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Polynomial and analytic methods for classifying complexity of planar graph homomorphisms

Jin-Yi Cai, Ashwin Maran

TL;DR

This work develops polynomial and analytic tools to classify the complexity of planar graph homomorphisms GH(M) for nonnegative real M, culminating in a complete three-way dichotomy for all full-rank 4×4 matrices. The authors couple eigenvalue-lattice analysis with confluence-based polynomial obstructions and analytic perturbations (stretching, thickening, and diagonal adjustments) to isolate the tensor-product forms defined by tensoring matchgates, which yield planar P-time solvable cases via FKT. For all other nonnegative 4×4 matrices, they prove #P-hardness by constructing reducts and gadgets that transfer Potts-model/Tutte-polynomial hardness to Pl-GH(M), and by a detailed Form(I)/(III)/(IV)/(VI) case analysis. The results establish that Valiant’s holographic framework is universal for planar tractability in this setting, and they provide a rigorous, constructive pathway to identify tractable versus intractable instances with potential implications for Planar counting CSPs and quantum isomorphism-related questions.

Abstract

We introduce some polynomial and analytic methods in the classification program for the complexity of planar graph homomorphisms. These methods allow us to handle infinitely many lattice conditions and isolate the new P-time tractable matrices represented by tensor products of matchgates. We use these methods to prove a complexity dichotomy for $4 \times 4$ matrices that says Valiant's holographic algorithm is universal for planar tractability in this setting.

Polynomial and analytic methods for classifying complexity of planar graph homomorphisms

TL;DR

This work develops polynomial and analytic tools to classify the complexity of planar graph homomorphisms GH(M) for nonnegative real M, culminating in a complete three-way dichotomy for all full-rank 4×4 matrices. The authors couple eigenvalue-lattice analysis with confluence-based polynomial obstructions and analytic perturbations (stretching, thickening, and diagonal adjustments) to isolate the tensor-product forms defined by tensoring matchgates, which yield planar P-time solvable cases via FKT. For all other nonnegative 4×4 matrices, they prove #P-hardness by constructing reducts and gadgets that transfer Potts-model/Tutte-polynomial hardness to Pl-GH(M), and by a detailed Form(I)/(III)/(IV)/(VI) case analysis. The results establish that Valiant’s holographic framework is universal for planar tractability in this setting, and they provide a rigorous, constructive pathway to identify tractable versus intractable instances with potential implications for Planar counting CSPs and quantum isomorphism-related questions.

Abstract

We introduce some polynomial and analytic methods in the classification program for the complexity of planar graph homomorphisms. These methods allow us to handle infinitely many lattice conditions and isolate the new P-time tractable matrices represented by tensor products of matchgates. We use these methods to prove a complexity dichotomy for matrices that says Valiant's holographic algorithm is universal for planar tractability in this setting.

Paper Structure

This paper contains 18 sections, 90 theorems, 461 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model of Computation
  4. Definitions
  5. Reduction from the Potts Model
  6. Lattice on Eigenvalues
  7. Proof of Hardness
  8. Lattice on Diagonal Entries
  9. Confluence and Pairwise Order Independence
  10. Diagonal and Pairwise Order Distinctness
  11. Hardness for Diagonal Distinct $4 \times 4$ matrices
  12. Hardness for non-Diagonal Distinct $4 \times 4$ matrices
  13. Forms (I), (III), (IV) and (VI)
  14. Form (I)
  15. Form (IV)
  16. ...and 3 more sections

Key Result

Lemma 1

For any fixed $M \in \mathbb{R}^{q \times q}$, $\mathop{\mathrm{\tt{Pl-GH}}}\nolimits(M)\equiv \text{\sf COUNT}(M)$.

Figures (7)

  • Figure 1: A graph $G$, the thickened graph $T_{3}G$, and the stretched graph $S_{2}G$.
  • Figure 2: The edge gadget $R_{n}$
  • Figure 3: Form (I)
  • Figure 3: The edge gadget $B$
  • Figure 4: Form (IV)
  • ...and 2 more figures

Theorems & Definitions (212)

  • Lemma 1
  • Definition 2
  • Remark
  • Lemma 3
  • proof
  • Remark
  • Lemma 4
  • proof
  • Remark
  • Definition 5
  • ...and 202 more