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Tight Inapproximability of Target Set Reconfiguration

Naoto Ohsaka

TL;DR

Minmax Target Set Reconfiguration studies reconfiguring target sets under irreversible activation with thresholds on a graph, aiming to minimize the maximum size of any intermediate target set. The paper proves NP-hard to approximate within $2 - o\left(\frac{1}{\operatorname{polylog} n}\right)$ and shows a simple 2-approximation, yielding a tight bound. The key method is a gap-preserving reduction from Target Set Selection that constructs a graph H with target sets X and Y of size $\ell$ using one-way gadgets, such that completeness and soundness map to the reconfiguration problem. This establishes a strong link between Target Set Selection hardness and reconfiguration and advances understanding of approximation limits in influence spread models.

Abstract

Given a graph $G$ with a vertex threshold function $τ$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $τ(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).

Tight Inapproximability of Target Set Reconfiguration

TL;DR

Minmax Target Set Reconfiguration studies reconfiguring target sets under irreversible activation with thresholds on a graph, aiming to minimize the maximum size of any intermediate target set. The paper proves NP-hard to approximate within and shows a simple 2-approximation, yielding a tight bound. The key method is a gap-preserving reduction from Target Set Selection that constructs a graph H with target sets X and Y of size using one-way gadgets, such that completeness and soundness map to the reconfiguration problem. This establishes a strong link between Target Set Selection hardness and reconfiguration and advances understanding of approximation limits in influence spread models.

Abstract

Given a graph with a vertex threshold function , consider a dynamic process in which any inactive vertex becomes activated whenever at least of its neighbors are activated. A vertex set is called a target set if all vertices of would be activated when initially activating vertices of . In the Minmax Target Set Reconfiguration problem, for a graph and its two target sets and , we wish to transform into by repeatedly adding or removing a single vertex, using only target sets of , so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of , where is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a -factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).
Paper Structure (7 sections, 4 theorems, 12 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background.
  3. Our Result.
  4. Additional Related Work.
  5. Proof of \ref{['thm:main']}
  6. Reduction
  7. Proof of Soundness

Key Result

Theorem 1.1

Minmax Target Set Reconfiguration is NP-hard to approximate within a factor of $2-o\left(\frac{1}{\mathop{\mathrm{\mathrm{polylog}}}\nolimits n}\right)$, where $n$ is the number of vertices.

Figures (1)

  • Figure 1: Construction of $H$ when $n=3$ and $\ell = 3$. One-way gadgets are denoted by .

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • proof : Proof of \ref{['thm:main']}
  • Definition 2.2: one-way gadget kyncl2017irreversiblecharikar2016approximatingbazgan2014parameterizeda
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Claim 2.6
  • proof
  • ...and 5 more