Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

Abstract

The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

Computing Subset Transversals in $H$-Free Graphs

“…Papadopoulos and Tzimas [32,33] proved that Subset Feedback Vertex Set is polynomial-time solvable for sP 1 -free graphs for any s ≥ 1, cobipartite graphs, interval graphs and permutation graphs, and thus P 4 -free graphs. Some of these results were generalized by Bergougnoux et al [2], who solved an open problem of Jaffke et al [23] by giving an n O(w 2 ) -time algorithm for Subset Feedback Vertex Set given a graph and a decomposition of this graph of mim-width w. This does not lead to new results for H-free graphs: a class of H-free graphs has bounded mim-width if and only if H ⊆ i P 4 [7].…”

Computing Subset Feedback Vertex Set via Leafage

“…For every connected component C of a chordal graph G, we compute its leafage and the corresponding tree model T (C) by using the O(n 3 )-time algorithm of Habib and Stacho [20]. If the leafage of C is less than two, then C is an interval graph and we can compute A (1) time by running the algorithm for SFVS on interval graphs given in [28]. Otherwise, we compute the expanded tree model T (C) from T (C) by Lemma 2 in O(n 2 ) time.…”

“…Among the stated classes, we stress that interval graphs is the only known subclass of chordal graphs for which Subset Feedback Vertex Set is solved in polynomial time. Related to the structural parameter mim-width, Bergougnoux et al [1] recently proposed an n O(w 2 ) -time algorithm that solves Subset Feedback Vertex Set given a decomposition of the input graph of mim-width w. As leaf power graphs admit a decomposition of mim-width one [23], from the later algorithm Subset Feedback Vertex Set can be solved in polynomial time on leaf power graphs if an intersection model is given as input. However, to the best of our knowledge, it is not known whether the intersection model of a leaf power graph can be constructed in polynomial time.…”

“…However, to the best of our knowledge, it is not known whether the intersection model of a leaf power graph can be constructed in polynomial time. Moreover, even for graphs of mim-width one that do admit an efficient construction of the corresponding decomposition, the exponent of the running time given in [1] is relative high.…”

“…They form a subclass of leaf powers and have unbounded leafage (through their underlying tree model). Although directed path graphs admit a decomposition of mim-width one [23] and such a decomposition can be constructed in polynomial time [10,31], the running time obtained through the bounded mim-width approach is rather unpractical, as it requires to store a table of size O(n 13 ) even in this particular case [1]. By analyzing further subsets of vertices at each intermediate step, we manage to derive an algorithm for Subset Feedback Vertex Set on directed path graphs that runs in O(n 2 m) time.…”