An Efficient Semi-Streaming PTAS for Tournament Feedback ArcSet with Few Passes

TL;DR

This paper introduces the first semi-streaming polynomial-time approximation scheme for the minimum feedback arc set problem in directed tournaments, achieving near-optimal solutions with few passes and limited memory.

Contribution

It presents a novel semi-streaming PTAS for tournament feedback arc set, improving pass and space efficiency over prior algorithms, and explores related streaming problems.

Findings

Achieves a (1+ε)-approximation in polynomial time with few passes

Provides a new time/space trade-off for 1-pass algorithms

Includes lower bounds and algorithms for related streaming problems

Abstract

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.