Asymptotically Smaller Encodings for Graph Problems and Scheduling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912435467714560 |
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| author | Subercaseaux, Bernardo |
| author_facet | Subercaseaux, Bernardo |
| contents | We show how several graph problems (e.g., vertex-cover, independent-set, $k$-coloring) can be encoded into CNF using only $O(|V|^2 / \lg |V|)$ many clauses, as opposed to the $Ω(|V|^2)$ constraints used by standard encodings. This somewhat surprising result is a simple consequence of a result of Erdős, Chung, and Spencer (1983) about biclique coverings of graphs, and opens theoretical avenues to understand the success of "Bounded Variable Addition'' (Manthey, Heule, and Biere, 2012) as a preprocessing tool. Finally, we show a novel encoding for independent sets in some dense interval graphs using only $O(|V| \lg |V|)$ clauses (the direct encoding uses $Ω(|V|^2)$), which we have successfully applied to a string-compression encoding posed by Bannai et al. (2022). As a direct byproduct, we obtain a reduction in the encoding size of a scheduling problem posed by Mayank and Modal (2020) from $O(NMT^2)$ to $O(NMT + M T^2 \lg T)$, where $N$ is the number of tasks, $T$ the total timespan, and $M$ the number of machines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically Smaller Encodings for Graph Problems and Scheduling Subercaseaux, Bernardo Logic in Computer Science Artificial Intelligence Data Structures and Algorithms We show how several graph problems (e.g., vertex-cover, independent-set, $k$-coloring) can be encoded into CNF using only $O(|V|^2 / \lg |V|)$ many clauses, as opposed to the $Ω(|V|^2)$ constraints used by standard encodings. This somewhat surprising result is a simple consequence of a result of Erdős, Chung, and Spencer (1983) about biclique coverings of graphs, and opens theoretical avenues to understand the success of "Bounded Variable Addition'' (Manthey, Heule, and Biere, 2012) as a preprocessing tool. Finally, we show a novel encoding for independent sets in some dense interval graphs using only $O(|V| \lg |V|)$ clauses (the direct encoding uses $Ω(|V|^2)$), which we have successfully applied to a string-compression encoding posed by Bannai et al. (2022). As a direct byproduct, we obtain a reduction in the encoding size of a scheduling problem posed by Mayank and Modal (2020) from $O(NMT^2)$ to $O(NMT + M T^2 \lg T)$, where $N$ is the number of tasks, $T$ the total timespan, and $M$ the number of machines. |
| title | Asymptotically Smaller Encodings for Graph Problems and Scheduling |
| topic | Logic in Computer Science Artificial Intelligence Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.14042 |