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Ineffectiveness for Search and Undecidability of PCSP Meta-Problems

Alberto Larrauri

TL;DR

This work analyzes the gap between search and decision for promise CSPs (PCSPs) by focusing on affine integer programming (AIP), basic linear programming (BLP), and their combination. It shows that rounding the relaxations produced by these algorithms to obtain a search certificate is TFNP-hard, and that natural meta-problems about these methods are undecidable, even for small templates. The authors develop a rich algebraic framework based on minions, interpretations, and minor conditions to carry out tiling-based reductions that transfer hardness, undecidability, and non-computability to sPCSPs. They also connect these results to classical algebraic tractability conditions (cyclic polymorphisms and weak near-unanimity) and demonstrate undecidability for the corresponding meta-questions in PCSPs. Overall, the paper highlights intrinsic obstacles to obtaining efficient search procedures from decision algorithms in the PCSP regime and outlines nuanced directions for future work on meta-problems and alternative relaxations.

Abstract

It is an open question whether the search and decision versions of promise CSPs are equivalent. Most known algorithms for PCSPs solve only their \emph{decision} variant, and it is unknown whether they can be adapted to solve \emph{search} as well. The main approaches, called BLP, AIP and BLP+AIP, handle a PCSP by finding a solution to a relaxation of some integer program. We prove that rounding those solutions to a proper search certificate can be as hard as any problem in the class TFNP. In other words, these algorithms are ineffective for search. Building on the algebraic approach to PCSPs, we find sufficient conditions that imply ineffectiveness for search. Our tools are tailored to algorithms that are characterized by minions in a suitable way, and can also be used to prove undecidability results for meta-problems. This way, we show that the families of templates solvable via BLP, AIP, and BLP+AIP are undecidable. Using the same techniques we also analyze several algebraic conditions that are known to guarantee the tractability of finite-template CSPs. We prove that several meta-problems related to cyclic polymorphims and WNUs are undecidable for PCSPs. In particular, there is no algorithm deciding whether a finite PCSP template (1) admits cyclic a polymorphism, (2) admits a WNU.

Ineffectiveness for Search and Undecidability of PCSP Meta-Problems

TL;DR

This work analyzes the gap between search and decision for promise CSPs (PCSPs) by focusing on affine integer programming (AIP), basic linear programming (BLP), and their combination. It shows that rounding the relaxations produced by these algorithms to obtain a search certificate is TFNP-hard, and that natural meta-problems about these methods are undecidable, even for small templates. The authors develop a rich algebraic framework based on minions, interpretations, and minor conditions to carry out tiling-based reductions that transfer hardness, undecidability, and non-computability to sPCSPs. They also connect these results to classical algebraic tractability conditions (cyclic polymorphisms and weak near-unanimity) and demonstrate undecidability for the corresponding meta-questions in PCSPs. Overall, the paper highlights intrinsic obstacles to obtaining efficient search procedures from decision algorithms in the PCSP regime and outlines nuanced directions for future work on meta-problems and alternative relaxations.

Abstract

It is an open question whether the search and decision versions of promise CSPs are equivalent. Most known algorithms for PCSPs solve only their \emph{decision} variant, and it is unknown whether they can be adapted to solve \emph{search} as well. The main approaches, called BLP, AIP and BLP+AIP, handle a PCSP by finding a solution to a relaxation of some integer program. We prove that rounding those solutions to a proper search certificate can be as hard as any problem in the class TFNP. In other words, these algorithms are ineffective for search. Building on the algebraic approach to PCSPs, we find sufficient conditions that imply ineffectiveness for search. Our tools are tailored to algorithms that are characterized by minions in a suitable way, and can also be used to prove undecidability results for meta-problems. This way, we show that the families of templates solvable via BLP, AIP, and BLP+AIP are undecidable. Using the same techniques we also analyze several algebraic conditions that are known to guarantee the tractability of finite-template CSPs. We prove that several meta-problems related to cyclic polymorphims and WNUs are undecidable for PCSPs. In particular, there is no algorithm deciding whether a finite PCSP template (1) admits cyclic a polymorphism, (2) admits a WNU.

Paper Structure

This paper contains 53 sections, 39 theorems, 154 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Search Problems
  4. Promise Constraint Satisfaction
  5. Efficient Algorithms and Rounding Problems
  6. Minions
  7. Minor Conditions
  8. Main Results
  9. Technical Overview
  10. Organization of the Paper
  11. Main Definitions
  12. Descriptions
  13. Interpretations over Minions
  14. The Exponential Minion
  15. Patterns
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let $\mathcal{Q}\in \{ \text{AIP}, \text{BLP}, \text{BLP}+\text{AIP}\}$, and let $\mathcal{S}^\mathcal{Q}$ be the family of finite templates $({\bm{A}}, {\bm{B}})$ such that $\mathcal{Q}$ solves $\mathop{\mathrm{PCSP}}\nolimits({\bm{A}}, {\bm{B}})$. Then, given any problem $\Lambda$ in the TFNP clas

Figures (2)

  • Figure 1: Main algorithmic results.
  • Figure 2: The quotient $\bm \Gamma/\sim_\mathcal{D}$

Theorems & Definitions (68)

  • Theorem 1.1: Main algorithmic result, informal
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.3
  • Theorem 4.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 58 more