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Matchgate signatures under variable permutations

Boning Meng, Yicheng Pan

TL;DR

This work studies when matchgate signatures remain valid under permutations of their input variables, introducing the notion of permutable matchgate signatures and giving a polynomial-time checkable condition based on the four-variable identity associated with matchgates ($MGI$). It shows that any nontrivial matchgate can be normalized and represented via star gadgets, and it characterizes permutable signatures by the relation $F(ab)F(cd)=F(ac)F(bd)=F(ad)F(bc)$ for all quadruples, enabling a decomposition into Pinning, Parity-type, and Matching-type classes. The paper then demonstrates how permutable signatures can model symmetric matchgate signatures through planar left-side gadgets, providing a route to erase the gap between Pl-$\#CSP$ and $\#CSP$ with planar embeddings and establishing a dichotomy for Pl-$\#R_D$-$CSP$ with $D\ge 3$. These results connect signatures, reductions, and holographic transformations to yield tractable and hard cases in both planar and nonplanar counting problems, and they open avenues for algebraic approaches to variants of the FKT algorithm. Overall, the work advances understanding of when planarity constraints are bypassable and how to engineer symmetric realizations of permutation-affected matchgates.

Abstract

In this article, we give a sufficient and necessary condition for determining whether a matchgate signature retains its property under a certain variable permutation, which can be checked in polynomial time. We also define the concept of permutable matchgate signatures, and use it to erase the gap between Pl-\#CSP and \#CSP on planar graphs in the previous study. We provide a detailed characterization of permutable matchgate signatures as well, by presenting their relation to symmetric matchgate signatures. In addition, we prove a dichotomy for Pl-$\#R_D$-CSP where $D\ge 3$ is an integer.

Matchgate signatures under variable permutations

TL;DR

This work studies when matchgate signatures remain valid under permutations of their input variables, introducing the notion of permutable matchgate signatures and giving a polynomial-time checkable condition based on the four-variable identity associated with matchgates (). It shows that any nontrivial matchgate can be normalized and represented via star gadgets, and it characterizes permutable signatures by the relation for all quadruples, enabling a decomposition into Pinning, Parity-type, and Matching-type classes. The paper then demonstrates how permutable signatures can model symmetric matchgate signatures through planar left-side gadgets, providing a route to erase the gap between Pl- and with planar embeddings and establishing a dichotomy for Pl-- with . These results connect signatures, reductions, and holographic transformations to yield tractable and hard cases in both planar and nonplanar counting problems, and they open avenues for algebraic approaches to variants of the FKT algorithm. Overall, the work advances understanding of when planarity constraints are bypassable and how to engineer symmetric realizations of permutation-affected matchgates.

Abstract

In this article, we give a sufficient and necessary condition for determining whether a matchgate signature retains its property under a certain variable permutation, which can be checked in polynomial time. We also define the concept of permutable matchgate signatures, and use it to erase the gap between Pl-\#CSP and \#CSP on planar graphs in the previous study. We provide a detailed characterization of permutable matchgate signatures as well, by presenting their relation to symmetric matchgate signatures. In addition, we prove a dichotomy for Pl--CSP where is an integer.

Paper Structure

This paper contains 23 sections, 27 theorems, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counting problems
  4. #PM
  5. #CSP problems
  6. Holant problems
  7. Reduction methods
  8. Constructing gadgets
  9. Holographic Transformation
  10. Matchgate signatures
  11. Dichotomies for #CSP
  12. Previous results for #CSP
  13. Dichotomy for Pl-#$R_D$-CSP
  14. Matchgate signatures under variable permutations
  15. Normalize the matchgate
  16. ...and 8 more sections

Key Result

Lemma 4

Let $\mathcal{C}$ be an arbitrary graph class, $\mathcal{F}$ be an arbitrary signature set and $D\ge 1$ be an integer. Then,

Figures (4)

  • Figure 1: The underlying graph of a $\text{\#CSP}(\mathcal{F})$ instance if $f,g,h\in \mathcal{F}$ and $x,y,z$ are 3 variables; or that of a $\text{Holant}(\mathcal{F}|\mathcal{EQ})$ instance if we treat $x,y,z$ as $=_3,=_2,=_1$ and the edges as the variables instead.
  • Figure 2: A visualization of the pairing $M=\{(1,6),(3,9),(4,7)\}$.
  • Figure 3: (a) The construction of a gadget appears in the Case 2 of Parity type. The vertex of degree 3 represented by a solid circle is assigned $[1,0,1,0]$, while each vertex of degree 2 represented by a hollowed circle is assigned $F$ with the second variable connecting to $[1,0,1,0]$. (b) The construction of the generalized mating gadget.
  • Figure 4: The construction of the generalized mating gadget.

Theorems & Definitions (35)

  • Definition 1
  • Remark 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Theorem 6: Holographic Transformationvaliant2008holographiccai2007valiant
  • Lemma 7
  • Definition 8
  • Lemma 9: cai2013matchgates
  • Theorem 10: MGI
  • ...and 25 more