Years and Authors of Summarized Original Work
2012; Chitnis, Cygan, Hajiaghayi, Pilipczuk, Pilipczuk
Problem Definition
The randomized contractions framework is often useful when designing fixed-parameter-tractable (FPT) algorithms for graph cut problems. Let us assume that we are given an undirected graph G with n vertices and m edges together with an integer k. The goal is to remove at most k edges or at most k vertices, in the edge- and vertex-deletion variants of a problem, respectively, to satisfy some problem-specific constraints. In this entry, for the sake of simplicity, we restrict our attention to edge-deletion variants only.
Examples of problems that fit in the above graph cut problem class include:
- Multiway Cut :
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Input: an undirected graph G, a set of terminals T ⊆ V (G), and an integer k.
Question: is there a set X ⊆ E(G) of at most k edges of G, so that in G ∖ X, no connected component contains more than one terminal from T?
- Steiner Cut :
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Input: an undirected graph G, a...
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Recommended Reading
Chen J, Liu Y, Lu S (2009) An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica 55(1):1–13. doi:10.1007/s00453-007-9130-6
Chitnis RH, Cygan M, Hajiaghayi M, Pilipczuk M, Pilipczuk M (2012) Designing FPT algorithms for cut problems using randomized contractions. In: 53rd annual IEEE symposium on foundations of computer science (FOCS 2012), New Brunswick, 20–23 Oct 2012. IEEE Computer Society, pp 460–469. doi:10.1109/FOCS.2012.29
Cygan M, Pilipczuk M, Pilipczuk M, Wojtaszczyk JO (2013) On multiway cut parameterized above lower bounds. TOCT 5(1):3. doi:10.1145/2462896.2462899
Kawarabayashi K, Thorup M (2011) The minimum k-way cut of bounded size is fixed-parameter tractable. In: Ostrovsky R (ed) IEEE 52nd annual symposium on foundations of computer science (FOCS 2011), Palm Springs, 22–25 Oct 2011. IEEE Computer Society, pp 160–169. doi:10.1109/FOCS.2011.53
Marx D (2006) Parameterized graph separation problems. Theor Comput Sci 351(3):394–406. doi:10.1016/j.tcs.2005.10.007
Naor M, Schulman LJ, Srinivasan A (1995) Splitters and near-optimal derandomization. In: 36th annual symposium on foundations of computer science, Milwaukee, 23–25 Oct 1995. IEEE Computer Society, pp 182–191. doi:10.1109/SFCS.1995.492475
Wahlström M (2014) Half-integrality, LP-branching and FPT algorithms. In: Chekuri C (ed) Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms (SODA 2014), Portland, 5–7 Jan 2014. SIAM, pp 1762–1781. doi:10.1137/1.9781611973402.128
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Cygan, M. (2015). Randomized Contraction. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27848-8_764-1
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DOI: https://doi.org/10.1007/978-3-642-27848-8_764-1
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