On the Complexity of Target Set Selection in Simple Geometric Networks

TL;DR

Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, this work restricts itself to networks that can be represented as geometric intersection graphs, such as interval graphs and grid graphs.

Abstract

We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.

Figures (8)

• Figure 1: Overview of our main results regarding Target Set Selection for unanimous, constant, majority, and unrestricted threshold function. Red squares correspond to NP-hardness, green correspond to polynomial-time solvability and yellow indicate an open question. Squares with borders correspond to results established in this work. Black squares indicate a main result while the gray indicate a direct or trivial corollary to a previously known result. An arrow from a class $\mathcal{A}$ to class $\mathcal{B}$ corresponds to the fact that $\mathcal{A}$ is a subclass of $\mathcal{B}$. The six classes in the picture are (by row) disk graphs, planar graphs, unit disk graphs, interval graphs, grid graphs, unit interval graphs.
• Figure 2: An example subdivision of the edge $e=\{u,v\}$ in the case of $4$-regular graphs. In this case, $q_{e}=1$ and $X_e=\{x_1^e,x_2^e,x_3^e,x_4^e,x_5^e,x_6^e,{x_2^e}',{x_2^e}",{x_5^e}',{x_5^e}"\}$. The half-edges going from $u$ and $v$ symbolize the rest of the graph. The filled vertices showcase the situation from Case 1 in proof of \ref{['lem:is:npc:onedir']}, as $u\notin I_{\ell - 1}$, we add $x_1^e$, $x_3^e$ and $x_5^e$ to $I_\ell$ and the set $I_\ell$ remains independent.
• Figure 3: Example of a construction of the unit disk representation of the graph $G'$ from the proof of \ref{['thm:is:npc:regular']} for $r=3$. On the left, the original $3$-regular graph $G$ is embedded into a grid (the half-edges represent the rest of the graph). On the right, the subdivision was made. The red disks correspond to the original vertices of the graph and blue disks correspond to the internal grid points contained in the polygonal chains representing the edges. These are the disks $D_2,\ldots,D_{g-1}$ for the corresponding edges. Consider the edge $e=\{a_1,a_4\}$. The red disks at $a_1$ and $a_4$ are the disks $D_1$ and $D_4$, respectively, and the blue disks at $b_1$ and $b_2$ are the disks $D_2$ and $D_3$, respectively. The empty disks correspond to the disks $E_j$ from \ref{['lem:is:npc:construction_technical_lemma1']}. The numbers next to the edges correspond to the values $w_i$ from \ref{['lem:is:npc:construction_technical_lemma2']} (and are equal to the number of empty disks $E_j$ between (filled) red and blue disks). For the edge $e$, we have $g=4$ grid points contained in the polygonal chain $\mathcal{E}(e)$, thus we are in the case $g=4 \mod 6$ from \ref{['lem:is:npc:construction_technical_lemma2']} thus $w_1=w_2=8$ and $w_3=6$ for this particular edge. The total number of disks on the subdivided edge $e$ is thus $2+8+8+6=24=0\mod 6$ and thus $q_e=4$.
• Figure 4: Schematic representation of the variable gadget $\operatorname{VG}(x)$ for variable $x$. The filled vertices have threshold $2$, while the white vertices have threshold $1$. Note also that the half-edges illustrate the fact that the gadget is connected with the rest of the graph only via $t_x^1,t_x^2$ and $f_x$.
• Figure 5: Transformation of a planar graph with maximum degree $3$ into a subgraph of a grid by subdividing edges at the internal points of the polygonal chains. Filled vertices correspond to the vertices of the original graph and the white ones are the newly created vertices.

Theorems & Definitions (52)

• Definition 1: Unit disk graph
• Definition 2: Grid graph
• Lemma 3
• proof
• Lemma 4
• Definition 5
• Theorem 6: Valiant1981
• Theorem 7
• Claim 8
• proof