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This algorithm is optimal up to o(1) factors assuming recent hardness conjectures: we show by a straightforward reduction that k-Cut on even a simple graph is as hard as -clique, establishing a lower bound of n(1−o(1))k for k-Cut

# Faster Minimum k-cut of a Simple Graph

2019 IEEE 60TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2019), pp.1056.0-1077, (2019)

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Abstract

We consider the (exact, minimum) k-CUT problem: given a graph and an integer k, delete a minimum-weight set of edges so that the remaining graph has at least k connected components. This problem is a natural generalization of the global minimum cut problem, where the goal is to break the graph into k = 2 pieces. Our main result is a (comb...More

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Introduction
• The authors consider the k-Cut problem: given a graph and an integer k, delete a minimumweight set of edges so that the remaining graph has at least k connected components.
• The authors apply a reduction in [9] which, at a multiplicative cost of O(nk) in the running time, produces a tree T such that exactly k − 1 edges of T have endpoints in different components of the minimum k-cut.
• Since s–t minimum cut is polynomial time solvable, the algorithm can detect which edges to contract.
Highlights
• We consider the k-Cut problem: given a graph and an integer k, delete a minimumweight set of edges so that the remaining graph has at least k connected components. This problem is a natural generalization of the global minimum 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 cut algorithms
• The fastest algorithm for general edge weights is due to Chekuri et al [4]
Results
• For each connected component C in H, there exists a spanning tree TC of C satisfying the following property: Let U be the set of endpoints of edges in TC (more formally, U := (u,v)∈E(TC){u, v}).
• Observe that if Conditions 4.5 and 4.6 hold, for each connected component Ci∗, there exists a set Ui∗ ∈ U of size |V (Ci∗)| such that each vertex vj∗ in Ci∗ belongs on a branch T (u) (u ∈ Ui∗).
• After running Algorithm 4.7, Conditions 4.5 and 4.6 hold, and for each connected component Ci∗, there exists a set Ui∗ ∈ U of size |V (Ci∗)| such that each vertex vj∗ in Ci∗ belongs on a branch T (u) (u ∈ U ).
• The ancestor cut has one edge sharing a maximal branch with each vertex in MinElts(Ui).
• The authors want a set of disjoint branches, one containing each minimal vertex, such that a variant of Lemma 4.4 still holds, so that the authors can set up a similar minimum s–t cut problem.
• For each connected component C in H, there exists a spanning tree TC of C satisfying the following property: Let U be the set of endpoints of edges in TC.
• Construct the graph Gb as follows: the add that edge with endpoints b(u), vertex b(v) in set is i Bi∗, and for each edge in Gb. Note that if the authors contract each
• Fix any component Ci∗, and let U be the set of endpoints in T ′ of edges in Ti∗.
Conclusion
• For each connected component C of H′, let C+ be the set of vertices in H contracted to a vertex in C.
• After running Algorithms 4.25 and 4.30, Conditions 4.23, 4.24, 4.26, 4.27, and 4.28 hold, and for each Ci+, there exists a set Ui+ ∈ U whose subtrees contain precisely all vertices vj∗ in Ci+.
• The minimum ancestor p-cut is the following problem: For each i ∈ [l], remove at least one edge in T (note: not T ′(ui)) such that after removal, no sj ∈ V (T) is in the same component as ui, and such that exactly p − 1 edges are removed in total.
Related work
• The k-Cut problem has been studied extensively in the approximate and fixed-parameter settings as well.

Approximation algorithms. The first approximation algorithm k-Cut was a 2(1−1/k)-approximation of Saran and Vazirani [23]. Later, Naor and Rabani [21], and also Ravi and Sinha [22] gave 2-approximation algorithms using tree packing and network strength respectively. Xiao et al [27] extended Kapoor [13] and Zhao et al [28] and generalized Saran and Vazirani to give an (2 − h/k)-approximation in time nO(h). On the hardness front, Manurangsi [18] showed that for any ǫ > 0, it is NP-hard to achieve a (2 − ǫ)-approximation algorithm in time poly(n, k) assuming the Small Set Expansion Hypothesis.
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