Restricted CSPs and F-free Digraph Algorithmics
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866909509414289408 |
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| author | Guzmán-Pro, Santiago Martin, Barnaby |
| author_facet | Guzmán-Pro, Santiago Martin, Barnaby |
| contents | In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Nešetřil. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted CSPs and F-free Digraph Algorithmics Guzmán-Pro, Santiago Martin, Barnaby Computational Complexity Discrete Mathematics Logic in Computer Science Combinatorics In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Nešetřil. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy. |
| title | Restricted CSPs and F-free Digraph Algorithmics |
| topic | Computational Complexity Discrete Mathematics Logic in Computer Science Combinatorics |
| url | https://arxiv.org/abs/2502.17596 |