Abstract
Let T = (V,E) be an undirected tree, in which each edge is associated with a non-negative cost, and let { s 1, t 1 }, ..., { s k , t k } be a collection of k distinct pairs of vertices. Given a requirement parameter t ≤ k, the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from T disconnects at least t out of these k pairs. This problem generalizes the well-known multicut on a tree problem, in which we are required to disconnect all given pairs.
The main contribution of this paper is an (\({\frac{8}{3}}+{\epsilon}\))-approximation algorithm for partial multicut on a tree, whose run time is strongly polynomial for any fixed ε > 0. This result is achieved by introducing problem-specific insight to the general framework of using the Lagrangian relaxation technique in approximation algorithms. Our algorithm utilizes a heuristic for the closely related prize-collecting variant, in which we are not required to disconnect all pairs, but rather incur penalties for failing to do so. We provide a Lagrangian multiplier preserving algorithm for the latter problem, with an approximation factor of 2. Finally, we present a new 2-approximation algorithm for multicut on a tree, based on LP-rounding.
Due to space limitations, several proofs are omitted from this extended abstract. We refer the reader to the full version of this paper [13], in which all missing proofs are provided.
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Levin, A., Segev, D. (2006). Partial Multicuts in Trees. In: Erlebach, T., Persinao, G. (eds) Approximation and Online Algorithms. WAOA 2005. Lecture Notes in Computer Science, vol 3879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671411_25
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DOI: https://doi.org/10.1007/11671411_25
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