Feedback Vertex Set Research Articles

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

For the Odd Cycle Transversal problem, the task is to find a small set S of vertices in a graph that intersects every cycle of odd length. The Subset Odd Cycle Transversal problem requires S to intersect only those odd cycles that include a vertex of a distinguished vertex subset T. If we are given weights for the vertices, we ask instead that S has small weight: this is the problem Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity dichotomy for Weighted Subset Odd Cycle Transversal for graphs that do not contain a graph H as an induced subgraph. In particular, our result shows that the complexities of the weighted and unweighted variant do not align on H-free graphs, just as Papadopoulos and Tzimas showed for Subset Feedback Vertex Set.

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

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

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(q^2log (q))}cdot n^{4}, and n^{O(k^2)} where rw is the rank-width, q the {mathbb {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 XP 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.

On 3-degree 4-chordal graphs

A graph [Formula: see text] is a 3-degree 4-chordal graph if every cycle of length at least five has a chord and the degree of each vertex is at most three. In this paper, we investigate the structure of 3-degree 4-chordal graphs, which is a subclass of 3-degree graphs. We present a structural characterization based on minimal vertex separators and bi-connected components. Many classical problems are known to be NP-complete for 3-degree graphs, and 3-degree 4-chordal graphs are a nontrivial subclass of 3-degree graphs. Using our structural results, we present polynomial-time algorithms for many classical problems which are grouped into (i) subset problems (vertex cover and its variants, feedback vertex set, Steiner tree), (ii) Hamiltonicity and its variants, (iii) graph embedding problems (treewidth, minimum fill-in), and (iv) Coloring problems(rainbow coloring).

Computing subset transversals in H-free graphs

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to H-free graphs, that is, to graphs that do not contain some fixed graph H as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on H-free graphs for every graph H except when H=sP1+P4 for some s≥1. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for (sP1+P4)-free graphs for every s≥1.

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.

Maximum weighted induced forests and trees: new formulations and a computational comparative review

AbstractGiven a graph with a weight associated with each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset of the vertices such that induces a forest. This NP‐hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem (MWIT), on the other hand, requires that the subset induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch‐and‐cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial mixed integer programming solver. More specifically, five formulations are compared, two compact ones (i.e., with a polynomial number of variables and constraints) and the three others with an exponential number of constraints. The experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the benchmark instances with up to 81 vertices. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to analyze the difference between the optimal solution values of the two problems when considering different classes of graphs.

On the complexity of solution extension of optimization problems

The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal vertex covers of a graph G=(V,E), one usually arrives at the problem to decide for a vertex set U⊆V (pre-solution), if there exists a minimal vertex cover S (i.e., a vertex cover S⊆V such that no proper subset of S is a vertex cover) with U⊆S (minimal extension of U). We propose a general, partial-order based formulation of such extension problems which allows to model parameterization and approximation aspects of extension, and also highlights relationships between extension tasks for different specific problems. As examples, we study a number of specific problems which can be expressed and related in this framework.In particular, we discuss extension variants of the problems dominating set and feedback vertex/edge set. All these problems are shown to be NP-complete even when restricted to bipartite graphs of bounded degree, with the exception of our extension version of feedback edge set on undirected graphs which is shown to be solvable in polynomial time. For the extension variants of dominating and feedback vertex set, we also show NP-completeness for the restriction to planar graphs of bounded degree. As non-graph problem, we also study an extension version of the bin packing problem. We further consider the parameterized complexity of all these extension variants, where the parameter is a measure of the pre-solution as defined by our framework.

Modeling the Pancreatic Cancer Microenvironment in Search of Control Targets.

Pancreatic ductal adenocarcinoma is among the leading causes of cancer-related deaths globally due to its extreme difficulty to detect and treat. Recently, research focus has shifted to analyzing the microenvironment of pancreatic cancer to better understand its key molecular mechanisms. This microenvironment can be represented with a multi-scale model consisting of pancreatic cancer cells (PCCs) and pancreatic stellate cells (PSCs), as well as cytokines and growth factors which are responsible for intercellular communication between the PCCs and PSCs. We have built a stochastic Boolean network (BN) model, validated by literature and clinical data, in which we probed for intervention strategies that force this gene regulatory network (GRN) from a diseased state to a healthy state. To do so, we implemented methods from phenotype control theory to determine a procedure for regulating specific genes within the microenvironment. We identified target genes and molecules, such that the application of their control drives the GRN to the desired state by suppression (or expression) and disruption of specific signaling pathways that may eventually lead to the eradication of the cancer cells. After applying well-studied control methods such as stable motifs, feedback vertex sets, and computational algebra, we discovered that each produces a different set of control targets that are not necessarily minimal nor unique. Yet, we were able to gain more insight about the performance of each process and the overlap of targets discovered. Nearly every control set contains cytokines, KRas, and HER2/neu, which suggests they are key players in the system's dynamics. To that end, this model can be used to produce further insight into the complex biological system of pancreatic cancer with hopes of finding new potential targets.

Network dismantling on factor graphs: break long loops and spare local structures

A new solution framework for the task of network dismantling is recently developed, based on a two-scale bipartite factor-graph representation of the original graph where local structures are abstracted as factor nodes. This technique leads to advancement of extant dismantling algorithms, among which the belief-propagation decimation (BPD) algorithm has an efficient counterpart (factor BPD, i.e., FBPD) on the factor graph, building upon a mean-field spin-glass theory developed for the underlying long-loop feedback vertex set (FVS) problem. In this paper, I (1) demonstrate the advantage as well as disadvantage of the new factor-graph approach, and investigate the varying choice of factors, (2) show that the method can be supported by an alternative microscopic picture, and the two distinct spin-glass theories derive equivalent outcomes, whose analytical results serve as lower bounds for the FVS size on random regular factor graphs, besides (3) an extra mathematical lower bound from the result on random regular (original) graphs. Performances of graph/factor-graph algorithms are compared on various real networks. It shows empirically and analytically that the factor-graph approach does not interfere with what we could achieve without applying this technique; the new approach does a good job where traditional algorithms may perform poorly.

(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.

Long-loop feedback vertex set and dismantling on bipartite factor graphs.

Network dismantling aims at breaking a network into disconnected components and attacking vertices that intersect with many loops has proven to be a most efficient strategy. Yet existing loop-focusing methods do not distinguish the short loops within densely connected local clusters (e.g., cliques) from the long loops connecting different clusters, leading to lowered performance of these algorithms. Here we propose a new solution framework for network dismantling based on a two-scale bipartite factor-graph representation, in which long loops are maintained while local dense clusters are simplistically represented as individual factor nodes. A mean-field spin-glass theory is developed for the corresponding long-loop feedback vertex set problem. The framework allows for the advancement of various existing dismantling algorithms; we developed the new version of two benchmark algorithms BPD (which uses the message-passing equations of the spin-glass theory as the solver) and CoreHD (which is fastest among well-performing algorithms). New solvers outperform current state-of-the-art algorithms by a considerable margin on networks of various sorts. Further improvement in dismantling performance is achievable by opting flexibly the choice of local clusters.

Finding Diverse Trees, Paths, and More

Mathematical modeling is a standard approach to solve many real-world problems and diversity of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding r solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as Vertex Cover, Feedback Vertex Set, d-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we further investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph G and an integer r, the problem of computing r spanning trees of G maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.

2-Approximating Feedback Vertex Set in Tournaments

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

We study algorithmic properties of the graph class {textsc {Chordal}}{-ke}, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from {textsc {Chordal}}{-ke}. More precisely, we identify a large class of optimization problems on {textsc {Chordal}}{-ke} solvable in time 2^{{mathcal{O}}(sqrt{k}log k)}cdot n^{{mathcal{O}}(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on {textsc {Chordal}}{-ke} when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of {textsc {Chordal}}{-ke} graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on {textsc {Chordal}}{-ke} graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on {textsc {Chordal}}{-ke} graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of {textsc {Chordal}}{-ke}, namely, {textsc {Interval}}{-ke} and {textsc {Split}}{-ke} graphs.

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.

Improved Analysis of Highest-Degree Branching for Feedback Vertex Set

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning. In this paper, we prove that this empirically fast algorithm runs in \(O(3.460^k n)\) time, where k is the solution size. This improves the previous best \(O(3.619^k n)\)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf Process Lett 114:556–560, 2014. https://doi.org/10.1016/j.ipl.2014.05.001).

Parameterized complexity of fair feedback vertex set problem

Given a graph G=(V,E), a subset S⊆V(G) is said to be a feedback vertex set of G if G−S is a forest. In the Feedback Vertex Set (FVS) problem, we are given an undirected graph G, and a positive integer k, the question is whether there exists a feedback vertex set of size at most k. In this paper, we study three variants of the FVS problem: Unrestricted Fair FVS, Restricted Fair FVS, and Relaxed Fair FVS. In Unrestricted Fair FVS, we are given a graph G and a positive integer ℓ, the question is does there exist a feedback vertex set S⊆V(G) (of any size) such that for every vertex v∈V(G), v has at most ℓ neighbours in S. First, we study Unrestricted Fair FVS from different parameterizations such as treewidth, treedepth, and neighbourhood diversity and obtain several results (both tractability and intractability). Next, we study Restricted Fair FVS, where we are also given an integer k in the input and we demand the size of S to be at most k. This problem is trivially NP-complete; we show that Restricted Fair FVS when parameterized by the solution size k and the maximum degree Δ of the graph G, admits a kernel of size O(Δk). Finally, we study the Relaxed Fair FVS problem, where we want that the size of S is at most k and for every vertex v outside S, v has at most ℓ neighbours in S. We give an FPT algorithm for Relaxed Fair FVS problem running in time cknO(1), for a fixed constant c.