Parametrized Complexity of Weak Odd Domination Problems

TL;DR

This paper investigates the computational complexity of weak odd domination problems in graphs, focusing on their parameterized intractability and approximation limits, with implications for quantum cryptography.

Contribution

It establishes the fixed-parameter intractability and approximation bounds for key graph parameters related to weak odd domination, advancing understanding of their computational hardness.

Findings

Proves W[1]-hardness for the problems involving ppa, ppa', and ppa_Q.

Shows constant factor approximation algorithms exist for these parameters.

Demonstrates no polynomial approximation scheme for ppa and ppa' unless P=NP.

Abstract

Given a graph $G=(V,E)$, a subset $B\subseteq V$ of vertices is a weak odd dominated (WOD) set if there exists $D \subseteq V {\setminus} B$ such that every vertex in $B$ has an odd number of neighbours in $D$. $\kappa(G)$ denotes the size of the largest WOD set, and $\kappa'(G)$ the size of the smallest non-WOD set. The maximum of $\kappa(G)$ and $|V|-\kappa'(G)$, denoted $\kappa_Q(G)$, plays a crucial role in quantum cryptography. In particular deciding, given a graph $G$ and $k>0$, whether $\kappa_Q(G)\le k$ is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems associated with the quantities $\kappa$, $\kappa'$ and $\kappa_Q$ are known to be NP-Complete. In this paper, we consider the approximation of these quantities and the parameterized complexity of the corresponding problems. We mainly prove the fixed-parameter intractability (W$[1]$-hardness) of these problems. Regarding the approximation, we show that $\kappa_Q$, $\kappa$ and $\kappa'$ admit a constant factor approximation algorithm, and that $\kappa$ and $\kappa'$ have no polynomial approximation scheme unless P=NP.