Subset Sum Research Articles

On primes and practical numbers

A number n is practical if every integer in [1, n] can be expressed as a subset sum of the positive divisors of n. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime k-tuples conjecture for practical numbers, proving the “correct” upper bound, and for pairs, improving on a lower bound of Melfi.

The subset sum game revisited

We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player. We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS (unless P=NP).

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Universal Nonlinear Spiking Neural P Systems with Delays and Weights on Synapses.

The nonlinear spiking neural P systems (NSNP systems) are new types of computation models, in which the state of neurons is represented by real numbers, and nonlinear spiking rules handle the neuron's firing. In this work, in order to improve computing performance, the weights and delays are introduced to the NSNP system, and universal nonlinear spiking neural P systems with delays and weights on synapses (NSNP-DW) are proposed. Weights are treated as multiplicative constants by which the number of spikes is increased when transiting across synapses, and delays take into account the speed at which the synapses between neurons transmit information. As a distributed parallel computing model, the Turing universality of the NSNP-DW system as number generating and accepting devices is proven. 47 and 43 neurons are sufficient for constructing two small universal NSNP-DW systems. The NSNP-DW system solving the Subset Sum problem is also presented in this work.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Fairness-Based Packing of Industrial IoT Data in Permissioned Blockchains

In recent years, blockchain has been broadly applied to industrial Internet of Things (IIoT) due to its features of decentralization, transparency, and immutability. In existing permissioned blockchain based IIoT solutions, transactions submitted by IIoT devices are arbitrarily packed into blocks without considering their waiting times. Hence, there will be a high deviation of the transaction response times, which is known as the lack of fairness. Unfair permissioned blockchain decreases the quality of experience from the perspective of the IIoT devices. Moreover, some transactions can get timeouts if not responded for a long time. In this article, we propose Fair-Pack, the first fairness-based transaction packing algorithm for permissioned blockchain empowered IIoT systems. First, we gain the insight that fairness is positively related to the sum of waiting times of the selected transactions through theoretical analysis. Based on this insight, we transform the fairness problem into the subset sum problem, which is to find a valid subset from a given set with subset sum as large as possible. However, it is time consuming to solve the problem using a brute-force approach because there is an exponential number of subsets for a given set. To this end, we propose a heuristic and a min-heap-based optimal algorithm for different parameter settings. Finally, we analyze the time complexity of Fair-Pack and conduct extensive experiments. The results reveal that Fair-Pack is time-efficient and outperforms the existing algorithms significantly in terms of both fairness and average transaction response time.

Optimality and Complexity Analysis of a Branch-and-Bound Method in Solving Some Instances of the Subset Sum Problem

Abstract In this paper we study the question of parallelization of a variant of Branch-and-Bound method for solving of the subset sum problem which is a special case of the Boolean knapsack problem. The following natural approach to the solution of this question is considered. At the first stage one of the processors (control processor) performs some number of algorithm steps of solving a given problem with generating some number of subproblems of the problem. In the second stage the generated subproblems are sent to other processors for solving (one subproblem per processor). Processors solve completely the received subproblems and return their solutions to the control processor which chooses the optimal solution of the initial problem from these solutions. For this approach we define formally a model of parallel computing (frontal parallelization scheme) and the notion of complexity of the frontal scheme. We study the asymptotic behavior of the complexity of the frontal scheme for two special cases of the subset sum problem.

Counting the decimation classes of binary vectors with relatively prime length and density

We present a method for determining the number of decimation classes of density $$\delta $$ binary vectors indexed by a finite abelian group G of size $$\ell $$ and exponent $${\ell ^*}$$ such that $$\delta $$ is relatively prime to $${\ell ^*}$$ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of $${{\mathbb {Z}}_{{\ell ^*}}^{\times }}$$ . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by $${\mathbb {Z}}_{\ell }$$ of odd length $$\ell $$ and density $$(\ell +1)/2$$ greatly increased the number of $$\ell $$ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.

Effective parallelization strategy for the solution of subset sum problems by the branch-and-bound method

Abstract An easily implementable recursive parallelization strategy for solving the subset sum problem by the branch-and-bound method is proposed. Two different frontal and balanced variants of this strategy are compared. On an example of a particular case of the subset sum problem we show that the balanced variant is more effective than the frontal one. Moreover, we show that, for the considered particular case of the subset sum problem, the balanced variant is also time optimal.

Inverse problems for certain subsequence sums in integers

Let A be a nonempty finite set of k integers. Given a subset B of A, the sum of all elements of B is called the subset sum of B and we denote it by s(B). For a nonnegative integer α (≤k), let Σα(A)≔{s(B):B⊂A,|B|≥α}.Now, let A=(a1,…,a1︸r1copies,a2,…,a2︸r2copies,…,ak,…,ak︸rkcopies) be a finite sequence of integers with k distinct terms, where ri≥1 for i=1,2,…,k. Given a subsequence B of A, the sum of all terms of B is called the subsequence sum of B and we denote it by s(B). For 0≤α≤∑i=1kri, let Σα(A)≔s(B):Bis a subsequence ofAof length≥α.Very recently, Balandraud obtained the minimum cardinality of Σα(A) in finite fields. Motivated by Balandraud’s work, we find the minimum cardinality of Σα(A) in the group of integers. We also determine the structure of the finite set A of integers for which |Σα(A)| is minimal. Furthermore, we generalize these results of subset sums to the subsequence sums Σα(A). As special cases of our results, we obtain some already known results for the usual subset and subsequence sums.

Smallest k-Enclosing Rectangle Revisited

Given a set of n points in the plane, and a parameter $$k$$ , we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing $$k$$ points. We present the first near quadratic time algorithm for this problem, improving over the previous near- $$O(n^{5/2})$$ -time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 2017). We provide an almost matching conditional lower bound, under the assumption that $$(\min ,+)$$ -convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for both perimeter and area) that can make the time bound sensitive to $$k$$ , giving near $$O(nk)$$ time. We also present a near linear time $$(1+\varepsilon )$$ -approximation algorithm to the minimum area of the optimal rectangle containing $$k$$ points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Explicit results on conditional distributions of generalized exponential mixtures

AbstractFor independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

Computing technical capacities in the European entry-exit gas market is NP-hard

As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete on trees and NP-hard in general. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes.

Constraint Solving for Synthesis and Verification of Threshold Logic Circuits

Threshold logic (TL) circuits gain increasing attention due to their feasible realization with emerging technologies and strong bind to neural network applications. In this work, we devise techniques for automatic synthesis and verification of TL circuits based on constraint solving. For synthesis, we formulate a fundamental operation to collapse TL functions, and derive a necessary and sufficient condition of collapsibility for linear combination of two TL functions. An approach based on solving the subset sum problem is proposed for fast circuit transformation. For verification, we propose a procedure to convert a TL function to a multiplexer (MUX) tree and to pseudo-Boolean (PB) constraints for formal Boolean and PB reasoning, respectively. Experiments on synthesis show that the collapse operation further reduces gate counts of synthesized TL circuits by an average of 18%. Experiments on verification demonstrate good scalability of the MUX-based method for equivalence checking of synthesized TL circuits, and efficiency of PB constraint conversion in cases where the conjunctive normal form (CNF) formula conversion and MUX tree conversion suffer from memory explosion.

Solutions for subset sum problems with special digraph constraints

The subset sum problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two extensions of this problem: The subset sum problem with digraph constraint (SSG) and subset sum problem with weak digraph constraint (SSGW). In both problems there is given a digraph with sizes assigned to the vertices. Within SSG we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if at least one of its predecessors is part of the solution. Within SSGW we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if all its predecessors are part of the solution. SSG and SSGW have been introduced recently by Gourvès et al. who studied their complexity for directed acyclic graphs and oriented trees. We show that both problems are NP-hard even on oriented co-graphs and minimal series-parallel digraphs. Further, we provide pseudo-polynomial solutions for SSG and SSGW with digraph constraints given by directed co-graphs and series-parallel digraphs.

On subset sum problem in branch groups

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting.

Yeni Bir Metahuristik Yaklaşımla Alt Küme Toplamı Probleminin Çözümü Ve Performans Analizi

Subset sum problem was solved with two different metaheuristic approaches in the study. After these approaches, which are simulated annealing and genetic algorithms, a hybrid model of two methods was created and better results were obtained. The observed results were compared with other methods in the literature and the best time cost results were yielded owing to the hybrid algorithm developed in the study. The algorithms used gave successful results in terms of Cost values too. Performance analyses were measured on the Subset Sum Problem, defined as NP-Complete problem in computer science, with different functions used in these methods. Thus, the success of the sub-functions of the commonly used Simulated Annealing and Genetic Algorithm methods were compared and the findings were yield that could guide the researchers in other studies.

Heuristic algorithms for diversity-aware balanced multi-way number partitioning

Number partitioning is a classic problem in artificial intelligence. And balanced multi-way number partitioning problem (BMNP) aims to partition a set of numbers into multiple subsets, such that (1) each subset contains the same number of numbers and (2) the subset sums are equal. The BMNP problem has various applications in real world scenarios, including task allocation, CPU scheduling, file placement in data center, multi-source data processing, etc. In this paper, we consider the problem of diversity-aware balanced multi-way number partitioning (DBMNP). DBMNP differs from BMNP, in that each number is associated with a type attribute. In addition to the two goals of BMNP, DBMNP also requires that the types of numbers in each subset are as diversified as possible. To solve the problem, we propose three heuristic algorithms to minimize the difference between subset sums and at the same time maximize diversify of each subset. Extensive experiments are conducted to evaluate the effectiveness of our proposed algorithms.

On Almost k-Covers of Hypercubes

In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0, 1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is uncovered? The $k=1$ case is the well-known Alon-F\"uredi theorem which says a minimum of $n$ affine hyperplanes is required, proved by the Combinatorial Nullstellensatz. We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integral version for fixed $n$ and $k \rightarrow \infty$. We also use a Punctured Combinatorial Nullstellensatz developed by Ball and Serra, to show that a minimum of $n+3$ affine hyperplanes is needed for $k=3$, and pose a conjecture for arbitrary $k$ and large $n$.