Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Resilience Problems via Valued Constraint Satisfaction

Manuel Bodirsky, Žaneta Semanišinová, Carsten Lutz

TL;DR

This article studies cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group whose automorphisms form an oligomorphic permutation group and conjecture that the hardness and tractability conditions match for resilience problems for UCQs.

Abstract

Valued constraint satisfaction problems (VCSPs) constitute a large class of computational optimization problems. It was shown recently that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions. In this article, we study cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group. Our results include a hardness condition based on a generalization of pp-constructability as known from classical CSPs and a polynomial-time tractability condition based on the concept of fractional polymorphisms. We then observe that the resilience problem for unions of conjunctive queries (UCQs) studied in database theory, under bag semantics, may be viewed as a special case of the VCSPs that we consider. We obtain a complexity dichotomy for the case of incidence-acyclic UCQs and exemplarily use our methods to determine the complexity of a conjunctive query that has been stated as an open problem in the literature. We conjecture that our hardness and tractability conditions match for resilience problems for UCQs. Further, we obtain a complete dichotomy for resilience problems for two-way regular path queries, under bag semantics.

The Complexity of Resilience Problems via Valued Constraint Satisfaction

TL;DR

This article studies cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group whose automorphisms form an oligomorphic permutation group and conjecture that the hardness and tractability conditions match for resilience problems for UCQs.

Abstract

Valued constraint satisfaction problems (VCSPs) constitute a large class of computational optimization problems. It was shown recently that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions. In this article, we study cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group. Our results include a hardness condition based on a generalization of pp-constructability as known from classical CSPs and a polynomial-time tractability condition based on the concept of fractional polymorphisms. We then observe that the resilience problem for unions of conjunctive queries (UCQs) studied in database theory, under bag semantics, may be viewed as a special case of the VCSPs that we consider. We obtain a complexity dichotomy for the case of incidence-acyclic UCQs and exemplarily use our methods to determine the complexity of a conjunctive query that has been stated as an open problem in the literature. We conjecture that our hardness and tractability conditions match for resilience problems for UCQs. Further, we obtain a complete dichotomy for resilience problems for two-way regular path queries, under bag semantics.
Paper Structure (21 sections, 38 theorems, 70 equations, 7 figures)

This paper contains 21 sections, 38 theorems, 70 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Outline.
  4. Preliminaries
  5. Valued Structures
  6. Valued Constraint Satisfaction
  7. Oligomorphicity
  8. Expressive Power
  9. Hardness from pp-Constructions
  10. Fractional Polymorphisms
  11. Tractability via Canonical Fractional Polymorphisms
  12. Fractional Polymorphims on Finite Domains
  13. Application: Resilience
  14. Connectivity
  15. Finite Duals
  16. ...and 6 more sections

Key Result

Lemma 3.3

Let $\Gamma$ be a valued structure with a countable domain $C$ and an oligomorphic automorphism group. Then for every instance $\phi(x_1,\dots,x_n)$ of $\mathop{\mathrm{VCSP}}\nolimits(\Gamma)$ there exists $a \in C^n$ such that the cost of $\phi$ equals $\phi^{\Gamma}(a)$.

Figures (7)

  • Figure 1: The resilience problem considered in this article.
  • Figure 2: The query $\mu$ from Example \ref{['expl:fin-dual']} (on the left) and the corresponding structure ${\mathfrak B}$ (on the right).
  • Figure 3: The dual structure ${\mathfrak B}$ from Example \ref{['expl:rpq-tract']}.
  • Figure 4: Example \ref{['expl:triad']}, visualization of $\mu$ and $\phi$. The thick edges cannot be removed.
  • Figure 5: Visualization of the query $\mu$ from \ref{['eq:mu1']}.
  • ...and 2 more figures

Theorems & Definitions (102)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Definition 4.1
  • Definition 4.2: valued relational clone
  • ...and 92 more