Increase In Difference Research Articles

Variation in labor skills and offshoring across time zones

AbstractWe check the role of time‐zone difference on offshoring of service tasks when the skill of labors varies between the partner countries. We frame a model where partner countries are located in non‐overlapping time zones, and the skill level of the partner‐country labors is lower. In our model, service production is divided into two sequential stages, and output is a supermodular function of the skill of labors and time. The problem of the producers is to choose between domestic production and offshoring. Domestic production employs high‐skill labors though the time management is inefficient. On the other hand, offshoring to a non‐overlapping time zone helps to work round the clock and reduce the cost, but the lower expertise of skilled labor lowers the output. In such a framework, we check conditions for beneficial offshoring. The condition derived in our analysis provides combinations of skill and time that can make offshoring beneficial. Results show that offshoring across time zones is beneficial even when the complexity of stages of production vary. However, only the relatively less‐critical task is offshored. We further find that availability of domestic low‐skill labor does not benefit the firm, while foreign low‐skill labor may be beneficially utilized through time‐zone exploitation.

Two-stage BP maximization under p-matroid constraint

The BP problem maximizes the sum of a suBmodular function and a suPermodular function(BP) subject to some constraints, where both functions are nonnegative and monotonic. This problem has been widely studied under the single-stage setting. In this paper, we consider a variant of the BP maximization problem. The problem is a two-stage BP maximization problem subject to a p-matroid constraint, for which we propose an approximation algorithm with constant approximation ratio parameterized by the curvatures of the two functions involved.

Interval-valued quasisupermodular function and monotone comparative statics

In this study, we consider monotone comparative statics under interval uncertainty. We introduce interval-valued supermodular functions and interval-valued quasisupermodular functions with respect to a partial order relation on intervals. In addition, we obtain some sufficient conditions for monotone comparative statics under interval uncertainty. We also apply our results to discuss the monotone comparative statics of interval game with strategic complements.

Evaluation of combinatorial optimisation algorithms for c-optimal experimental designs with correlated observations

We show how combinatorial optimisation algorithms can be applied to the problem of identifying c-optimal experimental designs when there may be correlation between and within experimental units and evaluate the performance of relevant algorithms. We assume the data generating process is a generalised linear mixed model and show that the c-optimal design criterion is a monotone supermodular function amenable to a set of simple minimisation algorithms. We evaluate the performance of three relevant algorithms: the local search, the greedy search, and the reverse greedy search. We show that the local and reverse greedy searches provide comparable performance with the worst design outputs having variance <10% greater than the best design, across a range of covariance structures. We show that these algorithms perform as well or better than multiplicative methods that generate weights to place on experimental units. We extend these algorithms to identifying modle-robust c-optimal designs.

Supermodularity in Two-Stage Distributionally Robust Optimization

In this paper, we solve a class of two-stage distributionally robust optimization problems that have the property of supermodularity. We exploit the explicit worst case expectation of supermodular functions and derive the worst case distribution for the robust counterpart. This enables us to develop an efficient method to obtain an exact optimal solution to these two-stage problems. Further, we provide a necessary and sufficient condition for checking whether any given two-stage optimization problem has the supermodularity property. We also investigate the optimality of the segregated affine decision rules when problems have the property of supermodularity. We apply this framework to several classic problems, including the multi-item newsvendor problem, the facility location problem, the lot-sizing problem on a network, the appointment-scheduling problem, and the assemble-to-order problem. Whereas these problems are typically computationally challenging, they can be solved efficiently under our assumptions. Finally, numerical examples are conducted to illustrate the effectiveness of our approach. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71971187], and Hong Kong Research Grants Council [Collaborative Research Fund (C6103-20GF), General Research Fund (14210821, 16204521, 16212419)]. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4748 .

Streaming algorithms for maximizing the difference of submodular functions and the sum of submodular and supermodular functions

Streaming algorithms for maximizing the difference of submodular functions and the sum of submodular and supermodular functions

From reflected Lévy processes to stochastically monotone Markov processes via generalized inverses and supermodularity

AbstractIt was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing, and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monotonicity results associated with a supermodular function of two coordinates of our Markov process, from which the above-mentioned monotonicity and convexity/concavity results directly follow, but now for the class of Markov processes considered rather than just reflected Lévy processes. In addition, various results for the transient case (when the Markov process is not in stationarity) are provided. The conditions imposed are natural, in that they are satisfied by various frequently used Markovian models, as illustrated by a series of examples.

Improved algorithms for non-submodular function maximization problem

The concept of submodularity finds wide applications in data science, artificial intelligence, and machine learning, providing a boost to the investigation of new ideas, innovative techniques, and creative algorithms to solve different submodular optimization problems arising from a diversity of applications. However pure submodular or supermodular problems only represent a small portion of the problems we are facing in real life applications. The main focus of this work is to consider a non-submodular function maximization problem subject to a cardinality constraint, where the objective function is the sum of a monotone γ-weakly submodular function and a supermodular function. This problem includes some previously studied problems as special cases, such as the submodular+supermodular maximization problem when γ=1, and the γ-weakly submodular function maximization problem when the supermodular function is void. We present greedy algorithms for this generalized problem under both offline and streaming models, improving existing results.

POSITIVE SKILL CLUSTERING IN ROLE‐ASSIGNMENT MATCHING MODELS

AbstractKremer and Maskin (1996, NBER Working Paper) explore matching when production involves defined roles. Despite underlying Cobb–Douglas production functions, the induced maximum production function is not supermodular (SPM), and positive sorting does not arise. This article introduces and solves a general class of role‐assignment matching models with general SPM production functions. The unique equilibrium entails a novel blend of positive sorting in the large, and locally negative sorting that I call positive clustering. I show how the equilibrium matching changes as the production function changes. In a dynamic extension, sorting, mobility, and wage inequality positively covary with changes in production across matching markets.

Optimizing network topology for average controllability

We address the problem of identifying a network topology of a networked system for maximizing a controllability measure, the average controllability under constraints on the number of links in the network. We consider networked systems consisting of subsystems with higher-order discrete-time linear time-invariant dynamics. We show that the average controllability is a monotone increasing supermodular function of a set of links in the networked system. Since maximizing such a function with cardinality constraints is an NP-hard problem, we analyze the performance guarantees obtained from the greedy algorithm for maximizing non-submodular set functions in terms of supermodular curvature. We show that the lower bound obtained for the greedy algorithm becomes trivial as the number of subsystems in the networked system increases. Hence, we propose two heuristic algorithms to solve the optimization problem and numerically demonstrate the efficiency of the proposed heuristics in terms of computational complexity and performance improvement in average controllability.

Stochastic greedy algorithms for maximizing constrained submodular + supermodular functions

SummaryThe problem of maximizing the sum of a constrained submodular and a supermodular function has many applications such as social networks, machine learning, and artificial intelligence. In this article, we study the monotone submodular + supermodular maximization problem under a cardinality constraint and a p‐system constraint, respectively. For each problem, we provide a stochastic algorithm and prove the approximation ratio of each algorithm theoretically. Since the algorithm of the latter problem can also solve the former problem, we do some numerical experiments of the two algorithms to compare the time as well as the quality of the two algorithms in solving the former problem.

Approximately Supermodular Scheduling Subject to Matroid Constraints

Control scheduling refers to the problem of assigning agents or actuators to act upon a dynamical system at specific times so as to minimize a quadratic control cost, such as the objectives of the linear-quadratic-Gaussian (LQG) or linear quadratic regulator problems. When budget or operational constraints are imposed on the schedule, this problem is in general NP-hard and its solution can therefore only be approximated even for moderately sized systems. The quality of this approximation depends on the structure of both the constraints and the objective. This article shows that greedy control scheduling is near-optimal when the constraints can be written as an intersection of matroids, algebraic structures that encode requirements such as limits on the number of agents deployed per time slot, total number of actuator uses, and duty cycle restrictions. To do so, it proves that the LQG cost function is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> -supermodular and provides new <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha /(\alpha + P)$</tex-math></inline-formula> -optimality certificates for the greedy minimization of such functions over an intersection of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula> matroids. These certificates are shown to approach the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1/(1+P)$</tex-math></inline-formula> guarantee of supermodular functions in relevant settings. These results support the use of greedy algorithms in nonsupermodular quadratic control problems as opposed to typical heuristics such as convex relaxations and surrogate figures of merit, e.g., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\log \det$</tex-math></inline-formula> of the controllability Gramian.

Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint

Arising from practical problems such as in sensor placement and influence maximization in social network, submodular and non-submodular maximization on the integer lattice has attracted much attention recently. In this work, we consider the problem of maximizing the sum of a monotone non-negative diminishing return submodular (DR-submodular) function and a supermodular function on the integer lattice subject to a cardinality constraint. By exploiting the special combinatorial structures in the problem, we introduce a decreasing threshold greedy algorithm with a binary search as its subroutine to solve the problem. To avoid introducing the diminishing return ratio and submodularity ratio of the objective function, we generalize the total curvatures of submodular functions and supermodular functions to the integer lattice version. We show that our algorithm has a constant approximation ratio parameterized by the new introduced total curvatures on integer lattice with a polynomial query complexity.

Online algorithms for BP functions maximization

BP maximization problem has many applications in machine learning and data science. It can be described as maximizing the sum of a suBmodular function and a suPermodular function (BP) under some constraints, where both functions are nonnegative and monotonic. In this paper, we consider two online cases. The first is a BP maximization problem subject to a uniform matroid constraint when the items arrive one-by-one, for which we offer an online algorithm with constant competitive ratio. The second is a BP maximization problem subject to a partition matroid constraint where each part of the partition arrives in a random order, for which we present two approximation algorithms of both constant competitive ratios, where one is randomized and the other is deterministic.

Packing of maximal independent mixed arborescences

This paper studies the following packing problem: Given a mixed graph F with vertex set V, a matroid M on a set S={s1,…,sk} along with a map π:S→V, find k mutually edge and arc-disjoint mixed arborescences T1,…,Tk in F with roots π(s1),…,π(sk), such that, for any v∈V, the set {si:v∈V(Ti)} is independent and its rank reaches the theoretical maximum. This problem was mentioned by Fortier, Király, Léonard, Szigeti and Talon in [Old and new results on packing arborescences in directed hypergraphs, Discrete Appl. Math. 242 (2018), 26-33]; Matsuoka and Tanigawa gave a solution to this in [On reachability mixed arborescence packing, Discrete Optimization 32 (2019) 1-10].In this paper, we give a new characterization for above packing problem. This new characterization is simplified to the form of finding a supermodular function that should be covered by an orientation of each strong component of a matroid-based rooted mixed graph. Our proofs come along with a polynomial-time algorithm. The technique of using components opens some new ways to explore arborescence packings.

Prioritizing transcriptomic and epigenomic experiments using an optimization strategy that leverages imputeddata.

MotivationSuccessful science often involves not only performing experiments well, but also choosing well among many possible experiments. In a hypothesis generation setting, choosing an experiment well means choosing an experiment whose results are interesting or novel. In this work, we formalize this selection procedure in the context of genomics and epigenomics data generation. Specifically, we consider the task faced by a scientific consortium such as the National Institutes of Health ENCODE Consortium, whose goal is to characterize all of the functional elements in the human genome. Given a list of possible cell types or tissue types (‘biosamples’) and a list of possible high-throughput sequencing assays, where at least one experiment has been performed in each biosample and for each assay, we ask ‘Which experiments should ENCODE perform next?’ResultsWe demonstrate how to represent this task as a submodular optimization problem, where the goal is to choose a panel of experiments that maximize the facility location function. A key aspect of our approach is that we use imputed data, rather than experimental data, to directly answer the posed question. We find that, across several evaluations, our method chooses a panel of experiments that span a diversity of biochemical activity. Finally, we propose two modifications of the facility location function, including a novel submodular–supermodular function, that allow incorporation of domain knowledge or constraints into the optimization procedure.Availability and implementationOur method is available as a Python package at https://github.com/jmschrei/kiwano and can be installed using the command pip install kiwano. The source code used here and the similarity matrix can be found at http://doi.org/10.5281/zenodo.3708538.Supplementary information Supplementary data are available at Bioinformatics online.

A short proof for stronger version of DS decomposition in set function optimization

Using a short proof, we show that every set function f can be decomposed into the difference of two monotone increasing and strictly submodular functions g and h, i.e., $$f=g-h$$ , and every set function f can also be decomposed into the difference of two monotone increasing and strictly supermodular functions g and h, i.e., $$f=g-h$$ .

A variation of DS decomposition in set function optimization

Any set function can be decomposed into the difference of two monotone nondecreasing submodular functions. This theorem plays an important role in the set function optimization theory. In this paper, we show a variation that any set function can be decomposed into the difference of two monotone nondecreasing supermodular functions. Meanwhile, we give an example in social network optimization and construct algorithmic solutions for the maximization problem of set functions with this variation of DS decomposition.

Approximate Supermodularity of Kalman Filter Sensor Selection

This work considers the problem of selecting sensors in a large scale system to minimize the error in estimating its states. More specifically, the state estimation mean-square error(MSE) and worst-case error for Kalman filtering and smoothing. Such selection problems are in general NP-hard, i.e., their solution can only be approximated in practice even for moderately large problems. Due to its low complexity and iterative nature, greedy algorithms are often used to obtain these approximations by selecting one sensor at a time choosing at each step the one that minimizes the estimation performance metric. When this metric is supermodular, this solution is guaranteed to be (1-1/e)-optimal. This is however not the case for the MSE or the worst-case error. This issue is often circumvented by using supermodular surrogates, such as the logdet, despite the fact that minimizing the logdet is not equivalent to minimizing the MSE. Here, this issue is addressed by leveraging the concept of approximate supermodularity to derive near-optimality certificates for greedily minimizing the estimation mean-square and worst-case error. In typical application scenarios, these certificates approach the (1-1/e) guarantee obtained for supermodular functions, thus demonstrating that no change to the original problem is needed to obtain guaranteed good performance.

On Submodular Search and Machine Scheduling

Suppose that some objects are hidden in a finite set S of hiding places that must be examined one by one. The cost of searching subsets of S is given by a submodular function, and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of S that finds all the objects with minimal expected cost. This problem is NP-hard, and we give an efficient combinatorial 2-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function of the completion times of subsets of jobs. We go on to give better approximations for submodular functions with low total curvature, and we give a full solution when the problem is what we call series-parallel decomposable. Next, we consider a zero-sum game between a cost-maximizing hider and a cost-minimizing searcher. We prove that the equilibrium mixed strategies for the hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.