A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866916113233739776 |
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| author | Apers, Simon Auza, Arinta Lee, Troy |
| author_facet | Apers, Simon Auza, Arinta Lee, Troy |
| contents | An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = Ω(n^2)$. In contrast, a randomized algorithm must make $Ω(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15587 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs Apers, Simon Auza, Arinta Lee, Troy Quantum Physics Computational Complexity Data Structures and Algorithms An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = Ω(n^2)$. In contrast, a randomized algorithm must make $Ω(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected. |
| title | A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs |
| topic | Quantum Physics Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2110.15587 |