Increase In Difference Research Articles

The even split rule for (concave) symmetric supermodular functions

This paper complements Jia (2019) by proving that the even split rule is the only Pareto efficient allocation that breaks down any concave symmetric supermodular function into two supermodular functions. It further provides an alternative proof for Theorem 1 of Jia (2019), which confirms that the even split rule is necessary to ensure any symmetric supermodular function, regardless its convexity or concavity, could be divided into two supermodular functions.

Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.

Convex hull representations of special monomials of binary variables

This paper gives the convex hull representation of any monomial in n binary variables $${\mathbf {x}}$$ wherein each variable is bounded above by an auxiliary binary variable y. The convex hull form is already known when the variable y is not present, but has not been considered for this more general case. Without y, the convex hull is obtained by replacing the monomial with a continuous variable, and then enforcing $$(n+2)$$ linear inequalities to ensure that the new variable equals the monomial value at all binary realizations. Specifically, these inequalities, together with the restrictions $${\mathbf {x}} \le {\mathbf {1}}$$ , give the convex hull of the corresponding set of $$2^n$$ points in $${\mathbb {R}}^{n+1}$$ that have the new variable equal to the monomial value. With y, we show that for the case in which $$n=2$$ , an implementation of a special-structure RLT gives the convex hull, while for $$n \ge 3$$ , a different level-1 RLT implementation accomplishes the same task. In fact, the argument for $$n \ge 3$$ allows us to obtain the convex hulls of various discrete and/or continuous sets, including those associated with certain supermodular functions, symmetric multilinear monomials in continuous variables over special box constraints, and the Boolean quadric polytope.

The cone of supermodular games on finite distributive lattices

In this article we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremality and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.

Optimizing Gradual SDN Upgrades in ISP Networks

Nowadays, there is a fast-paced shift from legacy telecommunication systems to novel software-defined network (SDN) architectures that can support on-the-fly network reconfiguration, therefore, empowering advanced traffic engineering mechanisms. Despite this momentum, migration to SDN cannot be realized at once especially in high-end networks of Internet service providers (ISPs). It is expected that ISPs will gradually upgrade their networks to SDN over a period that spans several years. In this paper, we study the SDN upgrading problem in an ISP network: which nodes to upgrade and when we consider a general model that captures different migration costs and network topologies, and two plausible ISP objectives: 1) the maximization of the traffic that traverses at least one SDN node, and 2) the maximization of the number of dynamically selectable routing paths enabled by SDN nodes. We leverage the theory of submodular and supermodular functions to devise algorithms with provable approximation ratios for each objective. Using real-world network topologies and traffic matrices, we evaluate the performance of our algorithms and show up to 54% gains over state-of-the-art methods. Moreover, we describe the interplay between the two objectives; maximizing one may cause a factor of 2 loss to the other. We also study the dual upgrading problem, i.e., minimizing the upgrading cost for the ISP while ensuring specific performance goals. Our analysis shows that our proposed algorithm can achieve up to 2.5 times lower cost to ensure performance goals over state-of-the-art methods.

The even split rule in positive assortative matching

Many economic models, especially in two-sided matching literature, involve breaking down a supermodular function into two supermodular functions according to a certain allocation rule. The conventional wisdom is to invoke the Nash bargaining solution, which boils down to an even split rule within the transferable utility framework. This paper rationalizes the use of the Nash bargaining solution in a two-sided matching model by showing that the even split rule is the only allocation which allows the net benefit functions to inherit supermodularity from the joint surplus function, which is necessary to ensure positive assortative matching.

Sociality and Mobility-Based Caching Strategy for Device-to-Device Communications Underlying Heterogeneous Networks

Proactive content caching at the wireless network edge, such as users and small base stations (SBSs), is an effective way to deal with high mobile traffic. In this paper, based on the user mobility and social relationships, we investigate the optimal caching strategy in device-to-device (D2D) communications underlying heterogeneous networks, where several SBSs are within the coverage of a macrobase station (MBS). Except for SBSs, important users (IUs) hired by an operator also cache files. First, assuming that user preference, i.e., the probability distribution of different file requests of users, are unknown, we cluster users and predict each user preference based on their history file requests by fitting to the Zipf distribution. Second, we derive the closed-form expression of the average system cost by jointly considering the mobility of users and the social relationship between them. With the purpose of minimizing the average system cost, we optimize the SBSs and IU caching strategies. The optimization problem is NP-complete. To solve the problem, we demonstrate that this problem belongs to the minimization of a supermodular function over a partition matroid, and thus, we provide a locally greedy caching algorithm with an approximation ratio of 2 to obtain the sub-optimal solution in polynomial time. Finally, since the operator can reduce the system cost by hiring more IUs, requiring higher payments, we reach a tradeoff by determining the number of IUs. The simulation results show that the proposed caching strategy outperforms the traditional caching strategy, and the suboptimal solution obtained by proposing the greedy algorithm is close to the optimal solution.

SDN Controller Placement With Delay-Overhead Balancing in Wireless Edge Networks

Fog architectures at the network edge are becoming a popular research trend to provide elastic resources and services to end-users, where the processing capacity resides at the network periphery as opposed to traditional data-centers. Despite their momentum, the control plane of these architectures remains complex and challenging to implement. To enhance control capability, in this paper, we propose to use software defined networking (SDN). SDN moves the control logic off data plane devices and onto external network entities, the controllers. We provide a proof-of-concept implementation of a multi-controller edge system and measure traffic delay and overheads. The results reveal the sensitivity of delay to the location of controllers and the magnitude of inter-controller and controller-node overheads. Guided by the above, we model the problem of determining the placement of controllers in the edge network. Using linearization and supermodular function techniques, we present approximation solutions which perform close to optimal and substantially better than state-of-the-art methods. Finally, we analyze the interplay between various performance and reliability objectives.

A Robust Privacy Preserving of Multiple and Binary Attribute by Using Super Modularity with Perturbation

With the increase of digital data on servers different approach of data mining is applied for the retrieval of interesting information in decision making. A major social concern of data mining is the issue of privacy and data security. So privacy preserving mining come in existence, as it validates those data mining algorithms that do not disclose sensitive information. This work provides privacy for sensitive rules that discriminate data on the basis of community, gender, country, etc. Rules are obtained by aprior algorithm of association rule mining. Those rules which contain sensitive item set with minimum threshold value are considered as sensitive. Perturbation technique is used for the hiding of sensitive rules. The age of large database is now a big issue. So researchers try to develop a high performance platform to efficiently secure these kind of data before publishing. Here proposed work has resolve this issue of digital data security by finding the relation between the columns of the dataset which is based on the highly relative association patterns. Here use of super modularity is also done which balance the risk and utilization of the data. Experiment is done on large dataset which have all kind of attribute for implementing proposed work features. The experiments showed that the proposed algorithms perform well on large databases. It work better as the Maximum lost pattern percentage is zero a certain value of support.

Inter-Cluster Cooperation for Wireless D2D Caching Networks

Proactive wireless caching and device to device (D2D) communication have emerged as promising techniques for enhancing users’ quality of service and network performance. In this paper, we propose a new architecture for D2D caching with inter-cluster cooperation. We study a cellular network in which users cache popular files and share them with other users either in their proximity via D2D communication or with remote users using cellular transmission. We characterize the network average delay per request from a queuing perspective. Specifically, we formulate the delay minimization problem and show that it is NP-hard. Furthermore, we prove that the delay minimization problem is equivalent to the minimization of a non-increasing monotone supermodular function subject to a uniform partition matroid constraint. A computationally efficient greedy algorithm is proposed which is proven to be locally optimal within a factor $(1 - e^{-1})\approx 0.63$ of the optimum. We analyze the average per request throughput for different caching schemes and conduct the scaling analysis for the average sum throughput. We show how throughput scaling depends on video content popularity when the number of files grows asymptotically large. Simulation results show a delay reduction of 45% to 80% compared to a D2D caching system without inter-cluster cooperation.

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds’ disjoint branchings theorem, characterize digraphs comprising k disjoint branchings Bi each having a specified number μi of arcs. (B) As an extension of Ryser’s maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.

Making Bipartite Graphs DM-Irreducible

The Dulmage--Mendelsohn decomposition (or the DM-decomposition) gives a unique partition of the vertex set of a bipartite graph reflecting the structure of all the maximum matchings therein. A bipa...

Submodular Optimization for Voltage Control

Voltage instability occurs when a power system is unable to meet the reactive power demand, and is typically corrected by switching on additional reactive power devices such as capacitor banks. Real-time monitoring and communication technologies can potentially improve voltage stability by enabling the rapid detection of low voltages and the implementation of corrective actions. These corrective actions, however, will only be effective in restoring stability if they are chosen in a timely, scalable manner. In this paper, we propose a submodular optimization approach for designing a control strategy that prevents voltage instability. Our key insight is that the voltage deviation from the desired level is a supermodular function of the set of reactive power injections that are employed, leading to computationally efficient control algorithms for stabilization with provable optimality guarantees. This submodular control framework is tested on the IEEE 300-bus transmission system.

Minimización de Funciones Supermodulares en un retículo finito relativamente complementado

This work presents two discarding principles to solve the problem of the minimization of a supermodular function over a relatively complemented finite lattice. This result generalizes the one presented in [1] for the case of a supermodular function defined in the class of subsets of a given finite set.

Optimal Approximation for Submodular and Supermodular Optimization with Bounded Curvature

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.

Multivariate analysis to research innovation complementarities

It is widely recognized that orthodox economics is obsessed with econometrics tools. However, econometrics techniques have a limited capacity to deal with qualitative variables coming from surveys. This paper presents a defence of the use of statistical methods, in particular multivariate analysis, which is the overall objective of the paper. Multivariate analysis is a set of methods that can be used when the problem that arises implies multiple dependent or interdependent variables of a qualitative nature. We considered an issue in the literature to probe multivariate analysis in a particular topic, namely: the question of innovation complementarities. We analyzed the presence of complementarities between internal and external innovation activities in 257 software firms from Argentina during the period 2008–2010, comparing the consideration of the problem of complementarities with the more modern complementarity econometrical tests, super and sub modularity tests arising from diverse firm-innovation function estimations (OProbit, Tobit and Probit), with the engagement of the same issue with multiple factor analysis and cluster techniques. The results show not only that the same results obtained by the econometrical tools can be reached by multivariate analysis techniques, but also that multiple factor analysis and cluster techniques allow for better exploitation of the richness of qualitative data.

Covering symmetric supermodular functions with graph edges: A short proof of a theorem of Benczúr and Frank

In this note we give a short and relatively simple algorithmic proof of a theorem of Benczúr and Frank on covering a symmetric crossing supermodular function with a minimum number of graph edges. Our proof method also implies a deficient form of the theorem.

Innovation, complementarity, and exporting. Evidence from German manufacturing firms

This paper assesses whether there might be complementarities between different types of innovation activities (product, process and organizational) and how these effects may be linked to the likelihood that a firm will export. Complementarity is addressed through the properties of supermodular functions, and firm heterogeneity by export destination is explored. A new econometric strategy to test for pairwise complementarity in a function with three independent variables and a binary dependent variable is proposed. Exogenous and endogenous innovation variables are considered by using bootstrapping for hypothesis testing, propensity score matching and treatment effects models. The empirical analysis shows that complementarity relationships between innovation strategies are more likely to exist when firms export to multiple foreign markets.

Partition Constrained Covering of a Symmetric Crossing Supermodular Function by a Graph

We are given a symmetric crossing supermodular set function $p$ on $V$ and a partition $\mathcal{P}$ of $V$. We solve the problem of finding a graph with vertex set $V$ having edges only between the classes of $\mathcal{P}$ such that for every subset $X$ of $V$ the cut of the graph defined by $X$ contains at least $p(X)$ edges. The objective is to minimize the number of edges of the graph. This problem is a common generalization of the global edge-connectivity augmentation of a graph with partition constraints, which was solved by Bang-Jensen et al. [SIAM J. Discrete Math., 12 (1999), pp. 160--207] and the problem of covering a symmetric crossing supermodular set function solved by Benczur and Frank [Math. Program., 84 (1999), pp. 483--503]. Our problem can be considered as an abstract form of the problem of global edge-connectivity augmentation of a hypergraph with partition constraints, which was earlier solved by the authors [J. Graph Theory, 72 (2013), pp. 291--312].

Polyhedral aspects of score equivalence in Bayesian network structure learning

This paper deals with faces and facets of the family-variable polytope and the characteristic-imset polytope, which are special polytopes used in integer linear programming approaches to statistically learn Bayesian network structure. A common form of linear objectives to be maximized in this area leads to the concept of score equivalence (SE), both for linear objectives and for faces of the family-variable polytope. We characterize the linear space of SE objectives and establish a one-to-one correspondence between SE faces of the family-variable polytope, the faces of the characteristic-imset polytope, and standardized supermodular functions. The characterization of SE facets in terms of extremality of the corresponding supermodular function gives an elegant method to verify whether an inequality is SE-facet-defining for the family-variable polytope. We also show that when maximizing an SE objective one can eliminate linear constraints of the family-variable polytope that correspond to non-SE facets. However, we show that solely considering SE facets is not enough as a counter-example shows; one has to consider the linear inequality constraints that correspond to facets of the characteristic-imset polytope despite the fact that they may not define facets in the family-variable mode.