The Complexity of Resilience for Digraph Queries
Manuel Bodirsky, Žaneta Semanišinová
TL;DR
The paper establishes a complete dichotomy for the resilience problem of unions of conjunctive digraph queries under bag semantics. It models resilience as a VCSP via a dual structure $\Delta_\mu$ and uses algebraic tools (pp-constructions, fractional polymorphisms) to certify tractability or hardness: tractability arises from a canonical pseudo cyclic fractional polymorphism, while hardness follows from pp-constructing the Boolean OIT template $(\{0,1\}; \mathrm{OIT})$. The authors prove disjointness of the two cases and provide detailed hardness proofs for directed-cycle oriented and finite-acyclic-dual scenarios, as well as reductions based on self-join variations. The results advance a broader program to classify resilience in bag semantics using VCSP techniques and model-complete cores, with implications for query evaluation and reverse data management.
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 $μ$ 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 $G \models \neg μ$? 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 $μ$ there exists a countably infinite ('dual') valued structure $Δ_μ$ which either primitively positively constructs 1-in-3-3-SAT, and hence the resilience problem for $μ$ is NP-complete by general principles, or has a pseudo cyclic canonical fractional polymorphism, and the resilience problem for $μ$ is in P.