GMSNP and Finite Structures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911926568615936 |
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| author | Guzmán-Pro, Santiago |
| author_facet | Guzmán-Pro, Santiago |
| contents | Given an (infinite) relational structure $\mathbb S$, we say that a finite structure $\mathbb C$ is a minimal finite factor of $\mathbb S$ if for every finite structure $\mathbb A$ there is a homomorphism $\mathbb S\to \mathbb A$ if and only if there is a homomorphism $\mathbb{C} \to \mathbb{A}$. In this brief note we prove that if CSP($\mathbb S$) is in GMSNP, then $\mathbb S$ has a minimal finite factor $\mathbb C$, and moreover, CSP($\mathbb C$) reduces in polynomial time to CSP($\mathbb S$). We discuss two nice applications of this result. First, we see that if a finite promise constraint satisfaction problem PCSP($\mathbb A,\mathbb B$) has a tractable GMSNP sandwich, then it has a tractable finite sandwich. We also show that if $\mathbb G$ is a non-bipartite (possibly infinite) graph with finite chromatic number, and CSP($\mathbb G$) is in GMSNP, then CSP($\mathbb G$) in NP-complete, partially answering a question recently asked by Bodirsky and Guzmán-Pro. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | GMSNP and Finite Structures Guzmán-Pro, Santiago Discrete Mathematics Combinatorics Logic 05C15, 03B70, 05C63 F.4.2; G.2.2 Given an (infinite) relational structure $\mathbb S$, we say that a finite structure $\mathbb C$ is a minimal finite factor of $\mathbb S$ if for every finite structure $\mathbb A$ there is a homomorphism $\mathbb S\to \mathbb A$ if and only if there is a homomorphism $\mathbb{C} \to \mathbb{A}$. In this brief note we prove that if CSP($\mathbb S$) is in GMSNP, then $\mathbb S$ has a minimal finite factor $\mathbb C$, and moreover, CSP($\mathbb C$) reduces in polynomial time to CSP($\mathbb S$). We discuss two nice applications of this result. First, we see that if a finite promise constraint satisfaction problem PCSP($\mathbb A,\mathbb B$) has a tractable GMSNP sandwich, then it has a tractable finite sandwich. We also show that if $\mathbb G$ is a non-bipartite (possibly infinite) graph with finite chromatic number, and CSP($\mathbb G$) is in GMSNP, then CSP($\mathbb G$) in NP-complete, partially answering a question recently asked by Bodirsky and Guzmán-Pro. |
| title | GMSNP and Finite Structures |
| topic | Discrete Mathematics Combinatorics Logic 05C15, 03B70, 05C63 F.4.2; G.2.2 |
| url | https://arxiv.org/abs/2406.13529 |