k-SUM Hardness Implies Treewidth-SETH
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908593439113216 |
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| author | Lampis, Michael |
| author_facet | Lampis, Michael |
| contents | We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|ϕ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)\frac{k}{2}}$ for some $δ>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$.
As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | k-SUM Hardness Implies Treewidth-SETH Lampis, Michael Computational Complexity Data Structures and Algorithms We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|ϕ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)\frac{k}{2}}$ for some $δ>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity. |
| title | k-SUM Hardness Implies Treewidth-SETH |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.08185 |