Related papers: Polynomial-time Method of Determining Subset Sum S…
This paper introduces a deterministic algorithm for solving an instance of the Subset Sum Problem based on a new method entitled the Bipartite Synthesis Method. The algorithm is described and shown to have worst-case limiting performance…
NP-Complete problems have an important attribute that if one NP-Complete problem can be solved in polynomial time, all NP-Complete problems will have a polynomial solution. The 3-CNF-SAT problem is a NP-Complete problem and the primary…
The question of whether the complexity class P is equal to the complexity class NP has been a seemingly intractable problem for over 4 decades. It has been clear that if an algorithm existed that would solve the problems in the NP class in…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
We consider the SUBSET SUM problem and its important variants in this paper. In the SUBSET SUM problem, a (multi-)set $X$ of $n$ positive numbers and a target number $t$ are given, and the task is to find a subset of $X$ with the maximal…
We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several…
Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of…
The subset sum problem over finite fields is a well-known {\bf NP}-complete problem. It arises naturally from decoding generalized Reed-Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a…
We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…
We propose a polynomially bounded, in time and space, method to decide whether a given 3-SAT formula is satisfiable or not. The tools we use here are, in fact, very simple. We first decide satisfiability for a particular 3-SAT formula,…
The CSP (constraint satisfaction problems) is a class of problems deciding whether there exists a homomorphism from an instance relational structure to a target one. The CSP dichotomy is a profound result recently proved by Zhuk (2020, J.…
We consider the robust version of items selection problem, in which the goal is to choose representatives from a family of sets, preserving constraints on the allowed items' combinations. We prove NP-hardness of the deterministic version,…
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must…
A polynomial algorithm is obtained for the NP-complete linear ordering problem.
We present a complexity reduction algorithm for a family of parameter-dependent linear systems when the system parameters belong to a compact semi-algebraic set. This algorithm potentially describes the underlying dynamical system with…
A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…