Sum Problem Research Articles

The Bauer–Ramanujan formula: historical analyses and perspectives

The formula 2 π = 1 − 5 ⋅ ( 1 2 ) 3 + 9 ⋅ ( 1 ⋅ 3 2 ⋅ 4 ) 3 − ⋯ was famously included as a discovery in Ramanujan's first letter to Hardy in 1913, and has been referred to as the Bauer–Ramanujan formula, in view of Bauer's 1859 proof of the above formula. There is a rich history associated with this formula and its many and dramatically different proofs, including a computer-based proof due to Zeilberger that may be seen as groundbreaking in the history of computer-assisted proofs. In addition to a complete survey we provide of all known proofs of the Bauer–Ramanujan formula, we introduce historical analyses based on these proofs, by arguing that the history of the Bauer–Ramanujan formula and our account of this history may be seen as being representative of much broader trends in the history of mathematics. In this regard, the earlier proofs tend to rely on one of the oldest and most basic tools in classical analysis, namely, interchanging the order of limiting operations. In contrast, the more modern proofs tend to rely on computer-related approaches toward summation problems, as in with Zeilberger-type and Gosper-type telescoping arguments.

Slalom at the Carnival: Privacy-preserving Inference with Masks from Public Knowledge

Machine learning applications gain more and more access to highly sensitive information while simultaneously requiring more and more computation resources. Hence, the need for outsourcing these computational expensive tasks while still ensuring security and confidentiality of the data is imminent. In their seminal work, Tramer and Boneh presented the Slalom protocol for privacy-preserving inference by splitting the computation into a data-independent preprocessing phase and a very efficient online phase. In this work, we present a new method to significantly speed up the preprocessing phase by introducing the Carnival protocol. Carnival leverages the pseudo-randomness of the Subset sum problem to also enable efficient outsourcing during the preprocessing phase. In addition to a security proof we also include an empirical study analyzing the landscape of the uniformity of the output of the Subset sum function for smaller parameters. Our findings show that Carnival is a great candidate for real-world implementations.

Counting vanishing matrix-vector products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let v∈Qd be a rational vector, (T1,T2…,Tm) a list of d×d rational matrices, S∈Qh×d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices Ti1,Ti2,…,Tik from the list such that STik⋯Ti1v=0∈Qh.In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d>3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

A REFINED CLASSIFICATION OF THE SUBSET SUM PROBLEM IN POLYNOMIAL TIME COMPLEXITY

The problem considered is the selection of at least one subset from a set (array) of distinct positive integers, such that the sum of the subset's elements exactly matches a given target sum (target certificate). According to R. M. Karp, this problem belongs to the class of NP-complete problems. Diophantine equations and an auxiliary problem, which facilitates the solution of the original problem and has independent scientific interest, have been introduced. A novel method has been developed, which includes proven lemmas and theorems. These results enable the development of efficient and straightforward algorithms for solving the subset sum problem. The time and space complexity for selecting the required subsets do not exceed the square of the length of the original set. An analytical framework has been proposed for managing indices within the original set. These algorithms are applicable to solving problems related to the independent set of cardinality k and the k-vertex cover problem. Additionally, we present examples to confirm claimed results. It should be noted that the time complexity of sorting an array of integers is proportional to the square of the array's size, and this problem belongs to class P. Therefore, based on the newly developed method, it can be inferred that the subset sum problem, originally classified as NP-complete within the NP class, also belongs to P.

Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling

Cost reduction and staff retention are important optimization objectives in home healthcare (HHC) systems. Home healthcare operators need to balance their objectives by optimizing resource use, service delivery and profits. Minimizing total travel distances to control costs is a common routing problem objective while minimizing total finishing time differences is a scheduling objective whose purpose is to enhance staff satisfaction. To optimize routing and scheduling, we propose mixed integer linear programming with a bi-objective function, which is a subset of the vehicle routing problem with time windows (VRPTWs). VRPTWs is a known NP-hard problem, and optimal solutions are very hard to obtain in practice. Metaheuristics offer an alternative solution to this type of problem. Our metaheuristic uses the simulated annealing algorithm and weighted sum approach to convert the problems to single-objective problems and is equipped with operators including swapping, moving, path exchange and ruin and recreate. The results show, firstly, that the algorithm can effectively find the Pareto front, and secondly, that minimizing total finishing time differences to balance the number of jobs per caretaker is an efficient way to tackle HHC scheduling. A statistical test shows that the algorithm can obtain the Pareto front with a lower number of weighted sum problems.

Solving the subset sum problem by the quantum Ising model with variational quantum optimization based on conditional values at risk

Solving the subset sum problem by the quantum Ising model with variational quantum optimization based on conditional values at risk

Signed zero sum problems for metacyclic group

Signed zero sum problems for metacyclic group

Dynamic threshold spiking neural P systems with weights and multiple channels

Membrane computing represents a sophisticated branch of computational science that assimilates the characteristics of cellular membranes and harnesses mathematical principles. It derives inspiration from the intricate structure and functional attributes exhibited by biological cell membranes. This exposition is dedicated to an in-depth exploration of spiking neural P systems (SN P systems), meticulously crafted to emulate the intricate signaling and interaction phenomena between cellular entities. Nevertheless, conventional spiking neural P systems confront inherent constraints pertaining to their excitation rules, which hinder their applicability to real-world challenges. In pursuit of overcoming these limitations, we introduce a novel paradigm encompassing the dynamic thresholds, synaptic weights, and the integration of multiple channels within synapses. This innovative framework culminates in the dynamic threshold spiking neural P systems with weights of synapses and multiple channels in synapses (DSNP-WM systems). It is noteworthy that DSNP-WM systems demonstrate Turing universality, effectively functioning as versatile entities capable of both number generation and acceptance, in addition to performing intricate computations. Importantly, these systems showcase the efficiency in tackling challenges posed by semi-uniform solutions, exemplified by the Subsets Sum problem—a fundamental member of the NP-complete problem class, which employs a nondeterministic approach.

Hypergraph-Based Numerical Spiking Neural Membrane Systems with Novel Repartition Protocols.

The classic spiking neural P (SN P) systems abstract the real biological neural network into a simple structure based on graphs, where neurons can only communicate on the plane. This study proposes the hypergraph-based numerical spiking neural membrane (HNSNM) systems with novel repartition protocols. Through the introduction of hypergraphs, the HNSNM systems can characterize the high-order relationships among neurons and extend the traditional neuron structure to high-dimensional nonlinear spaces. The HNSNM systems also abstract two biological mechanisms of synapse creation and pruning, and use plasticity rules with repartition protocols to achieve planar, hierarchical and spatial communications among neurons in hypergraph neuron structures. Through imitating register machines, the Turing universality of the HNSNM systems is proved by using them as number generating and accepting devices. A universal HNSNM system consisting of 41 neurons is constructed to compute arbitrary functions. By solving NP-complete problems using the subset sum problem as an example, the computational efficiency and effectiveness of HNSNM systems are verified.

Lower time bounds for parallel solving of the subset sum problem by a dynamic programming algorithm

SummaryIn the paper, we compute some lower bounds on time of parallel solving of the subset sum problem on a big number of processors by several versions of dynamic programming algorithm Balsub proposed before by Pisinger. Based on these lower bounds, we propose a version of Balsub which could be possibly effectively parallelized.

Sums of square roots that are close to an integer

Let k∈N and suppose we are given k integers 1≤a1,…,ak≤n. If a1+…+ak is not an integer, how close can it be to one? When k=1, the distance to the nearest integer is ≳n−1/2. Angluin-Eisenstat observed the bound ≳n−3/2 when k=2. We prove there is a universal c>0 such that, for all k≥2, there exists a ck>0 and k integers in {1,2,…,n} with0<‖a1+…+ak‖≤ck⋅n−c⋅k1/3, where ‖⋅‖ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for k=3, the problem appears hard.

Approximation algorithms for job scheduling with block-type conflict graphs

The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered in this paper. No two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine in this model.The presented model stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring. Recently, cluster graphs and their extensions like block graphs were given additional attention. We complement hardness results provided by other researchers for block graphs by providing approximation algorithms. In particular, we provide a 2-approximation algorithm for P|G=block graph|Cmax and a PTAS for the case when the jobs are unit time in addition. In the case of uniform machines, we analyze two cases. The first one is when the number of blocks is bounded, i.e. Q|G=k − block graph|Cmax. For this case, we provide a PTAS, improving upon results presented by D. Page and R. Solis-Oba. The improvement is two-fold: we allow richer graph structure, and we allow the number of machine speeds to be part of the input. Due to strong NP-hardness of Q|G=2 − clique graph|Cmax, the result establishes the approximation status of Q|G=k − block graph|Cmax. The PTAS might be of independent interest because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING WITH TARGET SUMS problem. The second case that we analyze is when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of unit time. In this case, we present an exact algorithm. In addition, we present an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines.The paper ends with extensive tests of the selected algorithms.

Optimization of Location-Routing for Multi-Vehicle Combinations with Capacity Constraints Based on Binary Equilibrium Optimizers

The Location-Routing Problem (LRP) becomes a more intricate subject when the limits of capacities of vehicles and warehouses are considered, which is an NP-hard problem. Moreover, as the number of vehicles increases, the solution to LRP is exacerbated because of the complexity of transportation and the combination of routes. To solve the problem, this paper proposed a Discrete Assembly Combination-Delivery (DACA) strategy based on, the Binary Equilibrium Optimizer (BiEO) algorithm, in addition, this paper also proposes a mixed-integer linear programming model for the problem of this paper. Our primary objective is to address both the route optimization problem and the assembly group sum problem concurrently. Our BiEO algorithm was designed as discrete in decision space to meet the requirements of the LRP represented by the DACA strategy catering to the multi-vehicle LRP scenario. The efficacy of the BiEO algorithm with the DACA strategy is demonstrated. through empirical analysis utilizing authentic data from Changchun City, China, Remarkably, the experiments reveal that the BiEO algorithm outperforms conventional methods, specifically GA, PSO, and DE algorithms, resulting in reduced costs. Notably, the results show the DACA strategy enables the simultaneous optimization of the LRP and the vehicle routing problem (VRP), ultimately leading to cost reduction. This innovative algorithm proficiently tackles both the assembly group sum and route optimization problems intrinsic to multi-level LRP instances.

Reimagining Niggli Numbers for modern data applications in petrology and exploration geochemistry

This study seeks to refamiliarize the geochemical community with Niggli Numbers and explore how Paul Niggli's largely forgotten scheme for representing major elements as equivalent molecular numbers that are combined as groups, sums and ratios. This scheme provides simple and effective tools for tackling many modern geochemical problems – particularly to reduce complexity and degrees of freedom where there are requirements to handle large volumes of data. Niggli Numbers provide an alternative to ratio-based methods for the long-recognised problem of the constant sum and data closure in geochemistry and offer a means to represent minerals and their relationships in a straightforward and consistent chemical framework. The Niggli calculation scheme follows simple rules that are easily automated, is flexible enough to accommodate missing elements or partial analyses, and is appropriate for all rock types. Following a short review of the most useful Niggli binary diagrams and where common minerals plot in Niggli space, a series of case studies are presented to illustrate potential applications of the scheme in modern petrology and resource exploration. These cover the following: magmatic evolution of a layered intrusion such as the Bushveld Complex; contamination of magmas by country rocks; tracing hydrothermal alteration in volcanogenic massive sulfide and porphyry CuMo deposits; and chemostratigraphy of sedimentary sequences.

Cube sum problem for integers having exactly two distinct prime factors

Cube sum problem for integers having exactly two distinct prime factors

Features of Creating Parallel Programs for the Parallel Dataflow Computing System "Buran"

The use of modern multi-core and multi-processor computer systems for actual tasks is not effective enough, even taking into account the use of parallel programming technologies. The solution to the problem of efficient loading of computing resources can be the transition to computing models and architectures that are inherently parallel. One of such architectures is the Parallel Dataflow Computing System (PDCS) "Buran", which implements a dataflow computing model with a dynamically formed context. A feature of the dataflow computing model is the activation of computations by data readiness, which affects both the architecture of the computing system and the creation of programs for such systems. Differences between imperative and dataflow programming paradigms are also reflected in the route of creating a program, especially its parallel implementation. The route of creating a dataflow parallel program differs markedly from the traditional one. Already at the first stage, a parallel algorithm is created and implemented (including the algorithm for generating initial data). Next, this algorithm is debugged when it is executed on an emulator or model of the system with one computing core. After that, the selection and configuration of function of computation distribution is performed, the program is executed (without changing its code) on the emulator or model of the system with several computing cores, and, finally, the program is debugged in multi-core mode. These stages of the route differ from similar stages of the traditional route, both in form and in essence. Unlike traditional parallel, and even more so sequential programs, it can be said that a dataflow program in equal parts consists of a program code that implements a task, an algorithm for generating initial data, and a function of computation distribution. This thesis is demonstrated by the example of solving the problem of finding sum of array elements, where the difference between the implementations is in the algorithm for generating initial data, which radically affects the nature of the dataflow program passing. Careful attention to each of the parts of the dataflow program is the key to the correct and efficient solution of problems on the PDCS, which provides support to the programmer at the hardware level.

Sum and difference patterns synthesis with common excitation amplitude

This paper focuses on the joint optimization problem of sum and difference patterns with common excitation amplitudes and separate excitation phases. The objective is to reduce the complexity of the antenna system while still maintaining a certain degree-of-freedom (DoF) in synthesizing dual-patterns. The problem is formulated with angular response and common excitation amplitude constraints, making it a nonconvex optimization problem. To address this, we introduce two groups of auxiliary variables to decouple the excitation vector from the complex nonconvex constraints. This allows the original problem to be effectively solved via alternately determining the closed-form solutions for several single-variable optimization subproblems. Additionally, our approach is extended to scenarios involving dynamic range ratio (DRR) reduction, aiming to lower the cost and difficulty of hardware implementation. Theoretical analysis and simulation results are provided to demonstrate that the proposed algorithm can converge to the Karush-Kuhn-Tucker points of the original problem. It is also observed that our approach is capable of generating desired dual-patterns with common excitation amplitudes, regardless of the presence or absence of DRR control.

A Modified Inexact SARAH Algorithm with Stabilized Barzilai-Borwein Step-Size in Machine learning

The Inexact SARAH (iSARAH) algorithm as a variant of SARAH algorithm, which does not require computation of the exact gradient, can be applied to solving general expectation minimization problems rather than only finite sum problems. The performance of iSARAH algorithm is frequently affected by the step size selection, and how to choose an appropriate step size is still a worthwhile problem for study. In this paper, we propose to use the stabilized Barzilai-Borwein (SBB) method to automatically compute step size for iSARAH algorithm, which leads to a new algorithm called iSARAH-SBB. By introducing this adaptive step size in the design of the new algorithm, iSARAH-SBB can take better advantages of both iSARAH and SBB methods. We analyse the convergence rate and complexity of the modified algorithm under the usual assumptions. Numerical experimental results on standard data sets demonstrate the feasibility and effectiveness of our proposed algorithm.

A study on the sum of K-frames in n-Hilbert space

In recent years, the concepts of frame in [Formula: see text]-Hilbert space and [Formula: see text]-frames in [Formula: see text]-Hilbert space have been introduced. The [Formula: see text]-frames in the [Formula: see text]-Hilbert space are more generalized than the ordinary frames. This paper deals with the problem of sum of [Formula: see text]-frames in [Formula: see text]-Hilbert space. First, the sufficient condition for the finite sum of [Formula: see text]-frames and Bessel sequences in [Formula: see text]-Hilbert space is given. Then, some results on the operator [Formula: see text] and pre-frame operator are given, and some new results on the sum of [Formula: see text]-frame are proved. We then discuss several results for the perturbation and stability of the [Formula: see text]-frames in [Formula: see text]-Hilbert space about the sum of the [Formula: see text]-frames.

Fishburn trees

The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call Fishburn trees. We give bijections between Fishburn trees and other well-known combinatorial structures that are counted by the Fishburn numbers, and by composing these new maps we obtain simplified versions of some of the known maps. Finally, we apply this new machinery to the so called flip and sum problems on modified ascent sequences.