An FPT Algorithm Beating 2-Approximation for <i>k</i>-Cut

Gupta, Anupam; Lee, Euiwoong; Li, Jason

doi:10.1137/1.9781611975031.179

Cited by 32 publications

(43 citation statements)

References 23 publications

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“…To circumvent this, Saran and Vazirani [32] devised two simple polynomial time (2 − 2/k)-approximation algorithms for the problem. In the ensuing years, different approximation algorithms [33][34][35][36][37] have been proposed for the problem, none of which are able achieve an approximation ratio of (2 − ε) for some ε > 0 in polynomial time. In fact, Saran and Vazirani themselves conjectured that (2 − ε)-approximation is intractible for the problem [32].…”

Section: Minimum K-cutmentioning

confidence: 99%

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

Manurangsi

2018

Algorithms

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Abstract:The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions.

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Section: Minimum K-cutmentioning

confidence: 99%

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

Manurangsi

2018

Algorithms

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“…Our approximate DP technique turns out to be useful to get a 1.81-approximation for k-Cut in FPT time, improving on our previous approximation of ≈ 1.9997 [GLL18]. In particular, the laminar cut problem from [GLL18] also has a tight T-tree structure, and hence we can use (a special case of) our approximate DP algorithm to get a (1 + ε)-approximation for laminar cut, instead of the 2 − ε-factor previously known. Combining with other ideas in the previous paper, this gives us the 1.81-approximation.…”

Section: Introductionmentioning

confidence: 58%

“…, S k be the components maintained by the algorithm. In [GLL18], a is defined to be the smallest value of k when both the weight of the min-cut, as well as one-third of the weight of the min-4-cut, becomes bigger than w(∂S * 1 )(1 − ε 1 /3), for some ε 1 > 0. Moreover, b ∈ [k] is the smallest number such that w(∂S * b ) > w(∂S * 1 )(1 + ε 1 /3).…”

Section: B An 181-fpt Approximation Algorithmmentioning

confidence: 99%

“…, S * b−1 ∩ S i , S * ≥b ∩ S i } as the desired solution. Moreover, in this r i -cut instance on G[S i ], the laminar structure of the cuts of weight Let us sketch the high-level idea of the rest of the proof for those who don't remember details of [GLL18]. In the paper, we assume that the min-cut always remains smaller than M := w(∂S * 1 ), else we can branch on having found one component.…”

Section: B An 181-fpt Approximation Algorithmmentioning

confidence: 99%

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Faster Exact and Approximate Algorithms for k-Cut

Gupta

Lee

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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In the k-Cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. The current best algorithms are an O(n (2−o(1))k ) randomized algorithm due to Karger and Stein, and anÕ(n 2k ) deterministic algorithm due to Thorup. Moreover, several 2-approximation algorithms are known for the problem (due to Saran and Vazirani, Naor and Rabani, and Ravi and Sinha).It has remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this paper we show an O(k O(k) n (2ω/3+o(1))k )-time algorithm for k-Cut. Moreover, we show an (1 + ε)-approximation algorithm that runs in time O((k/ε) O(k) n k+O(1) ), and a 1.81-approximation in fixed-parameter time O(2 OIn this paper we consider the k-Cut problem: given an edge-weighted graph G = (V, E, w) and an integer k, delete a minimum-weight set of edges so that G has at least k connected components. This problem is a natural generalization of the global min-cut problem, where the goal is to break the graph into k = 2 pieces. This problem has been actively studied in theory of both exact and approximation algorithms, where each result brought new insights and tools on graph cuts.It is not a priori clear how to obtain poly-time algorithms for any constant k, since guessing one vertex from each part only reduces the problem to the NP-hard Multiway Cut problem. Indeed, the first result along these lines was the work of Goldschmidt and Hochbaum [GH94] who gave an O(n (1/2−o(1))k 2 )-time exact algorithm for k-Cut. Since then, the exact exponent in terms of k has been actively studied. The current best runtime is achieved by an O(n 2(k−1) ) randomized algorithm due to Karger and Stein [KS96], which performs random edge contractions until the remaining graph has k nodes, and shows that the resulting cut is optimal with probability at least Ω(n −2(k−1) ). The asymptotic runtime ofÕ(n 2(k−1) ) was later matched by a deterministic algorithm of Thorup [Tho08]. His algorithm was based on tree-packing theorems; it showed how to efficiently find a tree for which the optimal k-cut crosses it 2k − 2 times. Enumerating over all possible 2k − 2 edges of this tree gives the algorithm.These elegant O(n 2k )-time algorithms are the state-of-the-art, and it has remained an open question to improve on them. An easy observation is that the problem is closely related to k-Clique, so we may not expect the exponent of n to go below (ω/3)k. Given the interest in fine-grained analysis of algorithms, where in the range [(ω/3)k, 2k − 2] does the correct answer lie?Our main results give faster deterministic and randomized algorithms for the problem.Theorem 1.1 (Faster Randomized Algorithm). Let W be a positive integer. There is a randomized algorithm for k-Cut on graphs with edge weights in [W ] with runtimethat succeeds with probability 1 − 1/poly(n).Theorem 1.2 (Even Faster Deterministic Algorithm). Let W be a positive integer. For any ε >...

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“…Also, to the best of our knowledge, prior to this work, there was no approximation algorithm with a ratio better than 2 for any major class of graphs. It is also worth mentioning that a recent work by Gupta et al [19] showed that using an FPT algorithm the approximation factor of 2 can be beaten in the k-cut problem. They showed that there exists a 2 − ǫ approximation algorithm that runs in time 2 O(k 6 )Õ (n 4 ).…”

Section: Introductionmentioning

confidence: 99%

Polynomial-time Approximation Scheme for Minimum k-cut in Planar and Minor-free Graphs

Bateni

Farhadi

Hajiaghayi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The k-cut problem asks, given a connected graph G and a positive integer k, to find a minimum-weight set of edges whose removal splits G into k connected components. We give the first polynomial-time algorithm with approximation factor 2 − ǫ (with constant ǫ > 0) for the k-cut problem in planar and minor-free graphs. Applying more complex techniques, we further improve our method and give a polynomial-time approximation scheme for the k-cut problem in both planar and minor-free graphs. Despite persistent effort, to the best of our knowledge, this is the first improvement for the k-cut problem over standard approximation factor of 2 in any major class of graphs. * Google Research.

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An FPT Algorithm Beating 2-Approximation for k-Cut

Cited by 32 publications

References 23 publications

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

Faster Exact and Approximate Algorithms for k-Cut

Polynomial-time Approximation Scheme for Minimum k-cut in Planar and Minor-free Graphs

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