low-degree orientations.

ABSTRACT

Dvorak (2013) gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for the parameters in any class of graphs with bounded expansion. We improve and clarify this dependence using an LP-based argument inspired by the work of Bansal and Umboh (2017).

Motivation & Objective

• To improve the understanding of the relationship between distance $r$-dominating sets and $2r$-independent sets in sparse graphs.
• To refine and clarify the dependence between $\gamma_r(G)$ and $\alpha_{2r}(G)$ using LP-based arguments.
• To establish tighter approximation guarantees for $\gamma_r(G)$ in graph classes with bounded expansion or low-degree orientations.
• To generalize prior results on domination and independence in sparse graphs using weak coloring numbers and graph augmentation.

Proposed method

• Leverages LP duality between the $r$-dominating set and $2r$-independent set problems, showing $\alpha^\star_{2r}(G) = \gamma^\star_r(G)$.
• Introduces $r$-augmentation of a graph $G$ to model distance constraints and control neighborhood sizes in the dual LP.
• Applies a transformation to convert $2r$-independent sets into $(2r,b)$-independent sets via vertex orientation and degree control.
• Uses a coloring argument on the induced subgraph of a $(2r,b)$-independent set to extract a $2r$-independent subset of size at least $|Y|/(2b\Delta_{2r}(\widehat{G}))$.
• Establishes a chain of inequalities linking $\gamma_r(G)$, $\alpha_{2r}(G)$, and $\alpha_{2r,b}(G)$ via weak coloring numbers and degree parameters.
• Applies the framework to derive bounds in terms of weak coloring numbers $\text{wcol}_k(G)$, generalizing results from bounded expansion and low-arboricity graphs.

Experimental results

Research questions

• RQ1 How can the ratio $\gamma_r(G)/\alpha_{2r}(G)$ be bounded in terms of sparsity parameters like weak coloring numbers?
• RQ2 Can LP-based techniques improve upon existing approximation factors for distance $r$-dominating sets in sparse graphs?
• RQ3 What is the relationship between $2r$-independent sets and $(2r,b)$-independent sets under graph augmentation?
• RQ4 How do weak coloring numbers and orientation-based parameters influence the approximation of $\gamma_r(G)$?
• RQ5 Can the bound $\gamma_r(G) \leq \text{wcol}_{2r}^2(G) \cdot \alpha_{2r}(G)$ be tightened or generalized using structural graph parameters?

Key findings

• The paper establishes $\gamma_r(G) \leq \text{wcol}_r^2(G) \cdot \alpha_{2r}(G)$, improving on Dvořák's earlier bound.
• It proves $\alpha^\star_{2r}(G) = \gamma^\star_r(G)$, enabling exact computation of the LP relaxation in polynomial time.
• For any graph $G$ with an orientation of maximum indegree $d$, the bound $\frac{1}{2d+1}\gamma(G) \leq \alpha^\star_2(G)$ holds, generalizing known results.
• The paper shows $\alpha_{2r,b}(G) \geq \alpha^\star_{2r}(G)/2$ for $b = (\Delta_{r-1}(\widehat{G})+1)\Delta_r(\widehat{G}) - \Delta_{r-1}(\widehat{G})$, enabling constant-factor approximation.
• It derives $\alpha_{2r}(G) \geq \frac{1}{2b\Delta_{2r}(\widehat{G})} \alpha_{2r,b}(G)$, linking $2r$-independent sets to $(2r,b)$-independent sets.
• The final chain of inequalities yields $\gamma_r(G) \leq 4\text{wcol}_r^2(G)\text{wcol}_{2r}(G) \cdot \alpha_{2r}(G)$, providing a tight approximation factor in bounded expansion classes.