The Complexity of Resilience for Digraph Queries

TL;DR

This paper establishes a clear complexity dichotomy for the resilience problem in directed graph queries, proving it is either efficiently solvable or NP-complete depending on the query structure.

Contribution

It verifies a conjecture by classifying the complexity of resilience problems for all unions of conjunctive digraph queries, introducing a dichotomy based on algebraic properties.

Findings

Resilience problem is in P or NP-complete for all unions of conjunctive digraph queries.

Existence of a 'dual' valued structure determines the complexity class.

The paper confirms a conjecture for directed graph resilience problems.

Abstract

We prove a complexity dichotomy for the resilience problem for unions of conjunctive digraph queries (i.e., for existential positive sentences over the signature $\{R\}$ of directed graphs). Specifically, for every union $\mu$ of conjunctive digraph queries, the following problem is in P or NP-complete: given a directed multigraph $G$ and a natural number $u$, can we remove $u$ edges from $G$ so that $G \models \neg \mu$? In fact, we verify a more general dichotomy conjecture from (Bodirsky et al., 2024) for all resilience problems in the special case of directed graphs, and show that for such unions of queries $\mu$ there exists a countably infinite ('dual') valued structure $\Delta_\mu$ which either primitively positively constructs 1-in-3-3-SAT, and hence the resilience problem for $\mu$ is NP-complete by general principles, or has a pseudo cyclic canonical fractional polymorphism, and the resilience problem for $\mu$ is in P.