Bilevel Knapsack Problem Research Articles

A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints

A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints

Models, Algorithms and Approximation Results for a Bi-Level Synchronized Knapsack Problem

We describe here a bi-level Knapsack problem which involves an Eater/Feeder interaction between 2 Knapsack systems and expresses the collaboration between a local resource provider and a consumer. We first describe the SFEK: Synchronized Feeder/Eater Knapsack problem in an accurate way, and set an ILP model, enhanced with additional valid cuts. Next we focus on the complexity issue. We design a Dynamic Programming algorithm which allows us to state that SFEK is pseudo-polynomial and derive for this problem a polynomial time approximation scheme. We conclude with a brief discussion about the Collaborative issue.

Exact methods for discrete {varGamma }-robust interdiction problems with an application to the bilevel knapsack problem

Developing solution methods for discrete bilevel problems is known to be a challenging task—even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a varGamma -robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone varGamma -robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the varGamma -robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches.

Solution techniques for Bi-level Knapsack Problems

Traditional funding mechanisms for healthcare projects involve ranking the projects and awarding funds based on their cost to benefit ratio. An alternative funding mechanism based on Bi-level programming was proposed in the literature. We refer to this as Donor-Recipient Bi-level Knapsack Problem (DR-BKP), which we explore further in this work. There are two participants, a leader (a donor agency) and a follower (recipient country) in this problem. Both the participants have their individual budgets. There is a set of projects, each having a certain cost and profit associated. The cost of projects are common to both the participants however the profits can be different for them. There is an external project that is of exclusive interest to the follower. The leader decides on cost subsidies to provide for the projects that is within her budget, while the follower solves a knapsack problem with the cost subsidised projects and the external project. Two enumerative algorithms were proposed in the literature for Bi-level problems with discrete upper level variables. We adapt them for DR-BKP that has continuous upper level variables having non-linear interaction with lower level variables. We first show the existence of a solution for DR-BKP and show the convergence of these algorithms. We provide evidence for Σ2P-hardness by showing that the problem is both NP-hard and Co-NP hard. Finally, we have implemented these two enumerative algorithms and shared the results and analyses of the computational experiments. A set of fifteen differing data sets each having randomly generated 10 instances have been solved to evaluate the performance of the proposed algorithms.

A faster algorithm for the continuous bilevel knapsack problem

We construct a fast algorithm with time complexity O(nlogn) for a continuous bilevel knapsack problem with interdiction constraints for n items. This improves on a recent algorithm from the literature with quadratic time complexity O(n2).

On the Stackelberg knapsack game

In this work we consider a bilevel knapsack problem, in which one player, the follower, decides on the optimal utilization of a bounded resource. The second player, the leader, can offer incentives, or shared profits, so that the follower chooses options attractive also for the leader. Formally, each of the two players is associated with a subset of the knapsack items. The leader may offer profits for its items as incentives to the follower, before the follower selects a subset of all items in order to maximize its overall profit. The leader receives as pay-off only the profits from those of its items that are included by the follower in the overall knapsack solution. This pay-off is then reduced by the profits offered to the follower. The resulting setting is a Stackelberg strategic game. The leader has to resolve the trade-off between offering high profits as incentives to the follower and offering low profits to gain high pay-offs.We analyze the problem for the case in which the follower solves the resulting knapsack problem to optimality and obtain a number of strong complexity results. Then we invoke a common assumption of the literature, namely that the follower’s computing power is bounded. Under this condition, we study several natural Greedy-type heuristics applied by the follower. The solution structure and complexity of the resulting problems are investigated and solution strategies are derived, in particular an Integer Linear Programming (ILP) model, but also pseudopolynomial and polynomial algorithms, when possible.

Uncertain bilevel knapsack problem based on an improved binary wolf pack algorithm

To address indeterminism in the bilevel knapsack problem, an uncertain bilevel knapsack problem (UBKP) model is proposed. Then, an uncertain solution for UBKP is proposed by defining the $${\mathcal{P}_E}$$ Nash equilibrium and $${\mathcal{P}_E}$$ Stackelberg-Nash equilibrium. To improve the computational efficiency of the uncertain solution, an evolutionary algorithm, the improved binary wolf pack algorithm, is constructed with one rule (wolf leader regulation), two operators (invert operator and move operator), and three intelligent behaviors (scouting behavior, intelligent hunting behavior, and upgrading). The UBKP model and the $${\mathcal{P}_E}$$ uncertain solution are applied to an armament transportation problem as a case study.

On Bilevel Optimization with Inexact Follower

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

An exact approach for the bilevel knapsack problem with interdiction constraints and extensions

We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0–1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower’s problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval min–max regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature.

A Stackelberg knapsack game with weight control

We address a bilevel knapsack problem where a set of items with weights and profits is given. One player, the leader, may control the weights of a given subset of items. The second player, the follower, outputs the actual solution of the resulting knapsack instance, maximizing the overall profit. The leader receives as payoff the weights from those items of its associated subset that were included in the solution chosen by the follower.We analyze the leader's payoff maximization problem for three different solution strategies of the follower and discuss the complexity of the corresponding problems. In particular, we show that, when the follower adopts a greedy strategy, setting the optimal weight values is NP-hard. Also, it is NP-hard to provide a solution within a constant factor of the best possible solution. However, a MIP-formulation can be given. Moreover, the truncated greedy strategy allows an easy answer for the revision of weights. For the additional case, in which the follower faces a continuous (linear relaxation) version of the above problems, the optimal strategies can be fully characterized and computed in polynomial time.

On exact solution approaches for bilevel quadratic 0–1 knapsack problem

We consider the bilevel quadratic knapsack problem (BQKP) that model settings where a leader appropriates a budget for a follower, who solves a quadratic 0–1 knapsack problem. BQKP generalizes the bilevel knapsack problem introduced by Dempe and Richter (Cent Eur J Oper Res 8(2):93–107, 2000), where the follower solves a linear 0–1 knapsack problem. We present an exact-solution approach for BQKP based on extensions of dynamic programs for QKP bounds and the branch-and-backtrack algorithm. We compare our approach against a two-phase method computed using an optimization solver in both phases: it first computes the follower’s value function for all feasible leader’s decisions, and then solves a single-level, value-function reformulation of BQKP as a mixed-integer quadratically constrained program. Our computational experiments show that our approach for solving BQKP outperforms the two-phase method computed by a commercial state-of-the-art optimization software package.

On topology optimization and canonical duality method

Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other artificial technique, the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly addressed. A brief review on the canonical duality theory for modeling multi-scale complex systems and for solving general nonconvex/discrete problems are given in Appendix. This paper demonstrates a simple truth: elegant designs come from correct model and theory.

A polynomial algorithm for a continuous bilevel knapsack problem

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity O(n2).

Uncertain bilevel knapsack problem and its solution

Uncertain bilevel knapsack problem and its solution

Improved approximation algorithms for a bilevel knapsack problem

We study the Stackelberg/bilevel knapsack problem as proposed by Chen and Zhang [1]: Consider two agents, a leader and a follower. Each has his own knapsack. (Knapsack capacities are possibly different.) As usual, there is a set of items i=1,…,n of given weights wi and profits pi. It is allowed to pack item i into both knapsacks, but in this case the corresponding profit for each player becomes pi+ai, where ai is a given (positive or negative) number. The objective is to find a packing for the leader such that the total profit of the two knapsacks is maximized, assuming that the follower acts selfishly. We present tight approximation algorithms for all settings considered in [1].

A Study on the Computational Complexity of the Bilevel Knapsack Problem

We analyze the computational complexity of three fundamental variants of the bilevel knapsack problem. All three variants are shown to be complete for the second level of the polynomial hierarchy. We also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack problem are encoded in unary: two of the considered bilevel variants become solvable in polynomial time, whereas the third becomes NP-complete. Furthermore, we design a polynomial time approximation scheme for this third variant, whereas the other two variants cannot be approximated in polynomial time within any constant factor (assuming P\;$\ne$\;NP).

One-level reformulation of the bilevel Knapsack problem using dynamic programming

In this paper, we consider the Bilevel Knapsack Problem (BKP), which is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions for a parametric Knapsack Problem. We introduce a new reformulation of the BKP into a one-level integer programming problem using dynamic programming. We propose an algorithm that allows the BKP to be solved exactly in two steps. In the first step, a dynamic programming algorithm is used to compute the set of follower reactions to leader decisions. In the second step, an integer problem that is equivalent to the BKP is solved using a branch-and-bound algorithm. Numerical results are presented to show the performance of our method.

Approximation algorithms for a bi-level knapsack problem

In this paper, we consider a variant of the knapsack problem. There are two knapsacks with probably different capacities, owned by two agents respectively. Given a set of items, each with a fixed size and a profit. The two agents select items and pack them into their own knapsacks under the capacity constraint. Same items can be packed simultaneously to different knapsacks. However, in this case the profit of such items may vary. One agent packs items into his knapsack to maximize the total profit, while another agent can only pack items into his knapsack as well but he cares about the total profits of items packed into two knapsacks. The latter agent is a leader while the former is a follower. We aim at designing an approximation algorithm for the leader assuming that the follower is selfish. For different settings we provide approximation results.

The bilevel knapsack problem with stochastic right-hand sides

We introduce the bilevel knapsack problem with stochastic right-hand sides, and provide necessary and sufficient conditions for the existence of an optimal solution. When the leader’s decisions can take only integer values, we present an equivalent two-stage stochastic programming reformulation with binary recourse. We develop a branch-and-cut algorithm for solving this reformulation, and a branch-and-backtrack algorithm for solving the scenario subproblems. Computational experiments indicate that our approach can solve large instances in a reasonable amount of time.

Integer Programming Formulation of the Bilevel Knapsack Problem

The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions of parametric Knapsack Problem. In this paper, we propose two stages exact method for solving the BKP. In the first stage, a dynamic programming algorithm is used to compute the set of reactions of the follower. The second stage consists in solving an integer program reformulation of BKP. We show that the integer program reformulation is equivalent to the BKP. Numerical results show the efficiency of our method compared with those obtained by the algorithm of Moore and Bard