On tropical knapsack-type problems

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products. It is also shown that for every co-context-free group, the knapsack problem is decidable. For the subset sum problem (also introduced by Myasnikov, Nikolaev, and Ushakov) we show that it belongs to the class NL (nondeterministic logspace) for every finitely generated virtually nilpotent group and that there exists a polycyclic group with an NP-complete subset sum problem.

Linear Time Algorithms for Knapsack Problems with Bounded Weights

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P \mathbf {P} -time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is N P \mathbf {NP} -complete.

Knapsack problems are typically concerned with selecting from a set of n given items, each with a specified weight and value, a subset of items whose weight sum does not exceed a prescribed capacity and whose value is maximum. This NP-hard problem arises in many applications and has been the focus of considerable research over the past two decades. A number of exact algorithms have been developed for the classical 0–1 Knapsack Problems and its variants. In this paper, exact algorithms are presented for the following variants and data types: the Subset Sum Problem; the Strongly Correlated 0–1 Knapsack Problem; the Inverse Strongly Correlated 0–1 Knapsack Problem; and the corresponding Bounded Strongly Correlated Knapsack Problem and Bounded Subset Sum Problem. All our algorithms consist of three stages: the first stage generates an initial solution by a greedy procedure; the second stage refines the approximate solution; and the final stage applies a partial lexicographic search procedure to generate an optimal solution. Extensive computational experiments show that our algorithms are able to solve large problems of size up to one million variables in less than 7 seconds CPU time on a Silicon Graphic Workstation (R 5000) running at a clock speed of 150 MHz. A comparative analysis with some recent effective algorithms is given.

After the seminal books by Martello and Toth (1990) and Kellerer, Pferschy, and Pisinger (2004), knapsack problems became a classical and rich research area in combinatorial optimization. The purpose of this survey, which is structured in two parts, is to cover the developments that appeared in this field after the publication of the latter volume. Part I is devoted to problems whose goal is to optimally assign items to a single knapsack. Besides the classical knapsack problems (binary, subset sum, bounded, unbounded, change-making), we review problems with special constraints (setups, multiple-choice, conflicts, precedences, sharing, compartments) as well as relatively recent fields of investigation, like robust and bilevel problems. The subsequent Part II covers multiple, multidimensional, and quadratic knapsack problems, and includes a succinct treatment of online and multiobjective knapsack problems.

SUMMARY The subset sum problem, which is often called as the knapsack problem, is known as an NP-hard problem, and there are several cryptosystems based on the problem. Assuming an oracle for shortest vector problem of lattice, the low-density attack algorithm by Lagarias and Odlyzko and its variants solve the subset sum problem efficiently, when the “density” of the given problem is smaller than some threshold. When we define the density in the context of knapsack-type cryptosystems, weights are usually assumed to be chosen uniformly at random from the same interval. In this paper, we focus on general subset sum problems, where this assumption may not hold. We assume that weights are chosen from different intervals, and make analysis of the effect on the success probability of above algorithms both theoretically and experimentally. Possible application of our result in the context of knapsack cryptosystems is the security analysis when we reduce the data size of public keys.

Computationally complex problems such as the subset sum problem (SSP) are difficult to solve on the electronic computer when the problem size scales up. Optical computing enables the acceleration of complex problems due to its low latency and multiple free degrees of parallelism. Here, we thoroughly exploit the programmability in silicon photonic circuits and construct a programmable microring resonator mesh to implement the solution of two types of computationally complex problems: the SSP and stochastic process simulation. We experimentally accomplish the solving of multiple different sets, which shows the great versatility and scalability of the SSP solver. The knapsack problems are also demonstrated as an example of practical optimization problems derived from SSP. Besides, two-state discrete time stochastic processes are also experimentally accomplished with our solver chip. Our proposed photonic solver shows the potential application of optical NP-complete problem-solving and quantum computing.

Let al,...,an and s be a set of integers. The knapsack (or subset sum) problem is to find a 0–1 vector (el,...,en) such that Σ eiai = s or to show that such a vector does not exist. The integers al,...,an are sometimes referred to as weights. The general knapsack problem is known to be NP complete [5,6]. Several cryptosystems based on the knapsack problem have been designed [9,12,16]. In April, 1982, Adi Shamir [14] announced a method for breaking the Merkle-Hellman cryptosystem. Since that time there has been a flurry of activity to extend his results to include all of the proposed knapsack based cryptosystems [1,2,3,7,13].

Observations on optimal parallelizations of two-list algorithm

On semigroups of matrices with nonnegative diagonals

This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly $${\mathcal{N}\mathcal{P}}$$ -hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem.

Sensitivity analysis of a greedy heuristic for knapsack problems

Approximate minimization algorithms for the 0/1 Knapsack and Subset-Sum Problem

A knapsack (or subset-sum) problem that is useful for cryptographic purposes, consists of a set of n positive integers a = {a1, a2, ... an}, called the knapsack a, and a sum s. The density d of a knapsack is defined to be n /log2 (ai)max. The knapsack problem then consists of finding the set, if any, of binary numbers x = {x1 , x2, ... , xn}, such that Σxi·ai = s.

In [1], Pall proved an interesting result on a certain class of 2 × 2 integral matrices. He showed that the semigroup of 2 × 2 matrices of determinant 1 and non-negative entries contains exactly 2 primes , and every other non-unit is expressible uniquely as products of these primes. Before formally stating this result, we need some notation. Let Gn denote the semigroup of n × n matrices with determinant 1 and nonnegative integral entries, In the n × n identity matrix, the n × n matrix with a 1 as its (i, j) element and zeros elsewhere, and let . When the dimension is clear, we shall drop the superscripts.