Stable Approximation Algorithms for Dominating Set and Independent Set

TL;DR

This work analyzes stable dynamic algorithms for Dominating Set and Independent Set in the vertex-arrival model, introducing the $k$-stable $\rho$-approximation framework that bounds per-update changes. It establishes strong impossibility results (no SAS) for both problems in tightly-constrained degree regimes, using expander-based lower bounds, and complements these with a set of constant-stability algorithms whose performance scales with arrival or average degree. The authors also connect stable approximation schemes to PTAS techniques via local-search methods on graph classes with sublinear separators, enabling SAS results in specific graph families. They further extend the Independent Set results to a fully dynamic model, achieving a 6-stable $O(d)$-approximation, illustrating how stability constraints shape algorithmic trade-offs in dynamic graphs and offering guidance for recourse-aware design in practice.

Abstract

We study the Dominating set problem and Independent Set Problem for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is $k$-stable when it makes at most $k$ changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter $k$ of the algorithm and the approximation ratio it achieves. We obtain the following results. 1. We show that there is a constant $\varepsilon^*>0$ such that any dynamic $(1+\varepsilon^*)$-approximation algorithm the for Dominating set problem has stability parameter $Ω(n)$, even for bipartite graphs of maximum degree 4. 2. We present algorithms with very small stability parameters for the Dominating set problem in the setting where the arrival degree of each vertex is upper bounded by $d$. In particular, we give a $1$-stable $(d+1)^2$-approximation algorithm, a $3$-stable $(9d/2)$-approximation algorithm, and an $O(d)$-stable $O(1)$-approximation algorithm. 3. We show that there is a constant $\varepsilon^*>0$ such that any dynamic $(1+\varepsilon^*)$-approximation algorithm for the Independent Set Problem has stability parameter $Ω(n)$, even for bipartite graphs of maximum degree $3$. 4. Finally, we present a $2$-stable $O(d)$-approximation algorithm for the Independent Set Problem, in the setting where the average degree of the graph is upper bounded by some constant $d$ at all times. We extend this latter algorithm to the fully dynamic model where vertices can also be deleted, achieving a $6$-stable $O(d)$-approximation algorithm.

Figures (2)

• Figure 1: The lower-bound construction for Dominating Set.
• Figure 2: Example showing that Directed-DomSet has approximation ratio $\Omega(d^2)$.

Theorems & Definitions (26)

• Theorem 2.1
• Corollary 2.2
• Proposition 3.1
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Lemma 3.6
• Lemma 3.7
• Theorem 3.8
• Lemma 3.9