A $5$-Approximation Analysis for the Cover Small Cuts Problem
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arXiv
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| Natura: | Preprint |
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2026
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| author | Simmons, Miles Bansal, Ishan Cheriyan, Joe |
| author_facet | Simmons, Miles Bansal, Ishan Cheriyan, Joe |
| contents | In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $λ$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $λ$ is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio $16$ for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio $6$. We show that the same algorithm achieves approximation ratio $5$, by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_01462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $5$-Approximation Analysis for the Cover Small Cuts Problem Simmons, Miles Bansal, Ishan Cheriyan, Joe Data Structures and Algorithms 68W25, 90C27 In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $λ$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $λ$ is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio $16$ for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio $6$. We show that the same algorithm achieves approximation ratio $5$, by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity. |
| title | A $5$-Approximation Analysis for the Cover Small Cuts Problem |
| topic | Data Structures and Algorithms 68W25, 90C27 |
| url | https://arxiv.org/abs/2602.01462 |