Minimum Feedback Vertex Set Research Articles

Toward Stabilizable Large-Scale Boolean Networks by Controlling the Minimal Set of Nodes

This paper studies <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">structural</i> controllers and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed</i> pinning controllers for the global stabilization of Boolean networks (BNs) by integrating the information on their network structures with nodal dynamics. The main contribution is that several computationally efficient procedures are presented to reduce the number of controlled nodes and to determine a minimal set of controlled nodes without using the brute-force searches. The primary objective is to identify a minimal set of nodes that need to be controlled in the structural controllers for the strong structural stabilization of BNs when network structures is available yet nodal dynamics are unknown. To this end, a theorem shows this minimum controlled node problem can be addressed by seeking the minimum feedback vertex set of network structure. The subsequent part concentrates on designing distributed pinning controllers that merely rely on the node-to-node information exchanges for the global stabilization of BNs with the full knowledge of the nodal dynamics. Several sufficient conditions are provided by utilizing the irreducibility and activation-inhibition network structures to reduce the conservatism. Notably, we claim that, for regulatory BNs without positive cycles, the minimal set of pinned nodes can be determined with a linear amount of time subject to the total number of logical operators in the nodal dynamics. Finally, the effectiveness of these results are validated by three case studies.

On the price of independence for vertex cover, feedback vertex set and odd cycle transversal

Let vc(G), fvs(G) and oct(G), respectively, denote the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let ivc(G), ifvs(G) or ioct(G), respectively, denote the minimum size of such a set. Similar to a recent study on the price of connectivity (Hartinger et al. EuJC 2016), we investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases). We also investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are equal to vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find a complete classification for vertex cover and almost complete classifications for feedback vertex set (subject to one open case) and odd cycle transversal (subject to three open cases).

Fixed Point Preserving Model Reduction of Boolean Networks Focusing on Complement and Absorption Laws

A Boolean network (BN) is well known as a discrete model for analysis and control of complex networks such as gene regulatory networks. Since complex networks are large-scale in general, it is important to consider model reduction. In this paper, we consider model reduction that the information on fixed points (singleton attractors) is preserved. In model reduction studied here, the interaction graph obtained from a given BN is utilized. In the existing method, the minimum feedback vertex set (FVS) of the interaction graph is focused on. The dimension of the state is reduced to the number of elements of the minimum FVS. In the proposed method, we focus on complement and absorption laws of Boolean functions in substitution operations of a Boolean function into other one. By simplifying Boolean functions, the dimension of the state may be further reduced. Through a numerical example, we present that by the proposed method, the dimension of the state can be reduced for BNs that the dimension of the state cannot be reduced by the existing method.

Maximum max-k-clique subgraphs in cactus subtree graphs

In this paper, we analyze the intersection graphs of subtrees in a cactus, called cactus subtree graphs. A max-k-clique subgraph of a graph is an induced subgraph whose maximum clique has at most [Formula: see text] vertices. We describe a polynomial time algorithm to find a maximum weight max-[Formula: see text]-clique subgraph in a cactus subtree graph when [Formula: see text] is fixed. In perfect graphs this problem is equivalent to the maximum weight [Formula: see text]-colorable subgraph problem. In many applications like scheduling production lines and communication networks on imperfect graphs, a maximum max-[Formula: see text]-clique subgraph solution is better than a maximum [Formula: see text]-colorable subgraph solution. In addition, we describe cactus subtree graphs polynomial time algorithms for recognition, maximum independent sets, maximum cliques, maximum induced bipartite graphs, maximum induced forests and minimum feedback vertex sets.

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.

Introducing lop-Kernels: A Framework for Kernelization Lower Bounds

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop -kernels. Informally, this type of kernel is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. Discrete Appl Math 196:62–71, 2015. https://doi.org/10.1016/j.dam.2014.06.001), and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on \(K_t\)-free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős–Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless \(\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}\).

Controlled generation of self-sustained oscillations in complex artificial neural networks.

Spatially distinct, self-sustained oscillations in artificial neural networks are fundamental to information encoding, storage, and processing in these systems. Here, we develop a method to induce a large variety of self-sustained oscillatory patterns in artificial neural networks and a controlling strategy to switch between different patterns. The basic principle is that, given a complex network, one can find a set of nodes-the minimum feedback vertex set (mFVS), whose removal or inhibition will result in a tree-like network without any loop structure. Reintroducing a few or even a single mFVS node into the tree-like artificial neural network can recover one or a few of the loops and lead to self-sustained oscillation patterns based on these loops. Reactivating various mFVS nodes or their combinations can then generate a large number of distinct neuronal firing patterns with a broad distribution of the oscillation period. When the system is near a critical state, chaos can arise, providing a natural platform for pattern switching with remarkable flexibility. With mFVS guided control, complex networks of artificial neurons can thus be exploited as potential prototypes for local, analog type of processing paradigms.

(In)approximability of maximum minimal FVS

We study the approximability of the NP-complete Maximum Minimal Feedback Vertex Set problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: Maximum Minimal Vertex Cover, for which the best achievable approximation ratio is n, and Upper Dominating Set, which does not admit any n1−ϵ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for Maximum Minimal Feedback Vertex Set with a ratio of O(n2/3), as well as a matching hardness of approximation bound of n2/3−ϵ, improving the previously known hardness of n1/2−ϵ. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r, produces an r-approximate solution in time nO(n/r3/2). This time-approximation trade-off is essentially tight under the ETH.

Minimal Feedback Vertex Sets in Honeycomb Networks

A vertex subset of a graph G is called a feedback vertex set if its removal results in an acyclic subgraph. The feedback vertex number (G) is the cardinality of the minimum feedback vertex set of G , which is an important parameter of interconnection network topology. In this paper, we determine the feedback numbers of Honeycomb mesh n HM and Honeycomb torus n HT , i.e. 2 ( ) 9 / 2 15n/2 4 n  HM  n   and 2 ( ) 9 / 2 9 / 2 2 n  HT  n  n  . Comparing these results with those of Zhou, we find that (H) is related to the boundary of Honeycomb network H .

Reducing graph transversals via edge contractions

For a graph invariant π, the Contraction(π) problem consists of, given a graph G and positive integers k,d, deciding whether one can contract k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied the case where π is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in H, in particular implying that Contraction(π) is co-NP-hard for fixed k=d=1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Weighted minimum feedback vertex sets and implementation in human cancer genes detection

BackgroundRecently, many computational methods have been proposed to predict cancer genes. One typical kind of method is to find the differentially expressed genes between tumour and normal samples. However, there are also some genes, for example, ‘dark’ genes, that play important roles at the network level but are difficult to find by traditional differential gene expression analysis. In addition, network controllability methods, such as the minimum feedback vertex set (MFVS) method, have been used frequently in cancer gene prediction. However, the weights of vertices (or genes) are ignored in the traditional MFVS methods, leading to difficulty in finding the optimal solution because of the existence of many possible MFVSs.ResultsHere, we introduce a novel method, called weighted MFVS (WMFVS), which integrates the gene differential expression value with MFVS to select the maximum-weighted MFVS from all possible MFVSs in a protein interaction network. Our experimental results show that WMFVS achieves better performance than using traditional bio-data or network-data analyses alone.ConclusionThis method balances the advantage of differential gene expression analyses and network analyses, improves the low accuracy of differential gene expression analyses and decreases the instability of pure network analyses. Furthermore, WMFVS can be easily applied to various kinds of networks, providing a useful framework for data analysis and prediction.

Target Set Selection on generalized pancake graphs

Target Set Selection (TSS) was initially proposed to study the problem of the spread of information, ideas or influence through a social network and had formulated many problems arising in various practical applications. We consider a particular type of graphs, namely n-dimensional m-sided pancake graph mPn, which is one class of Cayley graphs and is widely used in the symmetric interconnection networks. We establish a bound of TSS on mPn by the minimum feedback vertex set.

Recognition and Optimization Algorithms for P5-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

The Parameterised Complexity of Computing the Maximum Modularity of a Graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be textsf {NP}-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to textsf {FPT} when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

Parameterized complexity of fair deletion problems

Edge deletion problems are those where the goal is to find a subset of edges such that after its removal the graph satisfies the given graph property. Typically, we want to minimize the number of elements removed. In fair deletion problems, the objective is changed, so the maximum number of deletions in a neighborhood of a single vertex is minimized.We study the parameterized complexity of fair deletion problems concerning the structural parameters such as the tree-width, the path-width, the tree-depth, the size of minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G.We prove the W[1]-hardness of the fair FO vertex-deletion problem with respect to the combined size of the tree-depth and the minimum feedback vertex set number. Moreover, we show that there is no algorithm for fair FO vertex-deletion problem running in time no(k3), where n is the size of the graph and k is the sum of the mentioned parameters, provided that the Exponential Time Hypothesis holds.On the other hand, we present an FPT algorithm for the fair MSO edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion problem parameterized by the neighborhood diversity.

Min (a)cyclic feedback vertex sets and min ones monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS).We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on restricted versions of these problems. In particular, we define the 2-Choice version of an optimization problem to be its restriction where the optimum value is known to be either D or D+1 for some integer D, and the problem is reduced to decide which of D or D+1 is the optimum value. We show that the 2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth assignment minimize the number of variables set to true.Finally, we propose two classes of directed graphs for which Acyclic FVS is polynomially solvable, namely flow reducible graphs (for which MFVS is already known to be polynomially solvable) and C1P-digraphs (defined by an adjacency matrix with the Consecutive Ones Property).

Analysis of Critical and Redundant Vertices in Controlling Directed Complex Networks Using Feedback Vertex Sets.

Controlling complex networks through a small number of controller vertices is of great importance in wide-ranging research fields. Recently, a new approach based on the minimum feedback vertex set (MFVS) has been proposed to find such vertices in directed networks in which the target states are restricted to steady states. However, multiple MFVS configurations may exist and thus the selection of vertices may depend on algorithms and input data representations. Our attempts to address this ambiguity led us to adopt an existing approach that classifies vertices into three categories. This approach has been successfully applied to maximum matching-based and minimum dominating set-based controllability analysis frameworks. In this article, we present an algorithm as well as its implementation to compute and evaluate the critical, intermittent, and redundant vertices under the MFVS-based framework, where these three categories include vertices belonging to all MFVSs, some (but not all) MFVSs, and none of the MFVSs, respectively. The results of computational experiments using artificially generated networks and real-world biological networks suggest that the proposed algorithm is useful for identifying these three kinds of vertices for relatively large-scale networks, and that the fraction of critical and intermittent vertices is considerably small. Moreover, an analysis of the signal pathways indicates that critical and intermittent MFVSs tend to be enriched by essential genes.

A Graph-Theoretic Approach to Design of Probabilistic Boolean Networks

Abstract In this paper, a graph-theoretic design method for a probabilistic Boolean network (PBN) is proposed. A PBN is well known as a mathematical model of complex network systems such as gene regulatory networks. In Boolean networks, interactions between binary states are modeled by Boolean functions. In PBNs, Boolean functions are switched probabilistically. In this paper, after a polynomial representation of a PBN is briefly explained, a simplified representation is proposed. Here, the steady value of the expected value of the state is focused, and is characterized by a minimum feedback vertex set of an interaction graph expressing interactions between states. Using this representation, input selection and stabilization are discussed. The proposed method is demonstrated by a biological example.

Improved bounds for minimal feedback vertex sets in tournaments.

We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949n minimal FVS. This significantly improves the previously best upper bound of 1.6667n by Fomin et al. [STOC 2016] and 1.6740n by Gaspers and Mnich [J. Graph Theory 72(1):72–89, 2013]. Our new upper bound almost matches the best‐known lower bound of 21n/7≈1.5448n, due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time O(1.5949n).

TEST REGISTER INSERTION AT RTL BASED ON REDUCED BIST

Built-in self-test (BIST) method has high area overhead and long test application time. In this paper, a new BIST method is proposed at register transfer level (RTL) as a design for testability (DFT) method to modify a given RTL circuit to a reduced BIST-able RTL circuit. First, we introduce modelling method called extended R-graph to represent the register connectivity of an RTL circuit. The original register in the RTL circuit is modified into multiple input signature registers (MISRs) as test register. The selection of MISR is performed by extended minimum feedback vertex set (MFVS) algorithm that identifies a set of vertices (representing test register) which breaks all the loops of extended R-graph with minimal cost when vertices are removed. It has been proven through simulation that the proposed BIST method has lower area overhead of 32.9% on average and achieves comparably high fault coverage compared to the previous method, concurrent BIST using ITC'99 benchmark circuits.