General Knapsack Problem Research Articles

The Knapsack Problem with Conflict Pair Constraints on Bipartite Graphs and Extensions

In this paper, we study the knapsack problem with conflict pair constraints. After a thorough literature survey on the topic, our study focuses on the special case of bipartite conflict graphs. For complete bipartite (multipartite) conflict graphs, the problem is shown to be NP-hard but solvable in pseudo-polynomial time, and it admits an FPTAS. Extensions of these results to more general classes of graphs are also presented. Further, a class of integer programming models for the general knapsack problem with conflict pair constraints is presented, which generalizes and unifies the existing formulations. The strength of the LP relaxations of these formulations is analyzed, and we discuss different ways to tighten them. Experimental comparisons of these models are also presented to assess their relative strengths. This analysis disclosed various strong and weak points of different formulations of the problem and their relationships to different types of problem data. This information can be used in designing special purpose algorithms for KPCC involving a learning component.

Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma

We consider integer programming problems in standard form max { c T x : Ax = b , x ⩾ 0, x ∈ Z n } where A ∈ Z m × n , b ∈ Z m , and c ∈ Z n . We show that such an integer program can be solved in time ( m ⋅ Δ) O ( m ) ⋅ \Vert b\Vert ∞ 2 , where Δ is an upper bound on each absolute value of an entry in A . This improves upon the longstanding best bound of Papadimitriou [27] of ( m ⋅ Δ) O ( m 2 ) , where in addition, the absolute values of the entries of b also need to be bounded by Δ. Our result relies on a lemma of Steinitz that states that a set of vectors in R m that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by m . We also use the Steinitz lemma to show that the ℓ 1 -distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by m ⋅ (2, m ⋅ Δ +1) m . Here Δ is again an upper bound on the absolute values of the entries of A . The novel strength of our bound is that it is independent of n . We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.

A novel quantum-inspired binary wolf pack algorithm for difficult knapsack problem

0-1 knapsack problem is a classical combinatorial optimisation problem, which is often used to validate the search performance of intelligent algorithms. Although the Binary Wolf Pack Algorithm (BWPA) can provide an effective solution to the general knapsack problem, it cannot achieve satisfactory optimisation results for the difficult knapsack problems. In order to improve the BWPA's searching ability in a difficult knapsack problem which has not been well resolved, a novel Quantum-Inspired WPA (QWPA) based on quantum encoding to enhance the performance of BWPA is presented. In this paper, the detailed quantitative design with a modified form of quantum collapse for the three important behaviours in QWPA is proposed and a new diversity analysis mechanism to verify the diversity of the proposed algorithm is given. Numerical simulation is obtained to show that the QWPA has a comparative performance than other quantum-inspired algorithms in solving the difficult knapsack problems.

A novel quantum-inspired binary wolf pack algorithm for difficult knapsack problem

0-1 knapsack problem is a classical combinatorial optimisation problem, which is often used to validate the search performance of intelligent algorithms. Although the Binary Wolf Pack Algorithm (BWPA) can provide an effective solution to the general knapsack problem, it cannot achieve satisfactory optimisation results for the difficult knapsack problems. In order to improve the BWPA's searching ability in a difficult knapsack problem which has not been well resolved, a novel Quantum-Inspired WPA (QWPA) based on quantum encoding to enhance the performance of BWPA is presented. In this paper, the detailed quantitative design with a modified form of quantum collapse for the three important behaviours in QWPA is proposed and a new diversity analysis mechanism to verify the diversity of the proposed algorithm is given. Numerical simulation is obtained to show that the QWPA has a comparative performance than other quantum-inspired algorithms in solving the difficult knapsack problems.

Space-Efficient Approximations for Subset Sum

S ubset S um is a well-known NP-complete problem: given t ∈ Z + and a set S of m positive integers, output YES if and only if there is a subset S ′⊆ S such that the sum of all numbers in S ′ equals t . The problem and its search and optimization versions are known to be solvable in pseudopolynomial time in general. We develop a one-pass deterministic streaming algorithm that uses space O(log t / ε) and decides if some subset of the input stream adds up to a value in the range {(1 ± ϵ) t }. Using this algorithm, we design space-efficient fully polynomial-time approximation schemes (FPTAS) solving the search and optimization versions of S ubset S um . Our algorithms run in O(1 / ϵ m 2 ) time and O(1 / ϵ) space on unit-cost RAMs, where 1 + ϵ is the approximation factor. This implies constant space quadratic time FPTAS on unit-cost RAMs when ϵ is a constant. Previous FPTAS used space linear in m . In addition, we show that on certain inputs, when a solution is located within a short prefix of the input sequence, our algorithms may run in sublinear time. We apply our techniques to the problem of finding balanced separators, and we extend our results to some other variants of the more general knapsack problem. When the input numbers are encoded in unary, the decision version has been known to be in log space. We give streaming space lower and upper bounds for unary S ubset S um (USS). If the input length is N when the numbers are encoded in unary, we show that randomized s -pass streaming algorithms for exact S ubset S um need space Ω (√N/s) and give a simple deterministic two-pass streaming algorithm using O(√ N log N ) space. Finally, we formulate an encoding under which USS is monotone and show that the exact and approximate versions in this formulation have monotone O (log 2 t ) depth Boolean circuits. We also show that any circuit using ε-approximator gates for S ubset S um under this encoding needs Ω( n /log n ) gates to compute the disjointness function.

Optimality and stability of symmetric evolutionary games with applications in genetic selection.

Symmetric evolutionary games, i.e., evolutionary games with symmetric fitness matrices, have important applications in population genetics, where they can be used to model for example the selection and evolution of the genotypes of a given population. In this paper, we review the theory for obtaining optimal and stable strategies for symmetric evolutionary games, and provide some new proofs and computational methods. In particular, we review the relationship between the symmetric evolutionary game and the generalized knapsack problem, and discuss the first and second order necessary and sufficient conditions that can be derived from this relationship for testing the optimality and stability of the strategies. Some of the conditions are given in different forms from those in previous work and can be verified more efficiently. We also derive more efficient computational methods for the evaluation of the conditions than conventional approaches. We demonstrate how these conditions can be applied to justifying the strategies and their stabilities for a special class of genetic selection games including some in the study of genetic disorders.

Модификация метода Лагариаса — Одлыжко для решения обобщённой задачи о рюкзаке и систем задач о рюкзаках

Модификация метода Лагариаса — Одлыжко для решения обобщённой задачи о рюкзаке и систем задач о рюкзаках

Design of Multiunit Electronic Exchanges Through Decomposition

In this paper, we exploit the idea of decomposition to match buyers and sellers in an electronic exchange for trading large volumes of homogeneous goods, where the buyers and sellers specify marginal-decreasing piecewise constant price curves to capture volume discounts. Such exchanges are relevant for automated trading in many e-business applications. The problem of determining winners and Vickrey prices in such exchanges is known to have a worst-case complexity equal to that of as many as (1+m+n) NP-hard problems, where m is the number of buyers and n is the number of sellers. Our method proposes the overall exchange problem to be solved as two separate and simpler problems: 1) forward auction and 2) reverse auction, which turns out to be generalized knapsack problems. In the proposed approach, we first determine the quantity of units to be traded between the sellers and the buyers using fast heuristics developed by us. Next, we solve a forward auction and a reverse auction using fully polynomial time approximation schemes available in the literature. The proposed approach has worst-case polynomial time complexity and our experimentation shows that the approach produces good quality solutions to the problem. Note to Practitioners- In recent times, electronic marketplaces have provided an efficient way for businesses and consumers to trade goods and services. The use of innovative mechanisms and algorithms has made it possible to improve the efficiency of electronic marketplaces by enabling optimization of revenues for the marketplace and of utilities for the buyers and sellers. In this paper, we look at single-item, multiunit electronic exchanges. These are electronic marketplaces where buyers submit bids and sellers ask for multiple units of a single item. We allow buyers and sellers to specify volume discounts using suitable functions. Such exchanges are relevant for high-volume business-to-business trading of standard products, such as silicon wafers, very large-scale integrated chips, desktops, telecommunications equipment, commoditized goods, etc. The problem of determining winners and prices in such exchanges is known to involve solving many NP-hard problems. Our paper exploits the familiar idea of decomposition, uses certain algorithms from the literature, and develops two fast heuristics to solve the problem in a near optimal way in worst-case polynomial time

A relaxed projection method for general integer quadratic knapsack problem

The integer quadratic knapsack problem having a single constraint was solved in a previous paper, by a projection method that completely eliminated the need for any transformation of variables. Here we present a projection method for general integer quadratic knapsack problem consisting of any finite number of constraints with the help of Kuhn-Tucker conditions and Branch and Bound Method.

An approach to connection admission control in single-hop multiservice wireless networks with QoS requirements

We present an approach to connection admission control in single-hop multiservice wireless networks with quality of service (QoS) requirements. The approach consists of two steps: 1) the specification of an admission region that captures the QoS requirements; 2) the formulation of a generalized knapsack problem that captures the connection admission control. To illustrate approach 1), we determine an outage-based admission region; to illustrate approach 2), we investigate the performance of the greedy admission policy in a generalized knapsack problem.

On dominated terms in the general knapsack problem

A necessary and sufficient condition for the identification of dominated terms in a general knapsack problem is derived. By general, we mean a knapsack problem which is unbounded, equality constrained and which has a parametric right-hand side. The given condition yields recently published results in the literature. A report on computational experiments for large-scale knapsack problems, demonstrating the effectiveness of this approach, is included.

Generalized Knapsack problems and fixed degree separations

Generalized Knapsack problems and fixed degree separations

Solving the generalized knapsack problem with variable coefficients

In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean-value cross-decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented.

Generalized Knapsack problems and fixed degree separations

Generalized Knapsack problems and fixed degree separations

A parallel algorithm for the knapsack problem using a generation and searching technique

The purpose of this paper is to propose a new parallel algorithm for the knapsack problem. We develop a generation and searching technique to derive the desired two ordered lists in the preliminary process of the general knapsack problem. The proposed parallel algorithm is developed on the basis of an SIMD machine with shared memory. The algorithm needs O(2 n 4 ) memory and O(2 n 8 ) processors to find a solution for the knapsack problem of n components in time O(2 n 2 ) . The proposed parallel algorithm has a cost of O(2 5n 8 ) which is an improved result over the past researches.

A PARALLEL ALGORITHM FOR THE KNAPSACK PROBLEM WITH MEMORY/PROCESSOR TRADEOFF M2P = O(2n/2)

The purpose of this paper is to suggest a new parallel algorithm for the knapsack problem. The design of the proposed parallel algorithm is based on a shared memory SIMD machine with a heap tree data structure. It can solve a general n-component knapsack problem in time O(2n/2) with memory-processor tradeoff M2P = O(2n/2). The proposed technique can have better performance in terms of cost as the number of processors needed can be offset by increasing memory space. The performance is compared with those of other algorithms to demonstrate its superiority. We believe that the proposed algorithm is pragmatically feasible at the moment when multiprocessor systems are gradually becoming popular.

On the optimality of the greedy solutions of the general knapsack problems

Summary: In this paper we submit a unified discussion of some closely related results which were achieved independently in number theory and integer programming, and we partially generalize them. In the unified discussion we treat together two problems where the greedy method has different characters, in the first one it is an internal-point algorithm, in the second one it is an outer-point method. We call a knapsack problem pleasant if the greedy solution is optimal for every right-hand side. A sufficient and two finite necessary and sufficient conditions for the pleasantness of a problem are discussed. The sufficient condition can be checked very easily. The paper is finished with an error analysis of some nonpleasant problems

A solution procedure for general knapsack problems with a few constraints

A solution procedure has been developed for the general knapsack problems with a few constraints. The procedure uses a forward state generation scheme of the dynamic programming along with a facility to eliminate the nonpromising states by calculating an upper bound on the best return that can be achieved with the remaining variables. In addition, the state generation is started anew with the remaining activities when the size of the state set generated at a stage exceeds a threshold value. This results in partitioning of the problem into the subproblems for which the state sets are constructed separately. When the state generation is completed, the state subsets of the subproblems are combined in a multidimensional multiple choice zero-one integer model to yield the overall optimal solution. The solution procedure has been applied to a set of randomly generated problems up to 1000 variables.

Opening the knapsack by DFTs

A new simple algorithm which uses discrete Fourier transforms for breaking the general knapsack problem (also known as the subset sum problem) is presented along with the number of operations required for solving the problem. The method is applicable when the elements belong to a Galois field as well as that of a set of integers and is most effective for high-density knapsacks.

A note on a general non-linear knapsack problem

This paper discusses a general non-linear knapsack problem with a concave objective function and a single conves constraint. in particular, it includes an efficient procedure to find the continuous (relaxed) solution and a reduction process which computes tight lower and upper bounds on the integer variables. Some implicit enumeration criteria to be used in an enumeration algorithm are also suggested.