Breaking a Long-Standing Barrier: 2-$\varepsilon$ Approximation for Steiner Forest
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| Format: | Preprint |
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2025
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| _version_ | 1866917986327068672 |
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| author | Ahmadi, Ali Gholami, Iman Hajiaghayi, MohammadTaghi Jabbarzade, Peyman Mahdavi, Mohammad |
| author_facet | Ahmadi, Ali Gholami, Iman Hajiaghayi, MohammadTaghi Jabbarzade, Peyman Mahdavi, Mohammad |
| contents | The Steiner Forest problem, also known as the Generalized Steiner Tree problem, is a fundamental optimization problem on edge-weighted graphs where, given a set of vertex pairs, the goal is to select a minimum-cost subgraph such that each pair is connected. This problem generalizes the Steiner Tree problem, first introduced in 1811, for which the best known approximation factor is 1.39 [Byrka, Grandoni, Rothvoß, and Sanità, 2010] (Best Paper award, STOC 2010).
The celebrated work of [Agrawal, Klein, and Ravi, 1989] (30-Year Test-of-Time award, STOC 2023), along with refinements by [Goemans and Williamson, 1992] (SICOMP'95), established a 2-approximation for Steiner Forest over 35 years ago. Jain's (FOCS'98) pioneering iterative rounding techniques later extended these results to higher connectivity settings. Despite the long-standing importance of this problem, breaking the approximation factor of 2 has remained a major challenge, raising suspicions that achieving a better factor -- similar to Vertex Cover -- might indeed be hard. Notably, fundamental works, including those by Gupta and Kumar (STOC'15) and Groß et al. (ITCS'18), introduced 96- and 69-approximation algorithms, possibly with the hope of paving the way for a breakthrough in achieving a constant-factor approximation below 2 for the Steiner Forest problem.
In this paper, we break the approximation barrier of 2 by designing a novel deterministic algorithm that achieves a $2 - 10^{-11}$ approximation for this fundamental problem. As a key component of our approach, we also introduce a novel dual-based local search algorithm for the Steiner Tree problem with an approximation guarantee of $1.943$, which is of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_11398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Breaking a Long-Standing Barrier: 2-$\varepsilon$ Approximation for Steiner Forest Ahmadi, Ali Gholami, Iman Hajiaghayi, MohammadTaghi Jabbarzade, Peyman Mahdavi, Mohammad Data Structures and Algorithms The Steiner Forest problem, also known as the Generalized Steiner Tree problem, is a fundamental optimization problem on edge-weighted graphs where, given a set of vertex pairs, the goal is to select a minimum-cost subgraph such that each pair is connected. This problem generalizes the Steiner Tree problem, first introduced in 1811, for which the best known approximation factor is 1.39 [Byrka, Grandoni, Rothvoß, and Sanità, 2010] (Best Paper award, STOC 2010). The celebrated work of [Agrawal, Klein, and Ravi, 1989] (30-Year Test-of-Time award, STOC 2023), along with refinements by [Goemans and Williamson, 1992] (SICOMP'95), established a 2-approximation for Steiner Forest over 35 years ago. Jain's (FOCS'98) pioneering iterative rounding techniques later extended these results to higher connectivity settings. Despite the long-standing importance of this problem, breaking the approximation factor of 2 has remained a major challenge, raising suspicions that achieving a better factor -- similar to Vertex Cover -- might indeed be hard. Notably, fundamental works, including those by Gupta and Kumar (STOC'15) and Groß et al. (ITCS'18), introduced 96- and 69-approximation algorithms, possibly with the hope of paving the way for a breakthrough in achieving a constant-factor approximation below 2 for the Steiner Forest problem. In this paper, we break the approximation barrier of 2 by designing a novel deterministic algorithm that achieves a $2 - 10^{-11}$ approximation for this fundamental problem. As a key component of our approach, we also introduce a novel dual-based local search algorithm for the Steiner Tree problem with an approximation guarantee of $1.943$, which is of independent interest. |
| title | Breaking a Long-Standing Barrier: 2-$\varepsilon$ Approximation for Steiner Forest |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.11398 |