We present the first semi-streaming PTAS for the minimum feedback arc set problem on directed tournaments in a small number of passes. Namely, we obtain a $(1 + \varepsilon)$-approximation in polynomial time $O \left( \text{poly}(n) 2^{\text{poly}(1/\varepsilon)} \right)$, with $p$ passes in $n^{1+1/p} \cdot \text{poly}\left(\frac{\log n}{\varepsilon}\right)$ space. The only previous algorithm with this pass/space trade-off gave a $3$-approximation (SODA, 2020), and other polynomial-time algorithms which achieved a $(1+\varepsilon)$-approximation did so with quadratic memory or with a linear number of passes. We also present a new time/space trade-off for $1$-pass algorithms that solve the tournament feedback arc set problem. This problem has several applications in machine learning such as creating linear classifiers and doing Bayesian inference. We also provide several additional algorithms and lower bounds for related streaming problems on directed graphs, which is a mostly unexplored territory.
翻译：我们展示了第一个半流式 PTAS, 用于在少数通行证中直接举行的锦标赛上的最低反馈弧度问题。 也就是说, 我们获得一个$(1 +\ varepsilon) $- programmemotion 在多元时间 $O left( text{poly} (n) 2 ⁇ text{poly}(1/\ varepsilon)}\right) $, 以 $@ 1+1/ p}\ cdot\ text{polyleft (\ frac) nunt- varepslon_right) 空间。 唯一一个之前的通/ 空交易算法提供了$( +\ varepslon) $( +\ varepsilon) $- procol- pol- progmation 美元( programme) $( lease) $( $1 + + + + + + + + + + + + + + + + 1/ 1/ 1/ 1/ 1/ 1/ vulpl) ( pright) right) $。 $ 1 $ 1 $ $ 1 $ 1 $ 1 $ 1 $ $ $ $ $ 1 $ $ $ $ $ $ $ $ ( $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ lexxx lex lex lex) leftcrefal- lex lex) lections lections prefor lections lection- lection) lections pal lections pal) la la la la la lax lax lax lements lax lax lax lax lax lement lements la la la
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