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The study of combinatorial optimization problems with submodular objectives has attracted much attention in recent years. Such problems are important in both theory and practice because their objective functions are very general. Obtaining further improvements for many submodular maximization problems boils down to finding better algorithms for optimizing a relaxation of them known as the multilinear extension. In this work, we present an algorithm for optimizing the multilinear relaxation whose guarantee improves over the guarantee of the best previous algorithm (by Ene and Nguyen). Moreover, our algorithm is based on a new technique that is, arguably, simpler and more natural for the problem at hand. In a nutshell, previous algorithms for this problem rely on symmetry properties that are natural only in the absence of a constraint. Our technique avoids the need to resort to such properties, and thus seems to be a better fit for constrained problems.

• Knowledge of how water stress impacts the carbon and water cycles is a key uncertainty in terrestrial biosphere models. • We tested a new profit maximization model, where photosynthetic uptake of CO₂ is optimally traded against plant hydraulic function, as an alternative to the empirical functions commonly used in models to regulate gas exchange during periods of water stress. We conducted a multi-site evaluation of this model at the ecosystem scale, before and during major droughts in Europe. Additionally, we asked whether the maximum hydraulic conductance in the soil–plant continuum k max (a key model parameter which is not commonly measured) could be predicted from long-term site climate. • Compared with a control model with an empirical soil moisture function, the profit maximization model improved the simulation of evapotranspiration during the growing season, reducing the normalized mean square error by c. 63%, across mesic and xeric sites. We also showed that k max could be estimated from long-term climate, with improvements in the simulation of evapotranspiration at eight out of the 10 forest sites during drought. • Although the generalization of this approach is contingent upon determining k max, it presents a mechanistic trait-based alternative to regulate canopy gas exchange in global models.

Given a social network with diffusion probabilities as edge weights and a positive integer k, which k nodes should be chosen for initial injection of information to maximize the influence in the network? This problem is popularly known as the Social Influence Maximization Problem (SIM Problem). This is an active area of research in computational social network analysis domain, since one and half decades or so. Due to its practical importance in various domains, such as viral marketing, target advertisement and personalized recommendation, the problem has been studied in different variants, and different solution methodologies have been proposed over the years. This paper presents a survey on the progress in and around SIM Problem. At last, it discusses current research trends and future research directions as well.

Directional distance function (DDF) has been a commonly used technique for estimating efficiency and productivity over the past two decades, and the directional vector is usually predetermined in the applications of DDF. The most critical issue of using DDF remains that how to appropriately project the inefficient decision-making unit onto the production frontier along with a justified direction. This paper provides a comprehensive literature review on the techniques for selecting directional vector of the directional distance function. It begins with a brief introduction of the existing methods around the inclusion of the exogenous direction techniques and the endogenous direction techniques. The former commonly includes arbitrary direction and conditional direction techniques, while the latter involves the techniques for seeking theoretically optimized directions (i.e., direction towards the closest benchmark or indicating the largest efficiency improvement potential) and market-oriented directions (i.e., directions towards cost minimization, profit maximization, or marginal profit maximization benchmarks). The main advantages and disadvantages of these techniques are summarized, and the limitations inherent in the exogenous direction-selecting techniques are discussed. It also analytically argues the mechanism of each endogenous direction technique. The literature review is end up with a numerical example of efficiency estimation for power plants, in which most of the reviewed directions for DDF are demonstrated and their evaluation performance are compared.

Purpose This study aims to help develop “business principles for stakeholder capitalism” in two steps. First, the study defines internal logic of three theories of capitalism and two variants within each theory. Second, it examines approaches to integration into modern democratic capitalism. Treating the three theories as substitutes identifies relative strengths and weaknesses; complementarity and partial overlap approaches to integration study the institutional settings within which stakeholder capitalism operates. Empirical outcomes reflect competition between market and stakeholder businesses for participants, with institutional conditions determining the scope of collective action. Design/methodology/approach The approach aligns three typologies in a unique conceptual arrangement defining the three theories of capitalism: forms of capitalism, potential failures of each form and associated types of goods. The first method examines the internal logic of each theory of capitalism. The second draws on traditional narrative review of references documenting each theory of capitalism and variants together with modern Marxist anti-capitalism. Findings Three typologies align uniquely with the theories of capitalism, each having two variants. Both variants of stakeholder capitalism are compatible with compassionate capitalism, constitutional government or polycentric governance but not with self-interest capitalism, dictatorship or Marxism. A theory of modern democratic capitalism allocates roles for private, club and social goods with empirically variable mixes occurring across countries. Competition among different types of enterprises provides an empirical test for comparative advantages of stakeholder capitalism. Future research should consider approaches for testing the proposed conceptual scheme in practice concerning capacity to deal with grand challenges, wicked problems and black swan events. Research limitations/implications Research approach is limited to logical examination of theories and literature documentation without direct empirical confirmation. The study does not address practical implications for managers and public officials or social implications concerning private incentives, stakeholder cooperation or collective action. Originality/value Originality lies in shifting terms of debate about stakeholder capitalism from advocacy of substitute theories to understanding of its relationship to market capitalism and collective action capitalism. Value lies in explaining desirability of theoretical integration of three types of capitalism into a comprehensive framework for modern democratic capitalism.

The stochastic block model (SBM), a popular framework for studying community detection in networks, is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. The degree-corrected stochastic block model (DCSBM) is a natural extension of SBM that allows for degree heterogeneity within communities. To find the communities under DCSBM, this paper proposes a convexified modularity maximization approach, which is based on a convex programming relaxation of the classical (generalized) modularity maximization formulation, followed by a novel doubly-weighted ℓ₁-norm k-medoids procedure. We establish nonasymptotic theoretical guarantees for approximate and perfect clustering, both of which build on a new degree-corrected density gap condition. Our approximate clustering results are insensitive to the minimum degree, and hold even in sparse regime with bounded average degrees. In the special case of SBM, our theoretical guarantees match the best-known results of computationally feasible algorithms. Numerically, we provide an efficient implementation of our algorithm, which is applied to both synthetic and realworld networks. Experiment results show that our method enjoys competitive performance compared to the state of the art in the literature.

We consider the problem of maximizing a nonnegative submodular set function $f:2 perpendicular \\rightarrow {\\mathbb R}_+$ over a ground set $N$ subject to a variety of packing-type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular, when $f$ may be a nonmonotone function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully, it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize $F$ over a downward-closed polytope $P$ described by an efficient separation oracle. Previously this was known only for monotone functions. For nonmonotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints or a matroid polytope. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when $f$ is nonmonotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via linear programming duality we show that a contention resolution scheme for a constraint is related to the correlation gap of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope. Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a monotone increasing submodular function or minimize a monotone decreasing supermodular function with a bounded total curvature c . Intuitively, the parameter c represents how nonlinear a function f is: when c = 0, f is linear, while for c = 1, f may be an arbitrary monotone increasing submodular function. For the case of submodular maximization with total curvature c , we obtain a (1 − c/e )-approximation—the first improvement over the greedy algorithm of of Conforti and Cornuéjols from 1984, which holds for a cardinality constraint, as well as a recent analogous result for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and nonoblivious local search, and allows us to approximately maximize the sum of a nonnegative, monotone increasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show a (1 − c )-approximation for maximization and a 1/(1 − c )-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.

A line of recent work has analyzed the behavior of the Expectation-Maximization (EM) algorithm in the well-specified setting, in which the population likelihood is locally strongly concave around its maximizing argument. Examples include suitably separated Gaussian mixture models and mixtures of linear regressions. We consider over-specified settings in which the number of fitted components is larger than the number of components in the true distribution. Such mis-specified settings can lead to singularity in the Fisher information matrix, and moreover, the maximum likelihood estimator based on n i.i.d. samples in d dimensions can have a nonstandard O ( ( d / n ) 1 4 ) rate of convergence. Focusing on the simple setting of two-component mixtures fit to a d-dimensional Gaussian distribution, we study the behavior of the EM algorithm both when the mixture weights are different (unbalanced case), and are equal (balanced case). Our analysis reveals a sharp distinction between these two cases: in the former, the EM algorithm converges geometrically to a point at Euclidean distance of O ( ( d / n ) 1 2 ) from the true parameter, whereas in the latter case, the convergence rate is exponentially slower, and the fixed point has a much lower O ( ( d / n ) 1 4 ) accuracy. Analysis of this singular case requires the introduction of some novel techniques: in particular, we make use of a careful form of localization in the associated empirical process, and develop a recursive argument to progressively sharpen the statistical rate.

To address the problem of maximizing desired opinions in social networks, we present the Limited Opinion Maximization with Dynamic Propagation Optimization framework, which is grounded in information entropy theory. Innovatively, we introduce the concept of node expression capacity, which quantifies the uncertainty of users’ expression intentions via entropy and effectively identifies the impact of silent nodes on the propagation process. Based on this, in terms of seed node selection, we develop the Limited Opinion Maximization algorithm for multi-stage seed selection, which dynamically optimizes the seed distribution among communities through a multi-stage seeding approach. In terms of node opinion changes, we establish the LODP dynamic opinion propagation model, reconstructing the node opinion update mechanism and explicitly modeling the entropy-increasing effect of silent nodes on the information propagation path. The experimental results on four datasets show that LOMDP outperforms six baseline algorithms. Our research effectively resolves the problem of maximizing desired opinions and offers insights into the dynamics of information propagation in social networks from the perspective of entropy and information theory.