# Disjunctive Queries of Safety in Graphs (Model-Checking Problem)

## Description

Given a model of a system and an objective, the model-checking problem asks whether the model satisfies the objective.

In this case, the model is a standard graph, and the objective is safety: given a set of target vertices $T\subseteq V$, determine whether or not there is a path that does not visit any vertex in $T$ (i.e. you want to avoid all vertices in $T$).

Furthermore, in the disjunctive queries problem, you are given multiple safety objectives (i.e. multiple target sets $T_i$) and you are to determine whether or not there is a path where at least one of the safety objectives is satisfied (i.e. whether or not an infinite path avoids all vertices in at least one of the target sets).

Disjunctive queries coincide with disjunctive objectives on graphs.

## Related Problems

Generalizations: Safety in Graphs

## Parameters

$n$: number of vertices

$m$: number of edges

$MEC$: O(\min(n^2, m^{1.5}))

## Table of Algorithms

Currently no algorithms in our database for the given problem.

## Reductions FROM Problem

Problem Implication Year Citation Reduction
Triangle Detection assume: Strong Triangle
then: there is no combinatorial $O(n^{3-\epsilon})$ or $O((k \cdot n^{2})^{1-\epsilon})$ algorithm, for any $\epsilon > {0}$ for disjunctive safety (objectives or queries) in graphs.
then: there is no $O(m^{2-\epsilon})$ or $O((k \cdot m)^{1 - \epsilon})$ algorithm for any $\epsilon > {0}$ for disjunctive safety ovjectives/queries in MDPs.