Non-additive Functions Research Articles

Polynomial-time data reduction for weighted problems beyond additive goal functions

Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.

Non-Additive Cost Functions for JPEG Steganography Based on Block Boundary Maintenance

Recent advances show that a reasonable non-additive cost function can significantly improve the security level of additive cost based steganography. So far, there is only one principle, called block boundary continuity (BBC), that has been proposed to define the non-additive cost function for JPEG steganography, and it aims at synchronizing the modification direction of inter-block boundaries in the spatial domain. In this article, we found that JPEG steganography usually introduces more and larger modifications on the boundary than on the inside of each intra-block in the spatial domain, which is another important factor affecting security. Therefore, we present a new principle, called block boundary maintenance (BBM), to minimize the modifications on the spatial block boundaries. In theory, we deduce the BBM principle on how to modify a pair of DCT coefficients of the intra-block to reduce the modifications on the spatial block boundary. According to the BBM principle, we design a new strategy to define non-additive cost functions for JPEG steganography by exploiting the coefficient correlation of the intra-block in the DCT domain. The experimental results show that the BBM-based strategy can minimize modifications on the spatial block boundaries and thus achieve a high-security level when resisting modern JPEG steganalysis. Furthermore, the two principles of BBC and BBM can be fused to further improve the empirical security.

The distribution of sums and products of additive functions

The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if $g_1(n), \ldots, g_k(n)$ is a collection of functions satisfying certain mild hypotheses for which an Erdős--Kac-type normal distribution law holds, and if $Q(x_1, \ldots, x_k)$ is a polynomial with nonnegative real coefficients, then $Q(g_1(n), \ldots, g_k(n))$ also obeys a normal distribution law. We also show that a similar result can be obtained if the set of inputs $n$ is restricted to certain subsets of the natural numbers, such as shifted primes. Our proof uses the method of moments. We conclude by providing examples of our theorem in action.

Nonadditive strengthening functions for cold-worked cubic metals: Experiments and constitutive modeling

Strong metals are greatly desired for lightweight and energy-efficient industrial design. The strengthening of metals is traditionally accomplished by the additive contributions of various obstacle families (e.g., solid solutions, particles, and grain boundaries) and dislocation self-interactions that impede dislocation motion. In the present work, unlike a traditional additive understanding, a distinctive nonadditive strengthening mixture rule for obstacles and dislocations were validated based on experimental and modeling analyses in numerous cold-worked steels and aluminum alloys. Concretely, numerous well-annealed body-centered cubic steels and face-centered cubic aluminum alloys were prepared, in which the hierarchical strength levels of solid solutions, grain boundaries, and/or particles were estimated. The above specimens were then cold rolled to various strain levels. Dislocation densities were quantified by utilizing X-ray diffraction line-profile-analysis, and the dislocation density was found to increase faster with an increasing strain level when a high strength was presented in well-annealed specimens. When plotting the yield stresses as a function of the square roots of the dislocation densities in massive distorted samples based on the Taylor hardening law, it is interesting to note that an approximate single linearity was obtained. Individual dislocation strengthening was found to respond to the total strength in the deformed specimens, which indicated that the full nonadditive strengthening mixture rule was employed between the obstacle families and the dislocation contributions. The mechanisms of the observed nonadditive strengthening were also discussed by implementing additional experiments and transmission electron microscopy observations. Then, the modified constitutive models based on both one and two internal parameters’ Knocks-Mecking models were developed respectively, which excellently captured the effects of the various obstacle families on dislocation storage processes. The developed models also rationalized the observed nonadditive strengthening mixture rule.

Non-Additive Cost Functions for Color Image Steganography Based on Inter-Channel Correlations and Differences

Despite the strong presence of color images for communication, scholars have mainly devoted their attention to research on steganography for grayscale images. In contrast to grayscale images, color images have three interrelated color channels, and the relationships among the three channels have a strong impact on the steganography security. In this paper, we present a steganographic scheme for spatial color images by exploiting the correlations and differences between the color channels. We find that the G channel has a stronger correlation with R and B than the one between R and B, and thus, synchronizing the modification directions of the R and B channels with those from the G channel will have better resistance to detection. In addition, the payload capacity and the distribution of complex regions between channels are different. Based on these findings, we design a new strategy for defining non-additive costs for color image steganography, called G-channel-related Inter-channel Non-Additive (GINA) strategy. The GINA strategy can make the modification directions of the R and B channels consistent with those of the G channel and can adaptively distribute the embedding capacity between the three channels. Specifically, this strategy will not violate the Complexity Prior rule. The experimental results show that the proposed GINA strategy can significantly improve the performance in terms of resisting color image steganalysis compared with previous methods.

Identifying non-additive multi-attribute value functions based on uncertain indifference statements

Multi-criteria decision analysis (MCDA) requires an accurate representation of the preferences of decision-makers, for instance in the form of a multi-attribute value function. Typically, additivity or other stringent assumptions about the preferences are made to facilitate elicitation by assuming a simple parametric form. When relaxing such assumptions, parameters cannot be elicited easily with standard methods. We present a novel approach for identifying multi-attribute value functions which can have any shape. As preference information indifference statements are used that can be elicited by trade-off questions. Instead of asking one indifference statement for each pair of attributes, we ask for multiple trade-offs at different points in the attribute space. This allows inferring parameters of complex value functions despite the simplicity of the preference statements. Parameters are estimated by taking into account preference and elicitation uncertainty with a probabilistic model. Statistical inference supports identifying the most adequate preference model out of several candidate models through quantifying the uncertainty and assessing the need for non-additivity. The approach is elaborated for determining value functions by hierarchical aggregation. We apply it to an assessment of the ecological state of rivers, which is used to support environmental management decisions in Switzerland. Preference models of four experts were quantified, confirming the feasibility of the approach and the relevance of considering non-additive functions. The method suggests a promising direction for improving the representation of preferences.

Strategy-proof Cake Cutting Mechanisms for All-or-nothing Utility

The cake cutting problem must fairly allocate a divisible good among agents who have varying preferences over it. Recently, designing strategy-proof cake cutting mechanisms has caught considerable attention from AI and MAS researchers. Previous works assumed that an agent’s utility function is additive so that theoretical analysis becomes tractable. However, in practice, agents have non-additive utility functions over a resource. In this paper, we consider the all-or-nothing utility function as a representative example of non-additive utility because it can widely cover agents’ preferences for real-world resources, such as the usage of meeting rooms, time slots for computational resources, bandwidth usage, and so on. We first show the incompatibility between envy-freeness and Pareto efficiency when each agent has all-or-nothing utility. We next propose two strategy-proofmechanisms that satisfy Pareto efficiency, which are based on a serial dictatorship mechanism, at the sacrifice of envy-freeness. To address computational feasibility, we propose an approximation algorithm to find a near-optimal allocation in time polynomial in the number of agents, since the problem of finding a Pareto efficient allocation is NP-hard. As another approach that abandon Pareto efficiency, we develop anenvy-free mechanism and show that one of our serial dictatorship based mechanisms satisfies proportionality in expectation, which is a weaker definition of proportionality. Finally, we evaluate the efficiency obtained by our proposed mechanisms by computational experiments.

Security Game with Non-additive Utilities and Multiple Attacker Resources

There has been significant interest in studying security games for modeling the interplay of attacks and defenses on various systems involving critical infrastructure, financial system security, political campaigns, and civil safeguarding. However, existing security game models typically either assume additive utility functions, or that the attacker can attack only one target. Such assumptions lead to tractable analysis, but miss key inherent dependencies that exist among different targets in current complex networks. In this paper, we generalize the classical security game models to allow for non-additive utility functions. We also allow attackers to be able to attack multiple targets. We examine such a general security game from a theoretical perspective and provide a unified view. In particular, we show that each security game is equivalent to a combinatorial optimization problem over a set system ε, which consists of defender's pure strategy space. The key technique we use is based on the transformation, projection of a polytope, and the ellipsoid method. This work settles several open questions in security game domain and significantly extends the state-of-the-art of both the polynomial solvable and NP-hard class of the security game.

Security Game with Non-additive Utilities and Multiple Attacker Resources

There has been significant interest in studying security games for modeling the interplay of attacks and defenses on various systems involving critical infrastructure, financial system security, political campaigns, and civil safeguarding. However, existing security game models typically either assume additive utility functions, or that the attacker can attack only one target. Such assumptions lead to tractable analysis, but miss key inherent dependencies that exist among different targets in current complex networks. In this paper, we generalize the classical security game models to allow for non-additive utility functions. We also allow attackers to be able to attack multiple targets. We examine such a general security game from a theoretical perspective and provide a unified view. In particular, we show that each security game is equivalent to a combinatorial optimization problem over a set system ε, which consists of defender's pure strategy space. The key technique we use is based on the transformation, projection of a polytope, and the ellipsoid method. This work settles several open questions in security game domain and extends the state-of-the-art of both the polynomial solvable and NP-hard class of the security game.

On temporal aggregators and dynamic programming

This paper proposes dynamic programming tools for payoffs based on aggregating functions that depend on the current action and the future expected payoff. Some regularity properties are provided on the aggregator to establish existence, uniqueness and computation of the solution to the Bellman equation. Our setting allows to encompass and generalize many previous results based upon additive or non-additive payoff functions.

A survey on Privacy Preserving Data Aggregation Schemes in People Centric Sensing Systems and Wireless Domains

Background: The technological advances in mobile and wireless communication have paved way for a new era of large scale sensing network. Smart phones used by people are nowadays equipped with rich set of sensing capabilities. These mobile devices carried by people can be used together to form a network called people centric sensing network. Methods: Since people are the mobile custodians, privacy becomes a threat to the network. This paper focuses on a study about the various privacy preserving aggregation schemes existing in other domains of the wireless network including people centric sensing. The design goals and challenges that arise when attempting to protect the user’s sensitive data are analyzed. The techniques commonly used to achieve user’s privacy are the homomorphic property of encryption schemes, Data slicing and Mix schemes. Findings: A study about the existing privacy preserving aggregation schemes enables us to design a new method that is more suitable for people centric sensing network. The design considerations involved in developing a security and privacy framework are also identified. The privacy preserving aggregation scheme should support spatial and temporal aggregation with fault tolerance. The scheme has to be adaptable for both additive and non-additive statistical aggregation functions. Also maintaining the data integrity, data privacy and data accuracy with less computational and communication overhead is essential. Applications: Privacy protection is important for user’s participation which supports large set of cooperative sensing applications. Irrespective of various domains in wireless networks, the privacy preserving data aggregation model is discussed that is suitable for the context of the applications.

Using simulated annealing for computing cost-aware covering arrays

The configuration spaces of software systems are often too large to test exhaustively. Combinatorial interaction testing approaches, such as covering arrays, systematically sample the configuration space and test only the selected configurations. In an attempt to reduce the cost of testing, standard t-way covering arrays aim to cover all t-way combinations of option settings in a minimum number of configurations. By doing so, they simply assume that every configuration costs the same. When the cost varies from one configuration to another, however, minimizing the number of configurations is not necessarily the same as minimizing the cost. To overcome this issue, we have recently introduced cost-aware covering arrays. In a nutshell, a t-way cost-aware covering array is a standard t-way covering array that “minimizes” a given cost function modeling the actual cost of testing. In this work we develop a simulated annealing-based approach to compute cost-aware covering arrays, which takes as input a configuration space model enhanced with a cost function and computes a cost-aware covering array by using two alternating neighboring state generation strategies together with a fitness function expressed as a weighted sum of two objectives: covering all required t-way option setting combinations and minimizing the cost function. To the best of our knowledge, the proposed approach is the first approach that computes cost-aware covering arrays for general, non-additive linear cost functions with multiplicative interaction effects. We evaluate the approach both by conducting controlled experiments, in which we systematically vary the input models to study the sensitivity of the approach to various factors and by conducting experiments using real cost functions for real software systems. We also compare cost-aware covering arrays to standard covering arrays constructed by well-known algorithms and study how fast the construction costs are compensated by the cost reductions provided. Our empirical results suggest that the proposed approach is more effective and efficient than the existing approaches.

New method for assessment of melting points of organic azides using their molecular structures

Organic azides are unstable chemical compounds, which can be easily decomposed or exploded. This work introduces three novel correlations for the prediction of melting point of different types of organic azides containing alkyl azides, alkenyl azides, aryl azides, acyl azides and heteroazide compounds. These models are based on the elemental composition as additive parameter and the contribution of some specific polar groups/structural moieties as non-additive functions. The new correlations have been used for 259 different organic azide compounds including complex molecular structures. The new models were validated by internal and external validation tests where their applicability domains were defined and analyzed. Third correlation, in which the contribution of Specific Polar Groups (SPG) and Inefficient Packing Factor (IPF) beside the number of carbon, hydrogen and nitrogen atoms and the number of azide groups are considered, gives the best predictions. It shows good performances in robustness and predictive power through different validation techniques. It also provides good statistical results and more reliable predictions as compared with two complex methods of the best available computer codes.

Reliable method for safety assessment of melting points of energetic compounds

This work presents a widely applicable model for the prediction of melting points of energetic compounds including organic peroxides, organic azides, organic nitrates, polynitro arenes, polynitro heteroarenes, acyclic and cyclic nitramines, nitrate esters and nitroaliphatic compounds. This method is based on the elemental composition of an energetic compound and the contribution of some specific polar groups/structural moieties as additive and non-additive functions, respectively. The new model was applied for 288 different energetic compounds including complex molecular structures. The average and maximum deviations of the suggested method are 6.8 and 21.1, respectively, which are much lower than the predicted values of two well-known different methods.

Double Helix Value Functions, Ordinal/Cardinal Approach, Additive Utility Functions, Multiple Criteria, Decision Paradigm, Process, and Types (Z Theory I)

Z Utility Theory refers to a class of nonlinear utility functions for solving Risk and Multiple Criteria Decision-Making problems. Z utility functions are hybrids of additive and nonadditive (nonlinear) functions. This paper addresses the concepts and assessment methods for the additive part of Z-utility functions for multiple criteria problems that satisfy the efficiency (nondominancy) principle. We provide a decision paradigm and guidelines on how to approach, formulate, and solve decision-making problems. We, also, overview the modeling of decision process based on four types of decision-making styles. For multi-criteria problems, a new definition of convex efficiency is introduced. Also polyhedral efficiency is developed for presenting multi-criteria efficiency (nondominancy) graphically. New double helix quasi-linear value functions for multi-criteria are developed. Two types of double helix value functions for solving bi-criteria (Advantages versus Disadvantages) and also risk problems are introduced: Food–Fun curves for expected values and Fight-Flight curves for expected risk values. Ordinal/Cardinal Approach (OCA) for assessment of additive utility functions is developed. Simple consistency tests to determine whether the assessed utility function satisfies ordinal and/or cardinal properties are provided. We show that OCA can also be used to solve outranking problems. We provide a critique of Analytic Hierarchy Process (AHP) for assessing additive value functions and show that the developed Ordinal/Cardinal Approach overcomes the shortcomings of AHP. We also develop a unified/integrated approach for simultaneous assessment of nonlinear value and additive (multi-criteria) utility functions. These results in an additive utility function that can be concave, convex, or hybrid concave/convex based on the nonlinear value function. Finally, we show an interactive paired comparisons approach for solving nonadditive and nonlinear utility functions for bi-criteria decision-making problems. Several illustrative examples are provided. The paper provides reliable and robust approaches for modeling the utility preferences of heterogeneous economic agents in macro and micro-economics.

Reliable prediction of the condensed (solid or liquid) phase enthalpy of formation of organic energetic materials at 298 K through their molecular structures

A reliable new method is introduced to predict the condensed (solid or liquid) phase enthalpy of formation of organic energetic materials (at 298 K). This model is based on some additive and non-additive functions, which are based on elemental composition and some molecular fragments. The novel correlation is constructed on the basis of experimental data of 222 organic compounds containing different types of energetic bonds such as -O-O-, –N3, –ONO, –NO2, –ONO2 and –NNO2. For 144 compounds of this set, where the best available group additivity method can be used, the root mean square (rms) deviations of the new and group additivity methods from experimental data are 31.0 and 68.2 kJ/mol, respectively. For further 77 energetic compounds, where the computed results of quantum mechanical methods for simple organic energetic compounds were available and the best available group additivity method for some complex molecules can be applied, the rms deviations of the new, quantum mechanical (54 compounds) and group additivity (51 compounds) methods from the measured data are 35.9, 42.1 and 170.6 kJ/mol, respectively.

Resource Optimization of Non-Additive Utility Functions in Localized SC-FDMA Systems

Resource Optimization of Non-Additive Utility Functions in Localized SC-FDMA Systems

Simple approach for prediction of melting points of organic molecules containing hazardous peroxide bonds

In this work, a new and simple method is proposed to estimate the melting points of hazardous organic peroxide compounds including hydroperoxides, dialkyl peroxides, primary and secondary ozonides, peroxyacids, diacyl peroxides and alkyl peroxyesters compounds. This method can be applied for any peroxide compound with general formula CxHyOz to predict its melting point on the basis of elemental composition and specific structural moieties as additive and non-additive functions. It was applied for 104 different peroxide compounds including complex molecular structures. The predicted results give more reliable results in comparison to two of the best available methods. The average and maximum percent deviations of the new method are 6.8 and 21.1, respectively, which are lower than corresponding predicted values of Joback–Reid (24.1 and 101.9) and Jain–Yalkowsky (25.4 and 211.1).

On the absolute continuity of additive and non-additive functions

On the absolute continuity of additive and non-additive functions

On nonatomicity for non-additive functions

The notion of nonatomicity for set functions plays a key role in classical measure theory and its applications. For classical measures taking values in finite dimensional Banach spaces, it guarantees the connectedness of range. Even just replacing σ-additivity with finite additivity for measures requires some stronger nonatomicity property for the same conclusion to hold. In the present paper, we deal with non-additive functions – called ‘s-outer’ and ‘quasi-triangular’ – defined on rings and taking values in Hausdorff topological spaces. No algebraic structure is required on their target spaces. In this context, we make use of a notion of strong nonatomicity involving just the behavior of functions on ultrafilters of their underlying Boolean domains. This notion is proved to be equivalent to that proposed in earlier contributions concerning Lyapunov-types theorems in additive and non-additive frameworks. Thus, in particular, our analysis allows to generalize, improve and unify several known results on this topic.