NP-hardness of Harm Minimization for exponents greater than one

Establish NP-hardness of the Harm Minimization problem for every exponent p>1, extending the paper’s result that minimizing the vulnerability-weighted linear shortfall objective H_1(A) is NP-hard.

Background

The paper defines Harm Minimization by minimizing H_p(A)=∑_{i∈V} w_i(s_i(A))^p over seed sets A of size at most k, where s_i(A) is the terminal awareness shortfall of node i, w_i is its vulnerability weight, and p≥1 controls the penalty assigned to severe shortfalls.

The authors prove NP-hardness only for the case p=1 by reducing Influence Maximization under the Independent Cascade model to Harm Minimization. They explicitly note that pointwise monotonicity of z↦z^p does not preserve the ordering of seed sets, so the p=1 reduction does not immediately establish hardness for p>1.