Minimum Feedback Vertex Set Problem Research Articles

Stochastic simulated annealing for directed feedback vertex set

The minimum feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make the graph acyclic. This is a well known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing. In this paper, we explore the complementary problem in directed graphs, i.e., how to construct the maximum directed acyclic graph (max-DAG). We show that the max-DAG problem is Poly-APX complete, which implies that even trying to obtain approximation algorithms for this problems is likely to be unfeasible. In light of these considerations, we introduce a new algorithmic solution, based on Simulated Annealing (SA), which combines techniques such as kernelization, efficient data-structures, novel heuristics to initialize the search process, a global re-structuring procedure, and a neighbor re-ordering technique to speed-up the local search step. We present an extensive experimental study that validates the key design and implementation choices undertaken in our proposal and compares it to state of the art SA-based solutions Galinier et al. (2013) and Tang et al. (2017). The proposed algorithm provides significant performance gains by obtaining feedback vertex sets up to 13.3× closer to the optimal solution in a wide variety of synthetic and real-world graphs.

On Minimum Feedback Vertex Sets in Bipartite Graphs and Degree-Constraint Graphs

SUMMARY We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n 2 log 6 n) time for n-vertex graphs with maximum degree at

Feedback vertex set on graphs of low clique-width

The Feedback Vertex Set problem asks whether a graph contains q vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if q vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for problems based on non-local properties are usually more complicated. In this paper, given a graph G of clique-width cw and a cw-expression of G, we solve the Minimum Feedback Vertex Set problem in time O(n22O(cwlogcw)). Our algorithm applies dynamic programming on a so-called k-module decomposition of a graph, as defined by Rao (2008) [29], which is easily derivable from ak-expression of the graph. The related notion of module-width of a graph is tightly linked to both clique-width and NLC-width, and in this paper we give an alternative equivalent characterization of module-width.

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Feedback vertex sets in mesh-based networks

In this paper, we consider the minimum feedback vertex set problem in graphs, i.e., the problem of finding a minimal cardinality subset of the vertices, whose removal makes a graph acyclic. The problem is N P -hard for general topologies, but optimal and near-optimal solutions have been provided for particular networks. In this paper, the problem is considered for undirected graphs with the following topologies: two- and higher-dimensional meshes of trees, trees of meshes, and pyramid networks. For the two-dimensional meshes of trees the results are optimal; for the higher-dimensional meshes of trees and the tree of meshes the results are asymptotically optimal. For the pyramid networks, there remains a small factor between the upper and the lower bounds.

Two feedback problems for graphs with bounded tree-width

Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree-decomposition. In this paper, it is discussed how to solve the minimum feedback vertex set problem and the minimum vertex feedback edge set problem efficiently by using dynamic programming on a tree-decomposition.

Combinatorial algorithms for feedback problems in directed graphs

Given a weighted directed graph G=( V, A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A′⊆ A such that the directed graph ( V, A⧹ A′) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.

Minimum feedback vertex set and acyclic coloring

In the feedback vertex set problem, the aim is to minimize, in a connected graph G=( V, E), the cardinality of the set V(G)⊆V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2.

Hardware-optimal test register insertion

Implementing a built-in self-test by a "test per clock" scheme offers advantages concerning fault coverage, detection of delay faults, and test application time. Such a scheme is implemented by test registers, for instance built-in logic block observers (BILBO's) and concurrent BILBO's (CBILBO's), which are inserted into the circuit structure at appropriate places. An algorithm is presented which is able to find the cost optimal placement of test registers for nearly all the ISCAS'89 sequential benchmark circuits, and a suboptimal solution with slightly higher costs is obtained for all the circuits within a few minutes of computing time. The algorithm can also be applied to the Minimum Feedback Vertex Set problem in partial scan design, and an optimal solution is found for all the benchmark circuits. The provably optimal solutions for the benchmark circuits mainly use CBILBO's which can simultaneously generate test patterns and compact test responses. Hence, test scheduling is not required, test control is simplified, and test application time is reduced.

Minimizing the number of switchboxes for region definition and ordering assignment

For a building block placement, the routing space will be fully defined into channels and switchboxes because the definition of switchboxes releases all the cycles in a channel precedence graph and further yields a safe routing ordering. However, switchbox routing is more difficult than channel routing. Due to the requirements in the routing phase, it is important for us to minimize the number of switchboxes in the definition of channels and switchboxes. In this paper, a region definition and ordering assignment (RDAOA) algorithm on minimizing the number of switchboxes is proposed. First, the routing space is modeled as a floorplan graph. According to the routing precedence in one "T" type junction, the graph is further transformed into a channel precedence graph. Hence, the problem of minimizing the number of switchboxes in region definition will correspond to the minimum feedback vertex set (MFVS) problem in the channel precedence graph. Second, based on the cyclic properties in a channel precedence graph, the MFVS problem is solved by the minimal-cycle phase and the long-cycle phase. All the minimal cycles and most of the long cycles are broken in the minimal-cycle phase, and the remaining long cycles are further broken in the long-cycle phase. As a result, fewer switchboxes are defined, and these channels and switchboxes are assigned to guarantee a safe routing ordering. The time complexity of the algorithm is proved to be in O(n) time, where n is the number of line segments in a given floorplan graph. Finally, several examples have been tested on the proposed algorithm and other published algorithms, and the experimental results show that our algorithm defines fewer switchboxes than other algorithms.

A bad example for the minimum feedback vertex set problem

Let A be a fully indecomposable n x n (0,1) matrix with S(A) nonzero entries. Let M be a minimum feedback vertex set of some strongly connected digraph corresponding to a permuted form of A with nonzero diagonal. Them M has between 1 and S(A)-2n elements. We exhibit a family of matrices, essentially unique, for which any member has permuted forms realizing each of these extremes. The significance for solving sparse linear systems is discussed.