Solving the undirected feedback vertex set problem by local search

Qin, Shao-Meng; Zhou, Haijun

doi:10.1140/epjb/e2014-50289-7

Cited by 16 publications

(18 citation statements)

References 29 publications

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“…Although the minimum FVS problem is also NP-hard, a very convenient mapping of this optimization problem to a locally constrained spin glass model was achieved in 2013 [18]. By applying the replica-symmetric mean field theory of statistical mechanics to this spin glass model, the minimum FVS sizes and hence also the minimum numbers of targeted attack nodes are quantitatively estimated for random Erdös-Renyí (ER) and random regular (RR) network ensembles [18], which are in excellent agreement with rigorously derived lower bounds [19] and simulated-annealing results [20,21]. Inspired by the spin glass mean field theory, an efficient minimum-FVS construction algorithm, belief propagation-guided decimation (BPD), was also introduced in [18], which is capable of constructing close-to-minimum feedback vertex sets for single random network instances and also for correlated networks.…”

Section: Introductionsupporting

confidence: 56%

Identifying optimal targets of network attack by belief propagation

2016

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For a network formed by nodes and undirected links between pairs of nodes, the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components. This problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin glass theory and an associated belief propagation-guided decimation (BPD) algorithm [H.-J. Zhou, Eur. Phys. J. B 86 (2013) 455]. In the present work we apply the BPD alrogithm (which has approximately linear time complexity) to the network optimal attack problem, and demonstrate that it has much better performance than a recently proposed Collective Information algorithm [F. Morone and H. A. Makse, Nature 524 (2015) 63-68] for different types of random networks and real-world network instances. The BPD-guided attack scheme often induces an abrupt collapse of the whole network, which may make it very difficult to defend.

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Section: Introductionsupporting

confidence: 56%

Identifying optimal targets of network attack by belief propagation

2016

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“…Galinier et al [226] and Qin et al [227] proposed local search algorithms to solve the FVS problem in directed and undirected networks, respectively. Considering the case of an undirected network G, Qin and Zhou [227] construct a so-called legal list formed by N ordered vertices as L = (v 1 , v 2 , · · · , v N ). For any vertex v i ∈ L, only no more than one neighbor of v i is allowed to appear in front of v i .…”

Section: Heuristic Algorithmsmentioning

confidence: 99%

Vital nodes identification in complex networks

et al. 2016

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Real networks exhibit heterogeneous nature with nodes playing far different roles in structure and function. To identify vital nodes is thus very significant, allowing us to control the outbreak of epidemics, to conduct advertisements for e-commercial products, to predict popular scientific publications, and so on. The vital nodes identification attracts increasing attentions from both computer science and physical societies, with algorithms ranging from simply counting the immediate neighbors to complicated machine learning and message passing approaches. In this review, we clarify the concepts and metrics, classify the problems and methods, as well as review the important progresses and describe the state of the art. Furthermore, we provide extensive empirical analyses to compare well-known methods on disparate real networks, and highlight the future directions. In despite of the emphasis on physics-rooted approaches, the unification of the language and comparison with cross-domain methods would trigger interdisciplinary solutions in the near future.

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“…We leave the problem of determining the exact constant c as an open problem for future research. Finally, we would like to point out that local search was experimentally applied to the FVS problem with good results [27,31]. In a certain sense, our result helps justifying for them.…”

Section: Algorithm 1 Localsearch(g )mentioning

confidence: 65%

A Simple Local Search Gives a PTAS for the Feedback Vertex Set Problem in Minor-Free Graphs

Zheng

2019

Lecture Notes in Computer Science

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We show that a simple local search gives a PTAS for the Feedback Vertex Set (FVS) problem in minor-free graphs. An efficient PTAS in minor-free graphs was known for this problem by Fomin, Lokshtanov, Raman and Sauraubh [13]. However, their algorithm is a combination of many advanced algorithmic tools such as contraction decomposition framework introduced by Demaine and Hajiaghayi [10], Courcelle's theorem [9] and the Robertson and Seymour decomposition [29]. In stark contrast, our local search algorithm is very simple and easy to implement. It keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached. Our main contribution is to show that the local optimum only differs the global optimum by (1 + ) factor. A polynomial-time approximation scheme for a minimization problem is an algorithm that, given a fixed constant > 0, runs in polynomial time and returns a solution within 1 + of optimal.2 A PTAS is efficient if the running time is of the form 2 poly(1/ ) n O(1) .By generalizing the contraction decomposition of Demaine and Hajiaghayi to minor-free graphs, Fomin, Lokshtanov, Raman and Sauraubh [13] obtained a PTAS for the FVS problem in this class of graphs. A graph is H-minor-free, or simply minor-free, if it excludes some fixed graph H as a minor. In this work, we assume that |V (H)| is a constant. We note that the class of minor-free graphs are vastly bigger than planar graphs and bounded-genus graphs. A typical example is the complete bipartite graph K 3,n which has unbounded genus but is K 5 -minor-free. In Section 5, we show that in some sense, minor-free graphs are the limit for which we are still able to obtain a PTAS for this problem.A common theme in all known algorithms is complication in both implementation and analysis. The algorithm of Kleinberg and Kumar [19] is obtained by recursively applying the planar separator theorem by Lipton and Tarjan [22] and analyzing several special cases. The algorithm by Demaine and Hajiaghayi [10] employs the primal-dual relationship of planar graphs to decompose the graphs into several bounded treewidth instances, then applies dynamic programming (DP) to solve the FVS problem on bounded treewidth graphs. DP on bounded treewidth graphs is a very strong algorithmic tool. However, the implementation details typically are quite complicated. Additionally, the NP-hardness complexity of finding a tree decomposition of minimum width in planar graphs is still a long standing open problem. The algorithm of Cohen-Addad et al. [7] for bounded-genus graphs is not simpler and has worst running time; however, it can work with nodeweighted graphs. Given the complicated nature of the algorithms for planar and bounded-genus graphs, it is not surprising that the technical level of the algorithm by Fomin, Lokshtanov, Raman and Sauraubh [13] for minor-free graphs is much higher. It uses advanced tools such as Courcelle's theorem [9] and the Robertson and Seymour decomposition [29]. We note that the decomposition of Robertson and Seymo...

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Solving the undirected feedback vertex set problem by local search

Cited by 16 publications

References 29 publications

Identifying optimal targets of network attack by belief propagation

Identifying optimal targets of network attack by belief propagation

Vital nodes identification in complex networks

A Simple Local Search Gives a PTAS for the Feedback Vertex Set Problem in Minor-Free Graphs

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