Deterministic Near-Linear Time Minimum Cut in Weighted Graphs
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866929206502359040 |
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| author | Henzinger, Monika Li, Jason Rao, Satish Wang, Di |
| author_facet | Henzinger, Monika Li, Jason Rao, Satish Wang, Di |
| contents | In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time $O(m\log^3n)$ which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs.
Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs. The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an $m^{1+o(1)}$ deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed "almost-linear''. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, "derandomize'' the methods of Karger.
In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with $o(1)$ error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to $1+o(1/\log n)$-approximate so that they can be applied recursively as in [Li21] over $O(\log n)$ many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05627 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deterministic Near-Linear Time Minimum Cut in Weighted Graphs Henzinger, Monika Li, Jason Rao, Satish Wang, Di Data Structures and Algorithms In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time $O(m\log^3n)$ which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs. Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs. The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an $m^{1+o(1)}$ deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed "almost-linear''. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, "derandomize'' the methods of Karger. In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with $o(1)$ error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to $1+o(1/\log n)$-approximate so that they can be applied recursively as in [Li21] over $O(\log n)$ many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters. |
| title | Deterministic Near-Linear Time Minimum Cut in Weighted Graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2401.05627 |