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RE-completeness of entangled constraint satisfaction problems

Eric Culf, Kieran Mastel

TL;DR

This work shows that a broad class of NP-complete CSP languages becomes RE-complete when cast as entangled, succinct nonlocal games, implying undecidability for their non-succinct forms. It achieves this by developing the weighted algebra framework, crafting commutativity gadgets to simulate empty constraints, and proving a strengthened subdivision lemma that preserves constant soundness without parallel repetition. The main theorem covers boolean non-TVF CSPs and graph 3-coloring, with extensions to 2-CSP(k) and related CSP variants, revealing deep quantum hardness for classic CSPs in the MIP* setting. These results advance understanding of entangled CSP complexity and raise open questions about TVF CSPs beyond the boolean and 3-coloring cases, as well as imperfect completeness regimes.

Abstract

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem, and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable, or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. Due to the equality of complexity classes MIP$^\ast=$ RE, general succinctly-presented entangled CSPs are RE-complete. In this work, we show that a wide range of NP-complete CSPs become RE-complete in this setting, including all boolean CSPs, such as 3SAT, as well as $3$-colouring. This also implies that these CSP languages remain undecidable even when not succinctly presented. To show this, we work in the weighted algebra framework introduced by Mastel and Slofstra, where synchronous strategies for a nonlocal game are represented by tracial states on an algebra. Along the way, we improve the subdivision technique in order to be able to separate constraints in the CSP while preserving constant soundness, construct commutativity gadgets for all boolean CSPs, and show a variety of relations between the different ways of presenting CSPs as games.

RE-completeness of entangled constraint satisfaction problems

TL;DR

This work shows that a broad class of NP-complete CSP languages becomes RE-complete when cast as entangled, succinct nonlocal games, implying undecidability for their non-succinct forms. It achieves this by developing the weighted algebra framework, crafting commutativity gadgets to simulate empty constraints, and proving a strengthened subdivision lemma that preserves constant soundness without parallel repetition. The main theorem covers boolean non-TVF CSPs and graph 3-coloring, with extensions to 2-CSP(k) and related CSP variants, revealing deep quantum hardness for classic CSPs in the MIP* setting. These results advance understanding of entangled CSP complexity and raise open questions about TVF CSPs beyond the boolean and 3-coloring cases, as well as imperfect completeness regimes.

Abstract

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem, and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable, or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. Due to the equality of complexity classes MIP RE, general succinctly-presented entangled CSPs are RE-complete. In this work, we show that a wide range of NP-complete CSPs become RE-complete in this setting, including all boolean CSPs, such as 3SAT, as well as -colouring. This also implies that these CSP languages remain undecidable even when not succinctly presented. To show this, we work in the weighted algebra framework introduced by Mastel and Slofstra, where synchronous strategies for a nonlocal game are represented by tracial states on an algebra. Along the way, we improve the subdivision technique in order to be able to separate constraints in the CSP while preserving constant soundness, construct commutativity gadgets for all boolean CSPs, and show a variety of relations between the different ways of presenting CSPs as games.

Paper Structure

This paper contains 30 sections, 65 theorems, 121 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Proof techniques
  4. Related work and open problems
  5. Outline
  6. Acknowledgements
  7. Preliminaries and notation
  8. General notation
  9. Complexity theory
  10. Weighted algebra formalism
  11. Von Neumann algebras and stability
  12. Constraint system algebras
  13. Definitions
  14. Relations between CS algebras
  15. Constraint system languages
  16. ...and 15 more sections

Key Result

Theorem 2.1

Suppose $\texttt{G}$ is a synchronous game with a $C$-diagonally dominant question distribution. If $\mathfrak{w}_q(\texttt{G})$ (resp. $\mathfrak{w}_{qc}(\texttt{G})$) is $\geq 1-\varepsilon$, then $\mathfrak{w}_q^s(\texttt{G})$ (resp. $\mathfrak{w}_{qc}^s(\texttt{G})$) is $\geq 1 - O((\varepsilon/

Figures (7)

  • Figure 1: Transitions in complexity based on soundness parameter for entangled $3$-colouring
  • Figure 2: $C$-homomorphisms (solid arrows) and trace-dependent mappings (dashed arrows) between the weighted algebras considered, for a $k$-ary CS $S=(X,\{(V_i,C_i)\}_{i=1}^m)$. Here $\pi'(i)=\sum_j\pi(i,j)$, $L=\max_i|V_i|$, $P=\max_{i,j.\;V_i\cap V_j\neq\varnothing}\frac{\pi'(i)}{\pi(i,j)}$, and $B(S)$ is the BCS defined in \ref{['def:bs']}.
  • Figure 3: The basic commutativity gadget for TVF boolean constraint systems. Exactly one variable in each triangle must be assigned value 1. These constraints bound the commutator $[x,y]$, and any assignment to $x$ and $y$ may be extended to an assignment to all three constraints.
  • Figure 4: Basic commutativity gadgets with one, two, or three variables per constraint negated. The white vertices indicate the negated variables: note that negated variables must be only connected amongst themselves to construct the gadget.
  • Figure 5: The compressible cycle TVF graph from \ref{['lem:compressible']}.\ref{['lem:compressible-iii']}.
  • ...and 2 more figures

Theorems & Definitions (145)

  • Theorem 2.1: marrakchi2023synchronous
  • Theorem 2.2
  • Lemma 2.3: MS24
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: MS24
  • Lemma 2.7: chapman2023efficiently
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 135 more