Abstract
In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to a requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X 1, ⋯ ,X g ⊆ V, each with a requirement r i between 0 and |X i |. The goal is to find a minimum cost set of edges whose removal separates each group X i into at least r i disconnected components.
We give an O(log n log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is a natural Ω (log g) hardness of approximation for the requirement cut problem.
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Nagarajan, V., Ravi, R. (2005). Approximation Algorithms for Requirement Cut on Graphs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_18
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DOI: https://doi.org/10.1007/11538462_18
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