Undirected Feedback Vertex Set Research Articles

Wannabe Bounded Treewidth Graphs Admit a Polynomial Kernel for Directed Feedback Vertex Set

In the Directed Feedback Vertex Set (DFVS) problem, given a digraph D and a positive integer k , the goal is to check if there exists a set of at most k vertices whose deletion from D results in a directed acyclic graph. The existence of a polynomial kernel for DFVS , parameterized by the solution size k , is a central open problem in Kernelization. In this paper, we give a polynomial kernel for DFVS parameterized by k plus the size of a treewidth- η modulator (of the underlying undirected graph), where η is any fixed positive integer. Since the status of the existence of a polynomial kernel for DFVS (parameterized by the solution size) is open for a very long time now, and it is known to not admit a polynomial kernel when the parameter is the size of a treewidth-2 modulator, solution size plus the size of the treewidth- η modulator makes for an interesting choice of parameter to study. In fact, the polynomial kernelization complexity of DFVS parameterized by the size of the undirected feedback vertex set (treewidth-1 modulator) in the underlying undirected graph, has already been studied in literature. Our choice of parameter strictly encompasses previous positive kernelization results on DFVS . Our result is based on a novel application of the tool of important separators embedded in state-of-the-art machinery such as protrusion decompositions.

Kernelization for feedback vertex set via elimination distance to a forest

We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption NP⁄⊆coNP/poly, we prove that for any minor-closed graph class G, Feedback Vertex Set parameterized by the size of a modulator to G has a polynomial kernel if and only if G has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem. [Display omitted]

Expansion Lemma—Variations and Applications to Polynomial-Time Preprocessing

In parameterized complexity, it is well-known that a parameterized problem is fixed-parameter tractable if and only if it has a kernel—an instance equivalent to the input instance, whose size is just a function of the parameter. The size of the kernel can be exponential or worse, resulting in a quest for fixed-parameter tractable problems with polynomial-sized kernels. The developments in machinery (showing lower bounds for the sizes of the kernels) have led researchers to question what are the asymptotically optimum sizes for the kernels of fixed-parameter tractable problems. In this article, we surveyed a tool called expansion lemma that helps in reducing the size of the kernel. Its early origin was in the form of crown decomposition, i.e., to obtain the linear kernel for the Vertex Cover problem; the specific lemma was identified as the tool behind the optimal O(k2) kernel for the undirected feedback vertex set problem. Since then, several variations and extensions of the tool have been discovered. We surveyed them along with their applications in this article.

Structural Parameterizations of Undirected Feedback Vertex Set: FPT Algorithms and Kernelization

A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure in the input. In particular, we consider parameterizations where the parameter is (instead of the solution size), the distance to a class of graphs where the problem is polynomial time solvable, and sometimes smaller than the solution size. Here, by distance to a class of graphs, we mean the minimum number of vertices whose removal results in a graph in the class. Such a set of vertices is also called the ‘deletion set’. In this paper, we show that FVS is fixed-parameter tractable by an \({{\mathcal {O}}}(2^k n^{{{\mathcal {O}}}(1)})\) time algorithm, but is unlikely to have polynomial kernel when parameterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper. We also show that an algorithm with running time \({{\mathcal {O}}}((\sqrt{2} - \epsilon )^k n^{{{\mathcal {O}}}(1)})\) is not possible unless SETH fails. When parameterized by k, the number of vertices, whose deletion results in a split graph, we give an \({{\mathcal {O}}}(3.148^k n^{{{\mathcal {O}}}(1)})\) time algorithm. When parameterized by k, the number of vertices whose deletion results in a cluster graph (a disjoint union of cliques), we give an \({{\mathcal {O}}}(5^k n^{{{\mathcal {O}}}(1)})\) algorithm. Regarding kernelization results, we show that When parameterized by k, the number of vertices, whose deletion results in a pseudo-forest, FVS has an \({{\mathcal {O}}}(k^7)\) vertices kernel improving from the previously known \({{\mathcal {O}}}(k^{10})\) bound. When parameterized by the number k of vertices, whose deletion results in a mock-d-forest, we give a kernel with \({{\mathcal {O}}}(k^{3d+3})\) vertices. We also prove a lower bound of \(\varOmega (k^{d+2})\) size (under complexity theoretic assumptions). Mock-forest is a graph where each vertex is contained in at most one cycle. Mock-d-forest for a constant d is a mock-forest where each component has at most d cycles.

Solving the undirected feedback vertex set problem by local search

An undirected graph consists of a set of vertices and a set of undirected edges between vertices. Such a graph may contain an abundant number of cycles, in which case a feedback vertex set (FVS) is a set of vertices intersecting with each of these cycles. Constructing a FVS of cardinality approaching the global minimum value is an optimization problem in the nondeterministic polynomial-complete complexity class, and therefore it might be extremely difficult for some large graph instances. In this paper we develop a simulated annealing local search algorithm for the undirected FVS problem by adapting the heuristic procedure of Galinier et al. [P. Galinier, E. Lemamou, M.W. Bouzidi, J. Heuristics 19, 797 (2013)], which worked for the directed FVS problem. By defining an order for the vertices outside the FVS, we replace the global cycle constraints by a set of local vertex constraints on this order. Under these local constraints the cardinality of the focal FVS is then gradually reduced by the simulated annealing dynamical process. We test this heuristic algorithm on large instances of Erdös-Rényi random graph and regular random graph, and find that this algorithm is comparable in performance to the belief propagation-guided decimation algorithm.

An improved exact algorithm for undirected feedback vertex set

A feedback vertex set in an undirected graph is a subset of vertices removal of which leaves a graph with no cycles. Razgon (in: Proceedings of the 10th Scandinavian workshop on algorithm theory (SWAT 2006), pp. 160---171, 2006) gave a $$1.8899^n n^{O(1)}$$1.8899nnO(1)-time algorithm for finding a minimum feedback vertex set in an $$n$$n-vertex undirected graph, which is the first exact algorithm for the problem that breaks the trivial barrier of $$2^n$$2n. Later, Fomin et al. (Algorithmica 52:293---307, 2008) improved the result to $$1.7548^n n^{O(1)}$$1.7548nnO(1). In this paper, we further improve the result to $$1.7266^n n^{O(1)}$$1.7266nnO(1). Our algorithm is analyzed by the measure-and-conquer method. We get the improvement by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.

A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

We investigate the computational complexity of a general “compression task” centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of the following task: given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, find a smaller disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that a few cases are polynomial-time solvable, and the others are NP-hard. The considered class of vertex deletion problems includes Vertex Cover (where the compression task is polynomial time) and Undirected Feedback Vertex Set (where the compression task is NP-complete).

Wavelength Conversion in Optical Networks

Wavelength Conversion in Optical Networks

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.