Knapsack Constraints Research Articles

Opportunistic Mobile Data Offloading with Deadline Constraints

Due to the explosive proliferation of mobile cloud computing applications, much data needs to be transmitted between mobile users and clouds, incurring a huge traffic demand on cellular networks. Mobile offloading is a promising approach to address this challenge. In this paper, we focus on the problem of offloading many deadline-sensitive data items to some WiFi networks with capacity constraints; that is, how to schedule each data item to the WiFi networks, so that we can offload as many data items before their deadlines as possible, while taking the constraints of transmission capacity into consideration. This problem involves a probabilistic combination of multiple 0-1 knapsack constraints, which differs from existing problems. To solve this problem, we propose a greedy oFfline Data Offloading (FDO) algorithm, achieving an approximation ratio of 2. Also, we propose an oNline Data Offloading (NDO) algorithm, which has a competitive ratio of 2. Additionally, we extend our problem to a more general scenario where WiFi transmission costs are heterogeneous. We design a Heterogeneous Data Offloading (HDO) algorithm to solve the extended problem, and give its performance analysis. Finally, we demonstrate the significant performances of our algorithms through extensive simulations based on some real-world and synthetic WiFi datasets.

Single batch processing machine scheduling with two-dimensional bin packing constraints

We study the problem of minimizing the makespan for jobs on a single batch processing machine (BPM). A BMP can process several jobs as a batch simultaneously. Unlike the classic batch processing machine scheduling problems in which the capacity of a machine is modeled as one-dimensional knapsack constraints, in this research, the machine's capacity is represented by a two-dimensional rectangle and a job occupies a rectangle of its dimensions, i.e., width and height, while being processed on the machine. The processing time and the dimensions of the jobs are non-identical. A batch's processing time equals to the longest processing time of jobs in the batch. A batch is feasible only if the jobs can be placed in the machine without overlapping with each other. Arising from practical industrial applications such as advanced manufacturing and wafer fabrication, this problem is a generalization of both the classic single BPM scheduling problem and the two-dimensional bin packing problem (2D-BPP), which are both proven to be NP-hard. Thus, it is NP-hard as well. We present a mixed integer programming (MIP) model for this problem. We also develop heuristic algorithms to tackle this problem, including four single-sequence based heuristics, a biased random-key genetic algorithm (BRKGA), and a hybrid bin loading (HBL) algorithm. The performances of the MIP model and proposed algorithms are compared based on a set of random generated instances.

Robust combinatorial optimization with knapsack uncertainty

We study in this paper min max robust combinatorial optimization problems for an uncertainty polytope that is defined by knapsack constraints, either in the space of the optimization variables or in an extended space. We provide exact and approximation algorithms that extend the iterative algorithms proposed by Bertsimas and Sim (2003). We also study the limitation of the approach and point out NP-hard situations. Then, we approximate axis-parallel ellipsoids with knapsack constraints and provide an approximation scheme for the corresponding robust problem. The approximation scheme is also adapted to handle the intersection of an axis-parallel ellipsoid and a box.

A note on representations of linear inequalities in non-convex mixed-integer quadratic programs

In the literature on the quadratic 0–1 knapsack problem, several alternative ways have been given to represent the knapsack constraint in the quadratic space. We extend this work by constructing analogous representations for arbitrary linear inequalities for arbitrary non-convex mixed-integer quadratic programs with bounded variables.

Revenue maximization in incentivized social advertising

Incentivized social advertising, an emerging marketing model, provides monetization opportunities not only to the owners of the social networking platforms but also to their influential users by offering a "cut" on the advertising revenue. We consider a social network (the host) that sells ad-engagements to advertisers by inserting their ads, in the form of promoted posts, into the feeds of carefully selected "initial endorsers" or seed users: these users receive monetary incentives in exchange for their endorsements. The endorsements help propagate the ads to the feeds of their followers. Whenever any user engages with an ad, the host is paid some fixed amount by the advertiser, and the ad further propagates to the feed of her followers, potentially recursively. In this context, the problem for the host is is to allocate ads to influential users, taking into account the propensity of ads for viral propagation, and carefully apportioning the monetary budget of each of the advertisers between incentives to influential users and ad-engagement costs, with the rational goal of maximizing its own revenue. We show that, taking all important factors into account, the problem of revenue maximization in incentivized social advertising corresponds to the problem of monotone submodular function maximization, subject to a partition matroid constraint on the ads-to-seeds allocation, and submodular knapsack constraints on the advertisers' budgets. We show that this problem is NP-hard and devise two greedy algorithms with provable approximation guarantees, which differ in their sensitivity to seed user incentive costs. Our approximation algorithms require repeatedly estimating the expected marginal gain in revenue as well as in advertiser payment. By exploiting a connection to the recent advances made in scalable estimation of expected influence spread, we devise efficient and scalable versions of our two greedy algorithms. An extensive experimental assessment confirms the high quality of our proposal.

A methodology for the large-scale multi-period precedence-constrained knapsack problem: an application in the mining industry

This paper considers a generalization of the precedence-constrained knapsack problem known as multi-period precedence-constrained knapsack, in which the decision maker faces a horizon of several periods. Associated with each period is a capacity limit that cannot be exceeded by items chosen in that specific period. The motivation for studying this problem comes from a recognized problem in the mining industry, known as open pit mine production scheduling. An old, yet fast sequencing heuristic has been used in the literature to tackle similar combinatorial problems with precedence constraints. In this study, we first strengthen the LP relaxation formulation of the problem by adding inequalities derived from both precedence and knapsack constraints, and then use the LP solutions to generate efficient weights for the sequencing heuristic. Generating the heuristic's weights in this way significantly improves its output. Using this methodology, extremely large instances can be solved to near-optimum levels in minutes. The performance of this methodology is tested on a set of benchmark instances in the mining industry, where this problem is a major application.

Solving the Team Orienteering Arc Routing Problem with a column generation approach

The Team Orienteering Arc Routing Problem is a variation of the so-called Team Orienteering Problem where the customers are associated with the arcs of a directed graph. A fleet of uncapacitated vehicles, located at a depot, must give service to a set of regular customers, while another set of optional customers is available to be potentially serviced. Each optional customer generates a profit that is collected if it is serviced. Although the customers may be visited several times by one or more vehicles, its associated profit is collected at most once. The problem aims at designing vehicle routes with a time duration not longer than a pre-specified time limit, serving all regular customers and some of the optional customers to maximize the total profit collected.Very recently, a commodity-flow model and a branch-and-cut approach have been published to solve this problem. Motivated by the weak dual bounds produced by the Linear-Programing relaxation of the knapsack constraints establishing the time limit for the routes, we propose a new approach based on column generation. As a first step we transform the original customer-on-arc representation into a customer-on-vertex representation. This transformation leads to a model using binary variables rather than integer variables, thus simplifying the branching phase of our algorithm. It also simplifies the pricing phase since the contribution of a route to the objective function depends only on the customers in the route and not on their sequence. We propose a set-partitioning formulation and two column-generation algorithms.Our proposal has been shown to be useful in those cases where the previous branch-and-cut algorithm shows poor performance, that is, in those instances with a significant influence of the knapsack constraint. Thus, the column generation algorithm becomes a complementary approach to the branch-and-cut algorithm.

Approximability issues for unconstrained and constrained maximization of half-product related functions

We address the Boolean programming problem of maximizing a half-product function, with and without a linear knapsack constraint. Maximizing the half-product can be done in polynomial time. Adding a knapsack constraint makes the problem non-approximable within a constant factor, provided that the coefficients in the linear part of the function are negative. For maximizing a function with positive coefficients in the linear part we develop a fully polynomial-time approximation scheme.

Proportional cost buyback problem with weight bounds

In this paper, we study the proportional cost buyback problem. The input is a sequence of elements e1,e2,…,en, each of which has a weight w(ei). We assume that weights have an upper and a lower bound, i.e., l≤w(ei)≤u for any i. Given the ith element ei, we either accept ei or reject it with no cost, subject to some constraint on the set of accepted elements. During the iterations, we could cancel some previously accepted elements at a cost that is proportional to the total weight of them. Our goal is to maximize the profit, i.e., the sum of the weights of elements kept until the end minus the total cancellation cost occurred. We consider a matroid and the unweighted knapsack constraints. For either case, we construct optimal online algorithms and prove that they are the best possible.

Maximizing a class of submodular utility functions with constraints

Motivated by stochastic 0---1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: $$P = \big \{(w,x)\in \mathbb {R}\times \left\{ 0,1\right\} ^N: w \le f(a'x + d), b'x \le B\big \}$$P={(w,x)?R×0,1N:w≤f(a?x+d),b?x≤B} where N is a positive integer, $$f:\mathbb {R}\mapsto \mathbb {R}$$f:R?R is a concave function, $$a, b \in \mathbb {R}^N$$a,b?RN are nonnegative vectors, d is a real number and B is a positive real number. We propose a family of inequalities for the convex hull of P by exploiting submodularity of the function $$f(a'x + d)$$f(a?x+d) over $$\{0,1\}^N$${0,1}N and the knapsack constraint $$b'x \le B$$b?x≤B. Computational effectiveness of the proposed inequalities within a branch-and-cut framework is illustrated using instances of an expected utility capital budgeting problem.

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

This article considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP has been developed to solve this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is computed using the variable fixing method or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the dual function and computing the solution of the dual problem. Also, a new version of the variable fixing algorithm is developed that is convergent even when the objective Hessian is not strictly positive definite. The hybrid algorithm NAPHEAP that uses a Newton-type method (variable fixing method, secant method, or Newton's method) to bracket a root, followed by a heap-based monotone break point search, can be faster than a Newton-type method by itself, as demonstrated in the numerical experiments.

Robust efficiency measures for linear knapsack problem variants

Numerous combinatorial optimization applications, such as the mobile retailer product mix problem, utilize the multidemand, multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value. The purpose of this paper is to demonstrate that core variables and efficiency measures, concepts used in the most efficient solvers to-date for binary knapsack problems and some of its variants, can be extended to the multidemand, multidimensional knapsack problem variant. Specifically, new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated. The contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity. The applicability of these new measures is demonstrated through the development of three heuristic procedures: a Fixed-Core heuristic, a Genetic Algorithm, and a Kernel Search heuristic. The results from these tests are compared with the results from a commercial solver and an existing heuristic. The findings from these tests demonstrate that the Fixed-Core and Kernel Search heuristics developed for this paper are the most efficient solvers to-date for hard multidemand, multidimensional knapsack problems.

Knapsack constraint reformulation: A new approach that significantly reduces the number of sub-problems in the branch and bound algorithm

The paper presents a new approach to significantly reduce the number of sub-problems required to verify optimality in the branch and bound algorithm. The branch and bound algorithm is used to solve linear integer models and these models have application in areas such as scheduling, resource allocation, transportation, facility allocation and capital budgeting. The single constraint of the knapsack linear integer problem (KLIP) is reformulated in such a way that the number of standard branch and bound sub-problems required to verify optimality is significantly reduced. Computational results of the proposed approach on randomly generated KLIPs are also presented.

Linear Submodular Bandits with a Knapsack Constraint

Linear submodular bandits has been proven to be effective in solving the diversification and feature-based exploration problems in retrieval systems. Concurrently, many web-based applications, such as news article recommendation and online ad placement, can be modeled as budget-limited problems. However, the diversification problem under a budget constraint has not been considered. In this paper, we first introduce the budget constraint to linear submodular bandits as a new problem called the linear submodular bandits with a knapsack constraint. We then define an alpha-approximation unit-cost regret considering that submodular function maximization is NP-hard. To solve this problem, we propose two greedy algorithms based on a modified UCB rule. We then prove these two algorithms with different regret bounds and computational costs. We also conduct a number of experiments and the experimental results confirm our theoretical analyses.

On the mixing set with a knapsack constraint

We study a substructure appearing in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution, which we call the mixing set with a knapsack constraint. Recently, Luedtke et al. (Math. Program. 122(2):247---272, 2010) and Kucukyavuz (Math Program 132(1):31---56, 2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (when the knapsack constraint reduces to a cardinality constraint). In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case. Our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope. Moreover, they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance-constrained programs.

Maximizing expected utility over a knapsack constraint

The expected utility knapsack problem is to pick a set of items with random values so as to maximize the expected utility of the total value of the items picked subject to a knapsack constraint. We devise an approximation algorithm for this problem by combining sample average approximation and greedy submodular maximization. Our main result is an algorithm that maximizes an increasing submodular function over a knapsack constraint with an approximation ratio better than the well known (1−1/e) factor.

Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system ( U , Ω) and an edge-weighted graph G = ( U , E ) on the same universe U , find the set A ∈ Ω such that the Steiner tree cost with terminals A is as large as possible—“which set in Ω is the most difficult to connect up?” This is an example of a max-min problem : find the set A ∈ Ω such that the value of some minimization (covering) problem is as large as possible. In this article, we show that for certain covering problems that admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system Ω is given by a p -system or knapsack constraints or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p -systems and knapsack constraints.

Differential approximation schemes for half-product related functions and their scheduling applications

We address the problem of minimizing a half-product function, with and without a linear knapsack constraint. We show how to convert known fully polynomial-time approximation schemes to differential approximation schemes that handle the problems with and without an additive constant and with and without a linear knapsack constraint. Thereby, we resolve the issue of differential approximation for a range of scheduling problems.

Stochastic graph partitioning: quadratic versus SOCP formulations

We consider a variant of the graph partitioning problem involving knapsack constraints with Gaussian random coefficients. In this new variant, under this assumption of probability distribution, the problem can be traditionally formulated as a binary SOCP for which the continuous relaxation is convex. In this paper, we reformulate the problem as a binary quadratic constrained program for which the continuous relaxation is not necessarily convex. We propose several linearization techniques for latter: the classical linearization proposed by Fortet (Trabajos de Estadistica 11(2):111–118, 1960) and the linearization proposed by Sherali and Smith (Optim Lett 1(1):33–47, 2007). In addition to the basic implementation of the latter, we propose an improvement which includes, in the computation, constraints coming from the SOCP formulation. Numerical results show that an improvement of Sherali–Smith’s linearization outperforms largely the binary SOCP program and the classical linearization when investigated in a branch-and-bound approach.

A simulated annealing heuristic for the multiconstraint team orienteering problem with multiple time windows

This study proposes a simulated annealing with restart strategy (SA_RS) heuristic for the multiconstraint team orienteering problem with multiple time windows (MC-TOP-MTW), an extension of the team orienteering problem with time windows (TOPTW). A set of vertices is given in the MC-TOP-MTW. Each vertex is associated with a score, a service time, one or more time windows, and additional knapsack constraints. The goal is to maximize the total collected score using a predetermined number of tours. We develop two versions of SA_RS. The first version, SA_RSBF, uses Boltzmann function to determine the acceptance probability of a worse solution while the second version, SA_RSCF, accepts a worse solution based on the acceptance probability determined by Cauchy function. Results of the computational study indicate that both SA_RSBF and SA_RSCF can effectively solve MC-TOP-MTW. Moreover, in several cases, they find new best solutions to benchmark instances. The results also show that SA with restart strategy performs better than that without restart strategy.