The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms

Chen, Hubie; Curticapean, Radu; Dell, Holger

doi:10.1007/978-3-030-30786-8_28

“…The observant reader might notice that the 4-vertex path has a non-trivial involution, and thus, we cannot further reduce ⊕Ret(H) to ⊕Hom(H) in this case. 3 However, the construction works for any graph H with an induced path (o, s, i, x) such that s and i each only have two neighbours. The notion of a hardness gadget, which we formally introduce in Section 4, is essentially a generalization of the previous construction.…”

Section: Contributions and Techniquesmentioning

confidence: 99%

“…Let H be a K 4 -minor-free graph that contains F as a subgraph. We say that F has type V in H if one of the following is true (1,5) and Γ H\F (3,5) are non-empty and Γ H\F (2,4) and Γ H\F (2,6) are empty.…”

Section: Chordal Bipartite Componentsmentioning

confidence: 99%

“…If Γ H\F (1,5) and Γ H\F (2,4) are both non-empty, then the vertices v 1 , v 2 , v 4 and v 5 yield a K 4 -minor. If Γ H\F (2,6) and Γ H\F (3,5) are both non-empty, then the vertices v 2 , v 3 , v 5 and v 6 yield a K 4 -minor.…”

Section: Chordal Bipartite Componentsmentioning

confidence: 99%

“…Applications of this problem are discussed in [1]. The complexity of the problem has been the focus of much research (see, for example, [3,9,15,16,29]). 1 The complexity of counting homomorphisms was initiated by Dyer and Greenhill [9], who gave a complete dichotomy theorem.…”

Section: Introductionmentioning

confidence: 99%

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Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2

Focke,

Goldberg,

Roth

et al. 2020

Preprint

0

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We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph H and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class ⊕P of parity problems.We verify their conjecture for all graphs H that exclude the complete graph on 4 vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the ⊕P-complete cases, assuming the randomised Exponential Time Hypothesis.Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph H. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Göbel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2.

show abstract

“…Applications of this problem are discussed in [1]. The complexity of the problem has been the focus of much research (see, for example, [3,9,16,17,30]). 1 The complexity of counting homomorphisms was initiated by Dyer and Greenhill [9], who gave a complete dichotomy theorem.…”

Section: Introductionmentioning

confidence: 99%

Counting Homomorphisms to $K_4$-Minor-Free Graphs, Modulo 2

Focke¹,

Goldberg²,

Roth³

et al. 2021

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph H and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class ⊕P of parity problems.We verify their conjecture for all graphs H that exclude the complete graph on 4 vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the ⊕P-complete cases, assuming the randomised Exponential Time Hypothesis.Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph H. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Göbel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2.

show abstract

Exponential Time Complexity of the Complex Weighted Boolean #CSP

Liu

2023

Lecture Notes in Computer Science

0

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The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms

Cited by 7 publications

References 27 publications

Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2

Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2

Counting Homomorphisms to $K_4$-Minor-Free Graphs, Modulo 2

Exponential Time Complexity of the Complex Weighted Boolean #CSP

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