Knapsack Constraints Research Articles

Improved Approximation Algorithms for Matroid and Knapsack Means Problems

Both matroid means and knapsack means are variations of the classic [Formula: see text]-means problem in which we replace the cardinality constraint by matroid constraint or knapsack constraint respectively. In this paper, we give a 64-approximation algorithm for the matroid means problem and a [Formula: see text]-approximation algorithm for the knapsack means problem by using a simpler and more efficient rounding method. We improve previous 304 approximate ratio for the former and 20016 approximate ratio for the latter. In the rounding process, the application of integrality of the intersection of submodular (or matroid) polyhedra provides strong theoretical support. Moreover, we extend this method to matroid means problem with penalties, and give 64 and 880-approximate algorithms for uniform penalties and nonuniform penalties problem.

Greedy permanent magnet optimization

A number of scientific fields rely on placing permanent magnets in order to produce a desired magnetic field. We have shown in recent work that the placement process can be formulated as sparse regression. However, binary, grid-aligned solutions are desired for realistic engineering designs. We now show that the binary permanent magnet problem can be formulated as a quadratic program with quadratic equality constraints, the binary, grid-aligned problem is equivalent to the quadratic knapsack problem with multiple knapsack constraints, and the single-orientation-only problem is equivalent to the unconstrained quadratic binary problem. We then provide a set of simple greedy algorithms for solving variants of permanent magnet optimization, and demonstrate their capabilities by designing magnets for stellarator plasmas. The algorithms can a-priori produce sparse, grid-aligned, binary solutions. Despite its simple design and greedy nature, we provide an algorithm that compares with or even outperforms the state-of-the-art algorithms while being substantially faster, more flexible, and easier to use.

Resilient Service Provisioning for Edge Computing

We study the problem of resilient service provisioning for edge computing (RSPE), i.e., how to determine a service placement strategy to maximize the expected overall utility by service provisioning, in the presence of uncertain service failures. RSPE is extremely challenging to tackle, because the explicit expression of its objective function is difficult to obtain, and it is a resilient max–min problem subject to knapsack constraints, which is unexplored so far and cannot be addressed by existing resilient optimization techniques. We first explore the potential properties of the implicit objective function, and reveal that it is monotone submodular under certain conditions. We further prove that the knapsack constraints form a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -independence system constraint, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q&gt;0$ </tex-math></inline-formula> is a constant related to the constraints. We propose two novel solutions for the general RSPE and homogeneous case, respectively. First, for the general problem, we propose a “two-step greedy” algorithm achieving a constant approximation ratio within polynomial time. Second, for the homogeneous case where one of the knapsack constraints reduces to a matroid constraint, we propose an improved “first-greedy-then-local search” polynomial time algorithm achieving better approximation ratio than the previous one. Both extensive simulations and field experiments validate the effectiveness of our proposed algorithms.

QoE-Driven Adaptive Streaming for Point Clouds

With increasing popularity of virtual reality and augmented reality, application of point clouds is in critical demand as it enables users to freely navigate in an immersive scene with six degrees of freedom. However, point clouds usually comprise large amounts of data, and are thus difficult to stream in bandwidth-constrained networks. It is therefore important, yet challenging, to efficiently stream the resource-intensive point clouds, such that the user's quality of experience (QoE) is guaranteed on a high-level but with a low bandwidth consumption. To this end, we propose a QoE-driven adaptive streaming approach for the tile-based point cloud transmission, to maximize the user's QoE while reducing the transmission redundancy. By exploiting the perspective projection, we specifically model the QoE of a 3D tile as a function of the bitrate of its representation, user's view frustum and spatial position, occlusion between tiles, and the resolution of rendering device. Based on this QoE model, we then formulate the QoE-optimized rate adaptation problem as a multiple-choice knapsack problem, which allocates bitrates for different tiles under a given transmission capacity. It is equivalently converted to a submodular function maximization problem subject to knapsack constraints, and solved by a practical greedy-based algorithm with a theoretical worst-case performance guarantee. The proposed algorithm is able to achieve a near-optimal performance, but with a very low computational complexity. Experimental results further demonstrate superiority of the proposed rate adaptation algorithm over existing schemes, in terms of both user's visual quality and transmission efficiency.

Partial-monotone adaptive submodular maximization

Many AI/Machine learning problems require adaptively selecting a sequence of items, each selected item might provide some feedback that is valuable for making better selections in the future, with the goal of maximizing an adaptive submodular function. Most of existing studies in this field focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, Golovin and Krause (J Artif Intell Res 42:427–486, 2011) developed \((1-1/e)\) approximation solution subject to a cardinality constraint. For the cardinality-constrained non-monotone case, Tang (Theor Comput Sci 850:249–261, 2021) showed that a random greedy policy attains an approximation ratio of 1/e. In this work, we generalize the above mentioned results by studying the partial-monotone adaptive submodular maximization problem. To this end, we introduce the notation of adaptive monotonicity ratio \(m\in [0,1]\) to measure the degree of monotonicity of a function. Our main result is to show that for the case of cardinality constraints, if the utility function has an adaptive monotonicity ratio of m and it is adaptive submodular, then a random greedy policy attains an approximation ratio of \(m(1-1/e)+(1-m)(1/e)\). Notably this result recovers the aforementioned \((1-1/e)\) and 1/e approximation ratios when \(m = 1\) and \(m = 0\), respectively. We further extend our results to consider a knapsack constraint and develop a \((m+1)/10\) approximation solution for this general case. One important implication of our results is that even for a non-monotone utility function, we still can attain an approximation ratio close to \((1-1/e)\) if this function is “close” to a monotone function. This leads to improved performance bounds for many machine learning applications whose utility functions are almost adaptive monotone.

Practical Budgeted Submodular Maximization

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (Operat Res Lett 32:41–43, 2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $$\alpha =1-{1}/{e} \approx 0.632$$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko (Operat Res Lett 32:41–43, 2004) impractical as it leads to a time complexity of roughly $$O(n^5)$$ (this time complexity can be slightly improved using the thresholding technique of Badanidiyuru & Vondrák (in: SODA, 1497–1514, 2014) but only to roughly $$O(n^4)$$ ). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses (and using the thresholding technique of Badanidiyuru & Vondrák (in: SODA, 1497–1514, 2014), we get the same optimal approximation ratio of $$\alpha $$ with an improved time complexity of roughly $$O(n^3)$$ . Furthermore, by making only a single guess, we get an almost as good approximation ratio of $$0.6174 > 0.9767\alpha $$ in roughly $$O(n^2)$$ time. Prior to our work, the only approximation algorithms that were known to obtain an approximation ratio close to $$\alpha $$ for BSM were the algorithm of Sviridenko (Operat Res Lett 32:41–43, 2004) and an algorithm of Ene & Nguyen (in: ICALP, 53:1–53:12, 2019) that achieves $$(\alpha -\varepsilon )$$ -approximation. However, the algorithm of Ene & Nguyen (in: ICALP, 53:1–53:12, 2019) requires $${(1/\varepsilon )}^{O(1/\varepsilon ^4)}n\log ^2 n$$ time, and hence, is of theoretical interest only since $${(1/\varepsilon )}^{O(1/\varepsilon ^4)}$$ is huge even for moderate values of $$\varepsilon $$ . In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (in: Proc ACM Meas Anal Comput Syst, 5(1): 08:1–08:22, 2021) studied a simple greedy algorithm that already has a long research history, and proved that it admits an approximation ratio of at least 0.405 (without any guesses). The last part of this paper improves over the result of Tang et al. (in: Proc ACM Meas Anal Comput Syst, 5(1): 08:1–08:22, 2021) and shows that the approximation ratio of this algorithm is within the range [0.427, 0.462].

Truthful Mechanisms for Two-Sided Markets via Prophet Inequalities

We design novel mechanisms for welfare maximization in two-sided markets. That is, there are buyers willing to purchase items and sellers holding items initially, both acting rationally and strategically in order to maximize utility. Our mechanisms are designed based on a powerful correspondence between two-sided markets and prophet inequalities. They satisfy individual rationality, dominant-strategy incentive compatibility, and budget balance constraints and give constant factor approximations to the optimal social welfare. We improve previous results in several settings. Our main focus is on matroid double auctions. Here, sellers hold identical items, and the set of buyers that obtain an item needs to be independent in a matroid. We construct two mechanisms, the first being a 1/3 approximation of the optimal social welfare-satisfying strong budget balance and requiring the agents to trade in a customized order and the second being a 1/2 approximation weakly budget balanced and able to deal with online arrival determined by an adversary. In addition, we construct constant factor approximations in two-sided markets with identical items when buyers need to fulfill a knapsack constraint. Also, in combinatorial double auctions with heterogeneous items, where buyers have valuation functions over item bundles instead of being interested in only one item, using similar techniques, we design a mechanism that is a 1/2 approximation of the optimal social welfare, is strongly budget balanced, and can deal with the online arrival of agents in an adversarial order. Funding: A. Braun was funded by the Deutsche Forschungsgemeinschaft [Grant 437739576].

Beyond pointwise submodularity: Non-monotone adaptive submodular maximization subject to knapsack and k-system constraints

Although the knapsack-constrained and k-system-constrained non-monotone adaptive submodular maximization have been well studied in the literature, it has only been settled given the additional assumption of pointwise submodularity. In this paper, we remove the common assumption on pointwise submodularity and propose the first approximation solutions for both knapsack and k-system constrained adaptive submodular maximization problems. Inspired by two recent studies on non-monotone adaptive submodular maximization, we develop a sampling-based randomized algorithm that achieves a 110 approximation ratio for the case of a knapsack constraint and that achieves a 12k+4 approximation ratio for the case of a k-system constraint.

Streaming algorithms for monotone non-submodular function maximization under a knapsack constraint on the integer lattice

The study of non-submodular maximization on the integer lattice is an important extension of submodular optimization. In this paper, streaming algorithms for maximizing non-negative monotone non-submodular functions with knapsack constraint on integer lattice are considered. We first design a two-pass StreamingKnapsack algorithm combining with BinarySearch as a subroutine for this problem. By introducing the DR ratio γ and the weak DR ratio γw of the non-submodular objective function, we obtain that the approximation ratio is min⁡{γ2(1−ε)/2γ+1,1−1/γw2γ−ε}, the total memory complexity is O(Klog⁡K/ε), and the total query complexity for each element is O(log⁡Klog⁡(K/ε2)/ε). Then, we design a one-pass streaming algorithm by dynamically updating the maximal function value among unit vectors along with the currently arriving element. Finally, in order to decrease the memory complexity, we design an improved StreamingKnapsack algorithm and reduce the memory complexity to O(K/ε2).

Timeliness-Aware Incentive Mechanism for Vehicular Crowdsourcing in Smart Cities

Vehicular crowdsourcing is a promising paradigm that takes advantage of powerful onboard capabilities of vehicles to perform various tasks in smart cities. To fulfill this vision, a well-designed incentive mechanism is essential to stimulate the participation of vehicles. In this paper, we propose a timeliness-aware incentive mechanism for vehicular crowdsourcing by taking vehicle’s uncertain travel time into account. In view of the stochastic nature of traffic conditions, we derive a tractable expression for the probability distribution of task delay based on a discrete-time traffic model. By leveraging reverse auction framework, we model the utility of a service requester as a function in terms of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uncertain</i> task delay and incurred payment. To maximize the requester’s utility under a budget constraint, we cast the mechanism design as a non-monotone submodular maximization problem over a knapsack constraint. Based on this formulation, we develop a <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</u> ruthful <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</u> udgeted <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</u> tility <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</u> aximization <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</u> uction (TBUMA), which is truthful, budget feasible, profitable, individually rational and computationally efficient. Through extensive trace-based simulations, we demonstrate the effectiveness of our proposed incentive mechanism.

A Local Search Framework for Experimental Design

We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.

Fast Adaptive Non-Monotone Submodular Maximization Subject to a Knapsack Constraint

Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern-day applications can render existing algorithms prohibitively slow. Moreover, frequently those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a 5.83-approximation and runs in O(n log n) time, i.e., at least a factor n faster than other state-of-the-art algorithms. The versatility of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a (9 + ε)-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.

A Branch-and-Cut Algorithm for Submodular Interdiction Games

Many relevant applications from diverse areas such as marketing, wildlife conservation, and defending critical infrastructure can be modeled as interdiction games. In this work, we introduce interdiction games whose objective is a monotone and submodular set function. Given a ground set of items, the leader interdicts the usage of some of the items of the follower in order to minimize the objective value achievable by the follower, who seeks to maximize a submodular set function over the uninterdicted items subject to knapsack constraints. We propose an exact branch-and-cut algorithm for this kind of interdiction game. The algorithm is based on interdiction cuts, which allow the leader to capture the follower’s objective function value for a given interdiction decision of the leader and exploit the submodularity of the objective function. We also present extensions and liftings of these cuts and discuss additional preprocessing procedures. We test our solution framework on the weighted maximal covering interdiction game and the bipartite inference interdiction game. For both applications, the improved variants of our interdiction cut perform significantly better than the basic version. For the weighted maximal covering interdiction game for which a mixed-integer bilevel linear programming (MIBLP) formulation is available, we compare the results with those of a state-of-the-art MIBLP solver. Whereas the MIBLP solver yields a minimum of 54% optimality gap within one hour, our best branch-and-cut setting solves all but four of 108 instances to optimality with a maximum of 3% gap among unsolved ones.

Multi-cover inequalities for totally-ordered multiple knapsack sets: theory and computation

We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have a number of interesting properties. First, they generalize the well-known (1, k)-configuration inequalities. Second, they are not aggregation cuts. Third, they cannot be generated as rank-1 Chvátal-Gomory cuts from the inequality system consisting of the knapsack constraints and all their minimal cover inequalities. We also provide conditions under which the inequalities are facets for the convex hull of the totally-ordered knapsack set, as well as conditions for those inequalities to fully characterize its convex hull. We give an integer program to solve the separation and provide numerical experiments that showcase the strength of these new inequalities

Submodular Maximization Subject to a Knapsack Constraint Under Noise Models

The field of Submodular Maximization subject to a Knapsack constraint has recently expanded to a variety of application domains, which is facing some challenges such as data explosions or additional conditions. There exist plenty of objective functions that cannot be evaluated exactly in many real cases unless they are estimated with errors. It leads to solving the problem under noise models. Somewhat surprisingly, Submodular Maximization subject to a Knapsack constraint under Noise models ([Formula: see text]) has never been discussed a lot before. Hence, in this paper, we consider the problem with two kinds of noise models which are addition and multiplication. Inspired by the traditional Greedy algorithm, we first propose a Greedy algorithm under Noises with provable theoretical bounds. In order to find the solution when input data are extremely large, we then devise an efficient streaming algorithm that scans only a single pass over the data and guarantees theoretical approximations. Finally, we conduct some experiments on Influence Maximization problem under knapsack constraint, an instance of [Formula: see text] to show the performances of the proposed algorithms.

Monotone k-submodular secretary problems: Cardinality and knapsack constraints

In this paper, we consider the k-submodular secretary problem, in which the items or secretaries arrive one by one in a uniformly random order, and the goal is to select k disjoint sets of items, so as to maximize the expectation of a monotone non-negative k-submodular function. A decision for each item must be made immediately and irrevocably after its arrival: accept and assign it to one of the k dimensions, or reject it. We first show that, in the unconstrained setting, there is an offline algorithm that can be transformed to a k2k−1-competitive online algorithm, which is asymptotically the best possible. For the problem with cardinality constraints, we present online algorithms with provable performance guarantees for the total size constraint and individual size constraint settings that depend on the corresponding budget parameters. For the problem under a knapsack constraint, we provide a constant-competitive online algorithm.

SDP-based bounds for graph partition via extended ADMM

We study two NP-complete graph partition problems, k-equipartition problems and graph partition problems with knapsack constraints (GPKC). We introduce tight SDP relaxations with nonnegativity constraints to get lower bounds, the SDP relaxations are solved by an extended alternating direction method of multipliers (ADMM). In this way, we obtain high quality lower bounds for k-equipartition on large instances up to n =1000 vertices within as few as 5 min and for GPKC problems up to n=500 vertices within as little as 1 h. On the other hand, interior point methods fail to solve instances from n=300 due to memory requirements. We also design heuristics to generate upper bounds from the SDP solutions, giving us tighter upper bounds than other methods proposed in the literature with low computational expense.

Combining data reduction, MIP solver and iterated local search for generalized assignment

ABSTRACT This work is motivated by a memory allocation problem (MAP) in an embedded system which is modeled as a generalized assignment problem (GAP) with side constraints. Thus, this paper aims to design a fast and sufficiently accurate method for GAP that will be used in future research to obtain a solution for MAP. Since this solution barely satisfies the conflict constraints, it will be repaired and then improved using local search . In the current research, the proposed method for GAP combines data reduction, a MIP solver, and iterated local search (ILS). The generic data reduction procedure is based on useful information provided by applying subgradient optimization to the Lagrangian relaxation of the knapsack constraints. In practice, it can eliminate up to of GAP variables. The reduced GAP left by this procedure is small and sparse. Furthermore, its optimal solution can be easily extended to an optimal or near-optimal solution of GAP. An ILS metaheuristic is designed for solving the reduced GAP and the entire method is tested on a widely used benchmark of large and difficult instances. Its comparison with recently published methods shows that it is competitive in terms of solution quality; it is also by far the fastest.

One-pass streaming algorithm for DR-submodular maximization with a knapsack constraint over the integer lattice

Due to its broad applications, maximizing a diminishing return submodular function with a knapsack constraint has been extensively studied recently. In the paper, we mainly consider this problem on the integer lattice. Assuming the optimal value is known, we first design a one-pass algorithm and prove that its approximation ratio is (1/3−ɛ). Observing the difficult of actually knowing the optimal value, we design a streaming algorithm with two passes, where in the first round we find the maximum value of the unit vector to estimate the range of the OPT. Furthermore, in order to improve the performance of the algorithm, we design an online algorithm called DynamicMRT to reduce the number of rounds, eventually achieving an approximation ratio (1/3−ɛ), a memory complexity O(KlogK∖ε) and query complexity O(log2K∖ε) per element for the knapsack constraint K.

Approximation Algorithms for Matroid and Knapsack Means Problems

In this paper, we concentrate on studying the [Formula: see text]-means problem with a matroid or a knapsack constraint. In the matroid means problem, given an observation set and a matroid, the goal is to find a center set from the independent sets to minimize the cost. By using the linear programming [Formula: see text]-rounding technology, we obtain a constant approximation guarantee. For the knapsack means problem, we adopt a similar strategy to that of matroid means problem, whereas the difference is that we add a knapsack covering inequality to the relaxed LP in order to decrease the unbounded integrality gap.