Computing the frustration index of a signed graph is a key to solving problems in different fields of research including social networks, physics, material science, and biology. In social networks the frustration index determines network distance from a state of structural balance. Although the definition of frustration index goes back to 1960, an exact algorithmic computation method has not yet been proposed. The main reason seems to be the complexity of computing the frustration index which is closely related to well-known NP-hard problems such as MAXCUT. New quadratic and linear binary programming models are developed to compute the frustration index exactly. We introduce several speed-up techniques involving prioritised branching, local search heuristics, and valid inequalities inferred from graph structural properties. The computational improvements achieved by implementing the speed-up techniques allow us to calculate the exact values of the frustration index by running the optimisation models in Gurobi solver. The speed-up techniques make our models capable of processing graphs with thousands of nodes and edges in seconds on inexpensive hardware. The solve time and solution quality comparison against the literature shows the superiority of our models in both random and real signed networks.
For every fixed @math , there is a polynomial-time approximation scheme for the correlation clustering problem @cite_12 . For arbitrary @math , exact @cite_46 @cite_45 and heuristic methods @cite_13 @cite_44 @cite_14 are developed based on a mixed integer programming model @cite_31 . Denoting the order of a graph by @math , exact algorithms fail for @math @cite_46 and @math @cite_5 , while greedy algorithms @cite_13 and local search heuristics @cite_44 are capable of providing good solutions for @math and @math respectively.
@cite_31 @cite_14 @cite_44 @cite_45 @cite_5 @cite_46 @cite_13 @cite_12
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We construct a deterministic fully polynomial time approximationscheme (FPTAS) for computing the total number of matchings in abounded degree graph. Additionally, for an arbitrary graph, weconstruct a deterministic algorithm for computing approximately thenumber of matchings within running time exp(O(√n log2n)),where n is the number of vertices. Our approach is based on the correlation decay technique originating in statistical physics. Previously thisapproach was successfully used for approximately counting thenumber of independent sets and colorings in some classes of graphs [1, 24, 6].Thus we add another problem to the small, but growing, class of P-complete problems for whichthere is now a deterministic FPTAS. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . Abstract The paper studies effective approximate solutions to combinatorial counting and unform generation problems. Using a technique based on the simulation of ergodic Markov chains, it is shown that, for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time. It follows that, for self-reducible structures, polynomial time randomised algorithms for counting to within factors of the form (1 + n − β ) are available either for all β ϵ R or for no β ϵ R . A substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic Markov chains, and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established. Finally, the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour. We consider the following general correlation-clustering problem [N. Bansal, A. Blum, S. Chawla, Correlation clustering, in: Proc. 43rd Annu. IEEE Symp. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250]: given a graph with real nonnegative edge weights and a 〈+〉 〈-〉 edge labelling, partition the vertices into clusters to minimize the total weight of cut 〈+〉 edges and uncut 〈-〉 edges. Thus, 〈+〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈-〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be best possible by the problem definition.Correlation clustering was introduced by [Correlation clustering, in: Proc. 43rd Annu. IEEE Syrup. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250], motivated by both document clustering and agnostic learning. They proved NP-hardness and gave constant-factor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)-approximation algorithm for the general case based on a linear-programming rounding and the "region-growing" technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r3)-approximation algorithm for Kr, r-minor-free graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APX-hard and difficult to approximate better than Θ(log n).