subset sum

A directed graph D can be labeled antimagically by assigning different integers to its arcs, ensuring that the computed vertex weights are distinct. Antimagic labeling exists in a digraph D with h arcs and g vertices if it is possible to uniquely match each arc to an integer from 1 to h. For every vertex, the difference between the totals of the labels from incoming arcs and outgoing arcs is unique for each vertex. A directed graph that allows such antimagic labeling is referred to as an antimagic digraph. There are countless methods to create an antimagic digraph. These constructions are essential in fields such as network theory, coding theory, and combinatorial optimization. The subset sum problem is a well-recognized issue in both computer science and combinatorics. These problems play a decisive role in graph labeling, especially when it comes to the creation and evaluation of specific types of labels. In this paper, we connect the idea of subset sum problems with wheel digraphs-→ W n to represent it as antimagic. The Cartesian product of directed graphs is a fundamental concept in graph theory that holds both theoretical and practical importance. The applications of Cartesian products include network design and analysis, parallel computing, graph decomposition and construction, Game Theory, and Decision-Making. Additionally, in this paper, we have developed antimagic digraphs from the Cartesian product of directed path P n and-→ K 2 by traversing the directed path-→ P n in alternating and unidirectional ways.

Quantum computers are able to destroy most, if not absolutely all conventional cryptosystems that are widely used in practice, specifically, systems based on the problem of factoring integers (e.g., RSA). Some cryptosystems like RSA system with 4 000bit keys are considered useful to protect classic computers from attacks, but probably absolutely useless against attacks on quantum computers. One of the alternatives are post-quantum systems, systems based on lattices. These systems are known for high security levels based on the worst-case hardness. They are based on the complexity of the problems grids, the main of which is the problem of the shortest vector (SVP). In fact, we consider the approximate option when we find a lattice vector, with the length of (n) times more than the shortest nonzero vector. ±(n) is the approximation coefficient, n is the lattice size. The best known algorithm for grids problems is LLL algorithm. This algorithm needs polynomial time with approximation ...

Subset sum problems are a special class of difficult singly constrained zero-one integer programming problems. Several heuristics for solving these problems have been reported in the literature. In this paper we propose a new heuristic based on local search which improves upon the previous best.

In a traditional multiple subset sum problem (MSSP), there is a given set of items and a given set of bins (or knapsacks) with identical capacities. The objective is to select a subset of the items and pack them into the bins such that the total weight of the selected items is maximized. However, in many applications of the MSSP, the bins have assignment restrictions. In this article, we study the subset sum problem with inclusive assignment set restrictions, in which the assignment set of one item (i.e., the set of bins that the item may be assigned to) must be either a subset or a superset of the assignment set of another item. We develop an efficient 0.6492-approximation algorithm and test its effectiveness via computational experiments. We also develop a polynomial time approximation scheme for this problem.

In this paper, we present a new lattice-based public-key cryptosystem mixed with a factoring (RSA), which has reasonable key size and quick encryption and decryption. We consider the situation that the RSA secret key d is small and a sufficient amount of the LSBs (least significant bits) of d are known by the attacker. We show that our lattice construction is theoretically more efficient than known attacks proposed in (7, 14). Moreover, we can use the same module strategy to construct a framework for some GGH-type cryptosystems to improve their security.

Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer u in O({√(n)u,u^4/3,σ}), where σ is the sum of all elements in S and O hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in O({√(n)m,m^5/4}) time, where m is the order of the group.

Subset sum problems are a special class of di cult singly constrained zero-one integer programming problems. Several heuristics for solving these problems have been reported in the literature. In this paper we propose a new heuristic based on local search which improves upon the previous best.

We introduce a new combinatorial optimization problem in this article, called the minimum common integer partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n , and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S . Given a sequence of multisets S 1 , S 2 , …, S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset S i , 1 ≤ i ≤ k . The MCIP problem is thus defined as to find a common integer partition of S 1 , S 2 , …, S k with the minimum cardinality, denoted as MCIP( S 1 , S 2 , …, S k ). It is easy to see that the MCIP problem is NP-hard, since it generalizes the well-known subset sum problem. We can in fact show that it is APX-hard. We will also present a 5/4-...

We study a problem of model selection for data produced by two different context tree sources. Motivated by linguistic questions, we consider the case where the probabilistic context trees corresponding to the two sources are finite and share many of their contexts. In order to understand the differences between the two sources, it is important to identify which contexts and which transition probabilities are specific to each source. We consider a class of probabilistic context tree models with three types of contexts: those which appear in one, the other, or both sources. We use a BIC penalized maximum likelihood procedure that jointly estimates the two sources. We propose a new algorithm which efficiently computes the estimated context trees. We prove that the procedure is strongly consistent. We also present a simulation study showing the practical advantage of our procedure over a procedure that works separately on each dataset.

For n ∈ N, we consider the problem of partitioning the interval [0, n) into k subintervals of positive integer lengths 1 , . . . , k such that the lengths satisfy a set of simple constraints of the form i ij j where ij is one of <, >, or =. In the full information case, ij is given for all 1 i, j k. In the sequential information case, ij is given for all 1 < i < k and j = i ± 1. That is, only the relations between the lengths of consecutive intervals are specified. The cyclic information case is an extension of the sequential information case in which the relationship 1k between 1 and k is also given. We show that all three versions of the problem can be solved in time polynomial in k and log n.

For n ∈ N, we consider the problem of partitioning the interval [0, n) into k subintervals of positive integer lengths 1 , . . . , k such that the lengths satisfy a set of simple constraints of the form i ij j where ij is one of <, >, or =. In the full information case, ij is given for all 1 i, j k. In the sequential information case, ij is given for all 1 < i < k and j = i ± 1. That is, only the relations between the lengths of consecutive intervals are specified. The cyclic information case is an extension of the sequential information case in which the relationship 1k between 1 and k is also given. We show that all three versions of the problem can be solved in time polynomial in k and log n.

Given a multiset $S$ of $n$ positive integers and a target integer $t$, the subset sum problem is to decide if there is a subset of $S$ that sums up to $t$. We present a new divide-and-conquer algorithm that computes \emph{all} the realizable subset sums up to an integer $u$ in $\widetilde{O}\!\left(\min\{\sqrt{n}u,u^{4/3},\sigma\}\right)$, where $\sigma$ is the sum of all elements in $S$ and $\widetilde{O}$ hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in $O(nu)$ time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in $\widetilde{O}\!\left(\min\{\sqrt{n}m,m^{5/4}\}\right)$ time, where $m$ is the order of the group.

Subset sum problems are a special class of di cult singly constrained zero-one integer programming problems. Several heuristics for solving these problems have been reported in the literature. In this paper we propose a new heuristic based on local search which improves upon the previous best.