Faster fixed parameter tractable algorithms for finding feedback vertex sets

Abstract

A feedback vertex set (fvs) of a graph is a set of vertices whose removal results in an acyclic graph. We show that if an undirected graph on n vertices with minimum degree at least 3 has a fvs on at most 1 3 n 1− vertices, then there is a cycle of length at most 6 (for ≥ 1/2, we can even improve this to just 6).Using this, we obtain a O(( 12 log k log log k + 6) k n ω ) algorithm for testing whether an undirected graph on n vertices has a fvs of size at most k. Here n ω is the complexity of the best matrix multiplication algorithm. The previous best parameterized algorithm for this problem took O((2k + 1) k n 2 ) time.We also investigate the fixed parameter complexity of weighted feedback vertex set problem in weighted undirected graphs.

Improved algorithms for feedback vertex set problems

“…[13] O((2 log k + 2 log log k + 18) k n 2 ) Raman et al [15] O((12 log k/ log log k + 6) k n 2.376 ) Guo et al. [12] O((37.7) k n 2 ) Dehne et al [7] O( (10.6) Bodlaender [3] and by Downey and Fellows [8].…”

Isomorphism for Graphs of Bounded Feedback Vertex Set Number

“…From this it is immediate that if a problem is fixed-parameter tractable (FPT) with respect to some parameter then it is also FPT with respect to any larger (in Figure 1 higher) parameter: time O(f (k)n c ) with respect to k implies time O(f (g(k ))n c ) with respect to k (likewise for runtimes of the form O(n f (k) )). The feedback vertex set number, which has been extensively studied in various contexts [8,11,14,17,25,28], lies above other interesting parameters: As mentioned GI remains hard on graphs of bounded chromatic number, while being polynomially solvable for bounded treewidth. As the rooted tree distance width the feedback vertex set number is a measure For various graph parameters, the figure depicts the partial order given by the relation that defines a parameter to be lower than another parameter, if the former can be bounded by a function of the latter.…”

“…For reduced graphs, we can use a nice structural result by Raman, Saurabh and Subramanian [25], stating that graphs of minimum degree at least three must have a large feedback vertex set number or a cycle of length at most six. Thus, in contrast to the general bound of log n on the girth of a graph, there are few choices for the image of any feedback vertex under an isomorphism between two reduced graphs.…”

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

“…In both the directed and the undirected versions of the feedback vertex set problems, brute force can be used to check in time n O(k) if a solution of size at most k exists: one can go through all sets of size at most k. Thus the problem can be solved in polynomial time if the optimum is assumed to be small. In the undirected case, we can do significantly better: since the first FPT algorithm for FVS in undirected graphs by Mehlhorn [36] almost 30 years ago, there have been a number of papers [2,3,5,6,16,17,23,26,39,40] giving faster algorithms and the current fastest (randomized) algorithm runs in time O * (3 k ) [13] (the O * notation hides all factors that are polynomial in the size of input). That is, undirected FVS is fixed-parameter tractable parameterized by the size of the solution.…”