Geometric Bipartite Matching is in NC
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914815573753856 |
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| author | Bhore, Sujoy Equbal, Sarfaraz Gurjar, Rohit |
| author_facet | Bhore, Sujoy Equbal, Sarfaraz Gurjar, Rohit |
| contents | In this work, we study the parallel complexity of the Euclidean minimum-weight perfect matching (EWPM) problem. Here our graph is the complete bipartite graph $G$ on two sets of points $A$ and $B$ in $\mathbb{R}^2$ and the weight of each edge is the Euclidean distance between the corresponding points. The weighted perfect matching problem on general bipartite graphs is known to be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner, Gurjar, and Thierauf, 2016]. Both of these results work only when the weights are of $O(\log n)$ bits. It is a long-standing open question to show the problem to be in NC.
First, we show that for EWPM, a linear number of bits of approximation is required to distinguish between the minimum-weight perfect matching and other perfect matchings. Next, we show that the EWPM problem that allows up to $\frac{1}{poly(n)}$ additive error, is in NC. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric Bipartite Matching is in NC Bhore, Sujoy Equbal, Sarfaraz Gurjar, Rohit Computational Geometry Computational Complexity In this work, we study the parallel complexity of the Euclidean minimum-weight perfect matching (EWPM) problem. Here our graph is the complete bipartite graph $G$ on two sets of points $A$ and $B$ in $\mathbb{R}^2$ and the weight of each edge is the Euclidean distance between the corresponding points. The weighted perfect matching problem on general bipartite graphs is known to be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner, Gurjar, and Thierauf, 2016]. Both of these results work only when the weights are of $O(\log n)$ bits. It is a long-standing open question to show the problem to be in NC. First, we show that for EWPM, a linear number of bits of approximation is required to distinguish between the minimum-weight perfect matching and other perfect matchings. Next, we show that the EWPM problem that allows up to $\frac{1}{poly(n)}$ additive error, is in NC. |
| title | Geometric Bipartite Matching is in NC |
| topic | Computational Geometry Computational Complexity |
| url | https://arxiv.org/abs/2405.18833 |