Subset Sum Research Articles

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.

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

Two problems on subset sums

For a set A of positive integers, let P(A) denote the set of all finite subset sums of A. In this paper, we completely solve a problem of Chen and Wu by proving that if B={b1<b2<⋯} is a sequence of integers with b1≥11, 3b1+5≤b2≤4b1, 3b2+2≤b3≤3b2+b1 and 3bn−bn−2≤bn+1≤3bn(n≥3), then there exists a set of positive integers A for which P(A)=N∖B. We also partially answer a problem of Wu by determining the structure of B={b1<b2<⋯} with b1>10 and b2>3b1+4, for which there exists a set of positive integers A such that P(A∩[0,bk])=[0,2bk]∖{bi,2bk−bi:1≤i≤k}(k≥2).

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.

From approximate to exact integer programming

AbstractApproximate integer programming is the following: For a given convex body $$K \subseteq {\mathbb {R}}^n$$ K ⊆ R n , either determine whether $$K \cap {\mathbb {Z}}^n$$ K ∩ Z n is empty, or find an integer point in the convex body $$2\cdot (K - c) +c$$ 2 · ( K - c ) + c which is K, scaled by 2 from its center of gravity c. Approximate integer programming can be solved in time $$2^{O(n)}$$ 2 O ( n ) while the fastest known methods for exact integer programming run in time $$2^{O(n)} \cdot n^n$$ 2 O ( n ) · n n . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $$x^* \in (K \cap {\mathbb {Z}}^n)$$ x ∗ ∈ ( K ∩ Z n ) can be found in time $$2^{O(n)}$$ 2 O ( n ) , provided that the remainders of each component $$x_i^* \mod \ell $$ x i ∗ mod ℓ for some arbitrarily fixed $$\ell \ge 5(n+1)$$ ℓ ≥ 5 ( n + 1 ) of $$x^*$$ x ∗ are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a $$2^{O(n)}n^n$$ 2 O ( n ) n n algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ A x = b , 0 ≤ x ≤ u , x ∈ Z n . Such a problem can be reduced to the solution of $$\prod _i O(\log u_i +1)$$ ∏ i O ( log u i + 1 ) approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range$$0 \le x_i \le p(n)$$ 0 ≤ x i ≤ p ( n ) can be solved in time $$(\log n)^{O(n)}$$ ( log n ) O ( n ) . For these problems, the best running time so far was $$n^n \cdot 2^{O(n)}$$ n n · 2 O ( n ) .

Solving subset sum and SAT problems by reaction systems

Solving subset sum and SAT problems by reaction systems

Approximating subset sum ratio via partition computations

We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to 1 as possible. Our scheme makes use of exact and approximate algorithms for Partition, and clearly showcases the close relationship between the two algorithmic problems. Depending on the relationship between the size of the input set n and the error margin ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity O(n4/ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} (n^4 / \\varepsilon )$$\\end{document}. In particular, the exponent of n in our proposed scheme may decrease down to 2, depending on the Partition algorithm used.

CocoSketch: High-Performance Sketch-Based Measurement Over Arbitrary Partial Key Query

Sketch-based measurement has emerged as a promising solutions due to its high accuracy and resource efficiency. Prior sketches focus on measuring single flow keys and cannot support measurement on multiple keys. This work takes a significant step towards supporting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arbitrary partial key queries</i> , which aims to provide information for any key in the predefined range of possible flow keys. The designed system, casts arbitrary partial key queries to the subset sum estimation problem and makes the theoretical tools for subset sum estimation practical. utilizes two techniques: (1) stochastic variance minimization to significantly reduce per-packet update delay, and (2) removing circular dependencies in the per-packet update logic to make the implementation hardware-friendly. This paper extends the conference version by discussing how adapts to new measurement requirements, including: (1) collecting the exact information of specified flow keys, and (2) distributed measurement. is implemented on five popular platforms (CPU, Open vSwitch, Redis, P4, and FPGA). Experiment results show that compared to baselines that use traditional single-key sketches, improves average packet processing throughput by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$27.2\times$</tex-math> </inline-formula> and accuracy by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10.4\times$</tex-math> </inline-formula> when measuring six flow keys.

The Maximal Length of q-ary MDS Elliptic Codes Is Close to

Abstract Determining the maximal length of MDS codes with certain dimension is one of the central topics in coding theory and finite geometry. The MDS Main Conjecture states that the maximal length of a non-trivial $q$-ary MDS code of dimension $k$ is $q+1$ except when $q$ is even and $k=3$ or $k=q-1$. We prove that the maximal length of non-trivial $q$-ary MDS elliptic codes is close to $\frac{q}{2}$, which gives an affirmative answer to a conjecture of Li, Wan, and Zhang. Moreover, we apply our result to derive an answer to a question on subset sums in finite abelian groups from elliptic curves.

On arithmetic sums of Cantor-type sequences of integers

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate FS(B) the subset sum of sequence type B={⌊pnα⌋}n=0∞. When p=2, then we prove FS(B)+FS(B)=N by analogy with the Cantor set, and some structure theorem for p>2.

Spiking neural P systems with myelin and dendritic spines

Inspired by the dendritic and axonal morphology, some authors modeled the transmission speed of action potentials through of fixed dendritic and axonal delays, respectively. However, in practice, the exploration of dendrites and axons with variable delays potentially allows the improvement of advanced engineering applications. Taking inspiration from the structure of the dendritic spines and the process of myelination of axons, we introduce a new variant of the spiking neural P systems with variable axonal and dendritic delays named as a spiking neural P system with myelin and dendritic spines computation (MDSCSN P system). The inclusion of these experimentally proven biological features of axon and dendrites into standard spiking neural P systems allows to control the timing of dendritic/axonal transmissions. As a consequence, the soma processing of neurons is significantly reduced since it receives stable firing patterns. In contrast, standard spiking neural P systems and their recent variants use a large number of neurons/synapses to do the same. To demonstrate this, we generate a small universal SN P system by employing only neurons with standard rules to perform any Turing computable function and to figure out the Subset Sum problem in the uniform way.

Correction to: On subset sums of $$\mathbb {Z}_n^{\times }$$ which are equally distributed modulo n

Correction to: On subset sums of $$\mathbb {Z}_n^{\times }$$ which are equally distributed modulo n

Lower and upper bounding procedures for the bin packing problem with concave loading cost

We address the one-dimensional bin packing problem with concave loading cost (BPPC), which commonly arises in less-than-truckload shipping services. Our contribution is twofold. First, we propose three lower bounds for this problem. The first one is the optimal solution of the continuous relaxation of the problem for which a closed form is proposed. The second one allows the splitting of items but not the fractioning of bins. The third one is based on a large-scale set partitioning formulation of the problem. In order to circumvent the challenges posed by the non-linearity of the objective function coefficients, we considered the inner-approximation of the concave load cost and derived a relaxed formulation that is solved by column generation. In addition, we propose two subset-sum-based heuristics. The first one is a constructive heuristic while the second one is a local search heuristic that iteratively attempts to improve the current solution by selecting pairs of bins and solving the corresponding subset sum-problem. We show that the worst-case performance of any BPPC heuristic and any concave loading cost function is bounded by 2. We present the results of an extensive computational study that was carried out on large set of benchmark instances. This study provides empirical evidence that the column generation-based lower bound and the local search heuristic consistently exhibit remarkable performance.

An FPTAS for scheduling with resource constraints

We study a basic scheduling problem with resource constraints: A number of jobs need to be scheduled on two parallel identical machines with the objective of minimizing the makespan, subject to the constraint that jobs may require a unit of one of the given renewable resources during their execution. For this NP-hard problem, we develop a fully polynomial-time approximation scheme (FPTAS). Our FPTAS makes a novel use of existing algorithms for the subset-sum problem and the open shop scheduling problem.

Spiking neural P systems with weights and delays on synapses

Spiking neural P systems (SN P systems) are bio-inspired neural-like computational devices that mimic the communication between two nearby neurons and the spike changes in the neuron. This study incorporates concepts of the weight of synapses and the delay on synapses to increase the bio-explainability (the model can better simulate the communication between neurons), where the weight is a natural feature of synapses, and the delay is that in the communication between neurons, through synapses. Thus, an innovative type of spiking neural-like P system is defined, called spiking neural P systems with weights and delays on synapses (WDSN P systems). In WDSN P systems, synapses are assigned weights and delays, where weights are real numbers and delays are natural numbers. Results proved in this paper show that WDSN P systems can reach Turing universality in the generating and accepting mode. The semi-uniform solution for the Subset Sum problem demonstrates that WDSN P systems can solve NP-complete problems efficiently.

Some remarks on the Erdős Distinct subset sums problem

Let [Formula: see text] be a set of positive integers, [Formula: see text] denoting the largest element, so that for any two of the [Formula: see text] subsets the sum of all elements is distinct. Erdős asked whether this implies [Formula: see text] for some universal [Formula: see text]. We prove, slightly extending a result of Elkies, that for any [Formula: see text], [Formula: see text] with equality if and only if all subset sums are [Formula: see text]-separated. This leads to a new proof of the currently best lower bound [Formula: see text]. The main new insight is that having distinct subset sums and [Formula: see text] small requires the random variable [Formula: see text] to be close to Gaussian in a precise sense.

Pseudo-polynomial algorithms for solving the Knapsack Problem with dependencies between items

We consider a new variant of the Knapsack Problem with dependencies between items. In this variant, the set of items is partitioned into subsets with dependencies among them, and an item can be selected from a subset only if at least one item is selected from each of its dependent subsets. We develop pseudo-polynomial algorithms to solve this new constrained version in the cases where the dependencies (between the subsets of items rather than items) are represented by out-trees, in-trees, and directed acyclic graphs. We consider both cases, when the weight and profit of each item are similar, which is the classical Subset Sum problem, and the case when they take arbitrary non-negative values. The proposed algorithms run in O(nW) times and spaces for out-trees, while for in-trees and acyclic digraphs it runs in O(nW+mW2) and O(nW+max{mW2,m(nW)}), respectively, where n is the number of items, W is the knapsack capacity, and m is the number of nodes. Experiments on randomly generated knapsack instances with different graphs of dependency are carried out to assess algorithm efficiency, and show the running dependency on different instance parameters.