Distance Vector Domination
Gennaro Cordasco, Luisa Garagano, Adele A. Rescigno
TL;DR
The paper introduces Distance Vector Domination (DVD), a unifying framework that generalizes DS, VD, RD, and DD by enforcing per-vertex demands $t_v$ within a radius $d_v$, yielding a minimum-size immunizing set to curb misinformation diffusion. It provides a mixed picture of tractability: a negative W[1]-hardness result for the neighborhood-diversity family (and hence for modular-width/clique-width), alongside several fixed-parameter tractable algorithms parameterized by modular-width (for RD, VD with $mw,k$, and DVD with $mw,k$) and by treewidth (VD with $tw,\tau$ and RD with $tw,\delta$). The work employs modular- and tree-decomposition techniques to design DP-based FPT algorithms and derives concrete running times such as $O(mw\cdot 2^{mw}\cdot n)$, $O(mw\cdot k(k+1)^{mw}\cdot n^2)$, $O(mw^2\cdot k(k+1)^{2mw}\cdot n^2)$, $O(tw^2 2^{tw} (\tau+1)^{tw} n)$, and $O(tw (2\delta+1)^{tw} (n+tw^2) n^2 \log n)$. These results contribute both negative hardness insights and practical FPT algorithms for identifying robust immunizing sets in networks to mitigate fake information spread with applications in social and sensor networks.
Abstract
Identifying and mitigating the spread of fake information is a challenging issue that has become dominant with the rise of social media. We consider a generalization of the Domination problem that can be used to detect a set of individuals who, once immunized, can prevent the spreading of fake narratives. The considered problem, named {\em Distance Vector Domination} generalizes both distance and multiple domination, at individual (i.e., vertex) level. We study the parameterized complexity of the problem according to several standard and structural parameters. We prove the W[1]-hardness of the problem with respect to neighborhood diversity, even when all the distances are $1$. We also give fixed-parameter algorithms for some variants of the problem and parameter combinations.