Approximations for Fault-Tolerant Total and Partial Positive Influence Domination

TL;DR

The paper studies fault-tolerant total domination and weighted partial positive influence domination on graphs, introducing a general greedy framework that extends submodular techniques to non-submodular, fractional-valued functions. It derives the first $1+\ln(\Delta+m-1)$-approximation for fault-tolerant total domination and provides the first logarithmic approximations for simple, total, and connected PPIDS variants with rational edge weights, including extensions to degree percentage constraints. Central to the approach is building suitable submodular-like potential functions (and, when needed, ε-approximate or conditional submodularity) and proving that greedy selection achieves logarithmic guarantees quantified by δ-parameters derived from the function values. The framework unifies integer- and fractional-valued cases and extends to connectivity constraints, offering a versatile tool for designing resilient domination schemes in networks and diffusion models. The results have practical impact on network design and influence diffusion, enabling provable guarantees in settings with partial fault tolerance and weighted interactions.

Abstract

In $\textit{total domination}$, given a graph $G=(V,E)$, we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ has at least one neighbor in $S$. We define a $\textit{fault-tolerant}$ version of total domination, where we require any node in $V \setminus S$ to have at least $m$ neighbors in $S$. Let $Δ$ denote the maximum degree in $G$. We prove a first $1 + \ln(Δ+ m - 1)$ approximation for fault-tolerant total domination. We also consider fault-tolerant variants of the weighted $\textit{partial positive influence dominating set}$ problem, where we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ is either a member of $S$ or the sum of weights of its incident edges leading to nodes in $S$ is at least half of the sum of weights over all its incident edges. We prove the first logarithmic approximations for the simple, total, and connected variants of this problem. To prove the result for the connected case, we extend the general approximation framework for non-submodular functions from integer-valued to fractional-valued functions, which we believe is of independent interest.

Figures (1)

• Figure 1: An illustration of the "majority illusion" paradox for an edge-weighted network. The two networks are identical, except for the three initially active (circular) nodes. Initially, all other nodes are considered inactive. In the network on the right, all initially inactive nodes satisfy inequality \ref{['eq:majority']} for $\theta$ equal to 0.5, and therefore become active (square nodes). In the network on the left, however, there are inactive nodes not satisfying this threshold (triangular nodes), therefore they remain inactive.

Theorems & Definitions (68)

• definition thmcounterdefinition
• definition thmcounterdefinition: $\varepsilon$-approximately submodular
• corollary thmcountercorollary
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• proposition thmcounterproposition
• proof
• theorem thmcountertheorem