Target set selection with maximum activation time

07/10/2020
by   Lucas Keiler, et al.
0

A target set selection model is a graph G with a threshold function τ:V→ℕ upper-bounded by the vertex degree. For a given model, a set S_0⊆ V(G) is a target set if V(G) can be partitioned into non-empty subsets S_0,S_1,…,S_t such that, for i ∈{1, …, t}, S_i contains exactly every vertex v having at least τ(v) neighbors in S_0∪…∪ S_i-1. We say that t is the activation time t_τ(S_0) of the target set S_0. The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set S_0 that maximizes t_τ(S_0). That is, given a graph G, a threshold function τ in G, and an integer k, the objective of the TSS-time problem is to decide whether G contains a target set S_0 such that t_τ(S_0)≥ k. Let τ^* = max_v ∈ V(G)τ(v). Our main result is the following dichotomy about the complexity of TSS-time when G belongs to a minor-closed graph class C: if C has bounded local treewidth, the problem is FPT parameterized by k and τ^⋆; otherwise, it is NP-complete even for fixed k=4 and τ^⋆=2. We also prove that, with τ^*=2, the problem is NP-hard in bipartite graphs for fixed k=5, and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set S_0 in a given tree maximizing t_τ(S_0).

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

07/08/2020

Non-monotone target sets for threshold values restricted to 0, 1, and the vertex degree

We consider a non-monotone activation process (X_t)_t∈{ 0,1,2,…} on a gr...
12/27/2017

A Short Note on Parameterized Computation of Network Reliability with respect to Treewidth

We consider the classic problem of network reliability. A network is giv...
05/21/2021

Balancing the Spread of Two Opinions in Sparse Social Networks

Inspired by the famous Target Set Selection problem, we propose a new di...
04/25/2018

Incremental Optimization of Independent Sets under Reachability Constraints

We introduce a new framework for reconfiguration problems, and apply it ...
07/27/2018

Solving Target Set Selection with Bounded Thresholds Faster than 2^n

In this paper we consider the Target Set Selection problem. The problem ...
10/02/2017

Target Set Selection Parameterized by Clique-Width and Maximum Threshold

The Target Set Selection problem takes as an input a graph G and a non-n...
12/04/2018

An Inapproximability Result for the Target Set Selection Problem on Bipartite Graphs

Given an undirected graph G(V, E, τ) modeling a 'social network', where ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.