Abstract
Given a list of k source–sink pairs in an edge-weighted graph G, the minimum multicut problem consists in selecting a set of edges of minimum total weight in G, such that removing these edges leaves no path from each source to its corresponding sink. To the best of our knowledge, no non-trivial FPT result for special cases of this problem, which is APX-hard in general graphs for any fixed \(k \ge 3\), is known with respect to k only. When the graph G is planar, this problem is known to be polynomial-time solvable if \(k = O(1)\), but cannot be FPT with respect to k under the Exponential Time Hypothesis. In this paper, we show that, if G is planar and in addition all sources and sinks lie on the outer face, then this problem does admit an FPT algorithm when parameterized by k (although it remains APX-hard when k is part of the input, even in stars). To do this, we provide a new characterization of optimal solutions in this case, and then use it to design a “divide-and-conquer” approach: namely, some edges that are part of any such solution actually define an optimal solution for a polynomial-time solvable multiterminal variant of the problem on some of the sources and sinks (which can be identified thanks to a reduced enumeration phase). Removing these edges from the graph cuts it into several smaller instances, which can then be solved recursively.
Similar content being viewed by others
References
Bentz, C.: On the complexity of the multicut problem in bounded tree-width graphs and digraphs. Discrete Appl. Math. 156, 1908–1917 (2008)
Bentz, C.: A simple algorithm for multicuts in planar graphs with outer terminals. Discrete Appl. Math. 157, 1959–1964 (2009)
Bentz, C.: A polynomial-time algorithm for planar multicuts with few source–sink pairs. In: Proceedings IPEC, pp. 109–119 (2012)
Bentz, C., Costa, M.-C., Derhy, N., Roupin, F.: Cardinality constrained and multicriteria (multi)cut problems. J. Discrete Algorithms 7, 102–111 (2009)
Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: Proceedings STOC, pp. 459–468 (2011)
Chen, D.Z., Wu, X.: Efficient algorithms for \(k\)-terminal cuts on planar graphs. Algorithmica 38, 299–316 (2004)
Colin de Verdière, É.: Multicuts in planar and bounded-genus graphs with bounded number of terminals. Algorithmica 78, 1206–1224 (2017)
Colin de Verdière, É., Erickson, J.: Tightening nonsimple paths and cycles on surfaces. SIAM J. Comput. 39, 3784–3813 (2010)
Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)
Golovin, D., Nagarajan, V., Singh, M.: Approximating the \(k\)-multicut problem. In: Proceedings SODA, pp. 621–630 (2006)
Marx, D.: A tight lower bound for planar multiway cut with fixed number of terminals. In: Proceedings ICALP, pp. 677–688 (2012)
Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput. 43, 355–388 (2014)
Veblen, O.: Theory on plane curves in non-metrical analysis situs. Trans. Am. Math. Soc. 6, 83–98 (1905)
Yannakakis, M., Kanellakis, P., Cosmadakis, S., Papadimitriou, C.: Cutting and partitioning a graph after a fixed pattern. Proc. ICALP, Lect. Notes Comput. Sci. 154, 712–722 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bentz, C. An FPT Algorithm for Planar Multicuts with Sources and Sinks on the Outer Face. Algorithmica 81, 224–237 (2019). https://doi.org/10.1007/s00453-018-0443-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-018-0443-4