Knapsack Constraints Research Articles

New approximations for monotone submodular maximization with knapsack constraint

Given a monotone submodular set function with a knapsack constraint, its maximization problem has two types of approximation algorithms with running time O(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^2)$$\\end{document} and O(n5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^5)$$\\end{document}, respectively. With running time O(n5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^5)$$\\end{document}, the best performance ratio is 1-1/e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1-1/e$$\\end{document}. With running time O(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^2)$$\\end{document}, the well-known performance ratio is (1-1/e)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1-1/e)/2$$\\end{document} and an improved one is claimed to be (1-1/e2)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1-1/e^2)/2$$\\end{document} recently. In this paper, we design an algorithm with running O(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^2)$$\\end{document} and performance ratio 1-1/e2/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1-1/e^{2/3}$$\\end{document}, and an algorithm with running time O(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^3)$$\\end{document} and performance ratio 1/2.

Two-set inequalities for the binary knapsack polyhedra

This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.

Greedy algorithm for maximization of semi-monotone non-submodular functions with applications

The problem of maximizing submodular set functions has received increasing attention in recent years, and significant improvements have been made, particularly in relation to objective functions that satisfy monotonic submodularity. However, in practice, the objective function may not be monotonically submodular. While greedy algorithms have strong theoretical guarantees for maximizing submodular functions, their performance is barely guaranteed for non-submodular functions. Therefore, in this paper, we investigate the problem of maximizing non-monotone non-submodular functions under knapsack constraints based on the problem of infectious diseases and provides a more sophisticated analysis through the idea of segmentation. Since our definition characterizes the function more elaborately, a better bound, i.e., a tighter approximation guarantee, is achieved. Finally, we generalize the relevant results for the more general problems.

Deletion-Robust Submodular Maximization with Knapsack Constraints

Submodular maximization algorithms have found wide applications in various fields such as data summarization, recommendation systems, and active learning. In recent years, deletion-robust submodular maximization algorithms have garnered attention due to their significant implications in scenarios where some data points may be removed due to user preferences or privacy concerns, such as in recommendation systems and influence maximization. In this paper, we study the fundamental problem of submodular maximization with knapsack constraints and propose a robust streaming algorithm for it. To the best of our knowledge, our algorithm is the first to solve this problem for non-monotone submodular functions and can achieve an approximation ratio of 1/(6.82+2.63d)-ϵ under a near-optimal summary size of O(k+r), where k denotes the maximum cardinality of any feasible solution, d denotes the number of the knapsack constraints and r is the robustness parameter. For monotone submodular functions, our algorithm can achieve an approximation ratio of 1/(2+2d)-ϵ under a near-optimal summary size of O(k+r), significantly improving upon the best-known ratio of Ω((1/d-ϵ)^2). The empirical performance of our algorithm is extensively evaluated in several applications including influence maximization and recommendation systems, and the experimental results demonstrate the effectiveness of our algorithm.

Fusing Conditional Submodular GAN and Programmatic Weak Supervision

Programmatic Weak Supervision (PWS) and generative models serve as crucial tools that enable researchers to maximize the utility of existing datasets without resorting to laborious data gathering and manual annotation processes. PWS uses various weak supervision techniques to estimate the underlying class labels of data, while generative models primarily concentrate on sampling from the underlying distribution of the given dataset. Although these methods have the potential to complement each other, they have mostly been studied independently. Recently, WSGAN proposed a mechanism to fuse these two models. Their approach utilizes the discrete latent factors of InfoGAN for the training of the label models and leverages the class-dependent information of the label models to generate images of specific classes. However, the disentangled latent factor learned by the InfoGAN may not necessarily be class specific and hence could potentially affect the label model's accuracy. Moreover, the prediction of the label model is often noisy in nature and can have a detrimental impact on the quality of images generated by GAN. In our work, we address these challenges by (i) implementing a noise-aware classifier using the pseudo labels generated by the label model, (ii) utilizing the prediction of the noise-aware classifier for training the label model as well as generation of class-conditioned images. Additionally, We also investigate the effect of training the classifier with a subset of the dataset within a defined uncertainty budget on pseudo labels. We accomplish this by formalizing the subset selection problem as submodular maximization with a knapsack constraint on the entropy of pseudo labels. We conduct experiments on multiple datasets and demonstrate the efficacy of our methods on several tasks vis-a-vis the current state-of-the-art methods. Our implementation is available at https://github.com/kyrs/subpws-gan

Adaptive Influence Maximization: Adaptability via Nonadaptability

Adaptive influence maximization is an important research problem in computational social networks, which is also a typical problem in the study of adaptive processing of information and adaptive construction of objects. In this paper, we propose a new method that reduces the adaptive influence maximization problem into a nonadaptive one in a different social network, so that an adaptive optimization can be solved by those methods for nonadaptive optimization. In addition, we provide a new approximation algorithm for the submodular maximization problem with a knapsack constraint, which runs in [Formula: see text] time and has performance ratio [Formula: see text], where n is the number of nodes in the network. The ratio is better than the best known previous one with the same running time. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This research is supported in part by the National Natural Science Foundation of China [Grant U20A2068].

Addressing Heterogeneity in Federated Learning with Client Selection via Submodular Optimization

Federated learning (FL) has been proposed as a privacy-preserving distributed learning paradigm, which differs from traditional distributed learning in two main aspects: the systems heterogeneity, meaning that clients participating in training have significant differences in systems performance including CPU frequency, dataset size, and transmission power, and the statistical heterogeneity, indicating that the data distribution among clients exhibits Non-Independent Identical Distribution. Therefore, the random selection of clients will significantly reduce the training efficiency of FL. In this article, we propose a client selection mechanism considering both systems and statistical heterogeneity, which aims to improve the time-to-accuracy performance by trading off the impact of systems performance differences and data distribution differences among the clients on training efficiency. First, client selection is formulated as a combinatorial optimization problem that jointly optimizes systems and statistical performance. Then, we generalize it to a submodular maximization problem with knapsack constraint, and propose the Iterative Greedy with Partial Enumeration (IGPE) algorithm to greedily select the suitable clients. Then, the approximation ratio of IGPE is analyzed theoretically. Extensive experiments verify that the time-to-accuracy performance of the IGPE algorithm outperforms other compared algorithms in a variety of heterogeneous environments.

Fractionally Subadditive Maximization under an Incremental Knapsack Constraint with Applications to Incremental Flows

Fractionally Subadditive Maximization under an Incremental Knapsack Constraint with Applications to Incremental Flows

Approximation algorithm of maximizing non-monotone non-submodular functions under knapsack constraint

The maximization of non-negative monotone submodular functions under a certain constraint is intensively studied. However, there are few works considering the maximization of non-monotone non-submodular functions, which even might be negative. These functions also have many applications, such as optimal marketing for revenue maximization over social networks, budget allocation problems, and Epidemic transmission, etc. In our paper, we discuss the maximization of the non-monotone non-submodular functions under a knapsack constraint, which even might be negative, and explore the performance under the greedy algorithm. We obtain some improved approximate ratios, for the functions with different properties, such as the non-negative weak-monotone weak-submodular functions, and the weak-submodular weak-supermodular functions that might be negative. Moreover, we generalize the results to the functions with weak subadditivity and curvature.

Greedy+Singleton: An efficient approximation algorithm for k-submodular knapsack maximization

A k-submodular function takes k distinct, non-overlapping subsets of a ground set as input and outputs a value. It is a generalization of the well-known submodular function, which is the case when k=1 and takes a single subset as input. We study the problem of maximizing a non-negative k-submodular function under a knapsack constraint. Greedy+Singleton is an algorithm that chooses the better solution between the fully greedy solution and the best single-element solution, with query complexity and running time of O(n2k). We show that Greedy+Singleton has an approximation ratio of 0.273 for monotone functions, which improves the previous analysis of 0.158 in the literature. Moreover, we give the first analysis of Greedy+Singleton for non-monotone k-submodular functions, and prove an approximation ratio of 0.219.

Improved deterministic algorithms for non-monotone submodular maximization

Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization.In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283−o(1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1/6−ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints.

Randomized Strategies for Robust Combinatorial Optimization with Approximate Separation

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for our robust optimization problem if there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint.

Improved approximation algorithms for [formula omitted]-submodular maximization under a knapsack constraint

We investigate the problem of k-submodular maximization under a knapsack constraint over the ground set of size n. This problem finds many applications in various fields, such as multi-topic propagation, multi-sensor placement, cooperative games, etc. However, existing algorithms for the studied problem face challenges in practice as the size of instances increases in practical applications.This paper introduces three deterministic and approximation algorithms for the problem that significantly improve both the approximation ratio and query complexity of existing practical algorithms. Our first algorithm, FA, returns an approximation ratio of 1/10 within O(nk) query complexity. The second one, IFA, improves the approximation ratio to 1/4−ϵ in O(nk/ϵ) queries. The last one IFA+ upgrades the approximation ratio to 1/3−ϵ in O(nklog(1/ϵ)/ϵ) query complexity, where ϵ is an accuracy parameter. Our algorithms are the first ones that provide constant approximation ratios within only O(nk) query complexity, and the novel idea to achieve results lies in two components. Firstly, we divide the ground set into two appropriate subsets to find the near-optimal solution over these ones with O(nk) queries. Secondly, we devise algorithmic frameworks that combine the solution of the first algorithm and the greedy threshold method to improve solution quality. In addition to the theoretical analysis, we have evaluated our proposed ones with several experiments in some instances: Influence Maximization, Information Coverage Maximization, and Sensor Placement for the problem. The results confirm that our algorithms ensure theoretical quality as the cutting-edge techniques, including streaming and non-streaming algorithms, and also significantly reduce the number of queries.

Matching Mechanisms for Refugee Resettlement

Current refugee resettlement processes account for neither the preferences of refugees nor the priorities of hosting communities. We introduce a new framework for matching with multidimensional knapsack constraints that captures the (possibly multidimensional) sizes of refugee families and the capacities of communities. We propose four refugee resettlement mechanisms and two solution concepts that can be used in refugee resettlement matching under various institutional and informational constraints. Our theoretical results and simulations using refugee resettlement data suggest that preference-based matching mechanisms can improve match efficiency, respect priorities of communities, and incentivize refugees to report where they would prefer to settle. (JEL C78, D82, J15, J18)

Streaming Algorithms for Non-Submodular Functions Maximization with d-Knapsack Constraint on the Integer Lattice

We study the problem of maximizing a monotone non-submodular function under a [Formula: see text]-knapsack constraint on the integer lattice. We propose three streaming algorithms to approach this problem. We first design a two-pass [Formula: see text]-approximate algorithm with total memory complexity [Formula: see text], and total query complexity for each element [Formula: see text]. The algorithm relies on a binary search technique to determine the amount of the current elements to be added into the output solution. It also requires to have a good estimate of the optimal value, we use the maximum value of the unit standard vector which can be obtained by reading a round of data to construct a guess set of the optimal value. Then, we modify our algorithm to avoid a repetitive reading of data by dynamically update the maximum value of the unit vector along with the coming elements, and obtain a one-pass streaming algorithm with same approximate ratio. Moreover, we design an improved StreamingKnapsack algorithm to reduce the memory complexity to [Formula: see text].

User Features-Aware Content Delivery in Cache-Enabled Mobile MD2D Network

Device-to-device (D2D) communication is one of the most promising technologies for relieving the pressure of demands in the 5G mobile networks. However, due to randomness of user request, limitation of storage space and transmission capacity, it is still a challenge for channel allocation to optimize the successful delivery ratio of contents. Thus, in this paper, we first consider the link selection problem for mobile users in Multi-D2D (MD2D) based content delivery networks. We establish a content delivery utility that combines physical and social aspects, taking into account energy consumption, user mobility, and trust relationships. Second, we model the content delivery link selection as the maximum weighted matching problem. For this NP-hard problem, by relaxing the integer constraints, we propose a BnB-CDLS algorithm based on the branch-and-bound method for the proposed content delivery scheme. Furthermore, we prove that the problem can be computationally reduced to a monotone submodular problem subject to matroid and knapsack constraints, which can be solved by a greedy GA-CDLS algorithm. Numerical results show that as compared with the existing schemes, the proposed scheme can significantly improve the successful delivery ratio of contents.

Submodular Maximization under the Intersection of Matroid and Knapsack Constraints

Submodular maximization arises in many applications, and has attracted a lot of research attentions from various areas such as artificial intelligence, finance and operations research. Previous studies mainly consider only one kind of constraint, while many real-world problems often involve several constraints. In this paper, we consider the problem of submodular maximization under the intersection of two commonly used constraints, i.e., k-matroid constraint and m-knapsack constraint, and propose a new algorithm SPROUT by incorporating partial enumeration into the simultaneous greedy framework. We prove that SPROUT can achieve a polynomial-time approximation guarantee better than the state-of-the-art algorithms. Then, we introduce the random enumeration and smooth techniques into SPROUT to improve its efficiency, resulting in the SPROUT++ algorithm, which can keep a similar approximation guarantee. Experiments on the applications of movie recommendation and weighted max-cut demonstrate the superiority of SPROUT++ in practice.

Practical Parallel Algorithms for Submodular Maximization Subject to a Knapsack Constraint with Nearly Optimal Adaptivity

Submodular maximization has wide applications in machine learning and data mining, where massive datasets have brought the great need for designing efficient and parallelizable algorithms. One measure of the parallelizability of a submodular maximization algorithm is its adaptivity complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an (8+epsilon)-approximation under O(log n) adaptive complexity, which is optimal up to a factor of O(loglog n). Moreover, under slightly larger adaptivity, we also propose approximation algorithms with nearly optimal query complexity of O(n), while achieving better approximation ratios. We show that our algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.

Nonrobust Strong Knapsack Cuts for Capacitated Location Routing and Related Problems

“Nonrobust Strong Knapsack Cuts for Capacitated Location Routing and Related Problems,” by Liguori et al., presents a novel BCP algorithm for the CLRP and for other problems that share a nested knapsack structure. It outperforms existing exact algorithms in the literature, making it a powerful tool for solving instances with a large number of depot locations. A key methodological contribution is the introduction of RLKCs, a family of nonrobust cuts derived from the “outer” knapsack constraints. These cuts are strong in the sense that they contain all facets of the master knapsack polytope, dominating the cover cuts by Dabia et al. (2019) . By exploring their monotonicity and superadditivity properties, it is possible to adapt the labeling algorithm for handling RLKCs efficiently. The overall positive impact of RLKCs on the BCP performance varies depending on the problem and instance characteristics, but they have proven particularly effective for CLRP instances with tight depot capacities, making the final BCP algorithm more reliable.

Approximating single- and multi-objective nonlinear sum and product knapsack problems

We present a fully polynomial-time approximation scheme (FPTAS) for a very general version of the well-known knapsack problem. This generalization covers, with few exceptions, all versions of knapsack problems that have been studied in the literature so far and allows for an objective function consisting of sums or products of possibly nonlinear, separable item profits, while the knapsack constraint states an upper bound on the sum of possibly nonlinear, separable item weights. Moreover, we extend our FPTAS to a multi-objective fully polynomial-time approximation scheme (MFPTAS) for the multi-objective version of the problem.As applications of our general algorithms, we obtain the first FPTAS for the recently-introduced 0–1 time-bomb knapsack problem as well as FPTASs for a variety of robust knapsack problems. Moreover, we extend our FPTAS to the minimization version of our general problem, which, in particular, allows us to explicitly state an FPTAS for the classical minimization knapsack problem, which has been missing in the literature so far.