Related papers: On the Hardness of Average-case k-SUM
This paper presents universal algorithms for clustering problems, including the widely studied $k$-median, $k$-means, and $k$-center objectives. The input is a metric space containing all potential client locations. The algorithm must…
Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting…
This paper presents a new solution for choosing the K parameter in the k-nearest neighbor (KNN) algorithm, the solution depending on the idea of ensemble learning, in which a weak KNN classifier is used each time with a different K,…
The recent emergence of fine-grained cryptography strongly motivates developing an average-case analogue of Fine-Grained Complexity (FGC). This paper defines new versions of OV, $k$SUM and zero-$k$-clique that are both worst-case and…
Lagarias and Odlyzko (J.~ACM~1985) proposed a polynomial time algorithm for solving ``\emph{almost all}'' instances of the Subset Sum problem with $n$ integers of size $\Omega(\Gamma_{\text{LO}})$, where $\log_2(\Gamma_{\text{LO}}) > n^2…
The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this…
Clustering is one of the most fundamental tools in data science and machine learning, and k-means clustering is one of the most common such methods. There is a variety of approximate algorithms for the k-means problem, but computing the…
Given a basic compact semi-algebraic set $\K\subset\R^n$, we introduce a methodology that generates a sequence converging to the volume of $\K$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear…
In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand,…
Given an array A of n real numbers, the maximum subarray problem is to find a contiguous subarray which has the largest sum. The k-maximum subarrays problem is to find k such subarrays with the largest sums. For the 1-maximum subarray the…
We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation of study for this problem is the…
The $k$-means method is an iterative clustering algorithm which associates each observation with one of $k$ clusters. It traditionally employs cluster centers in the same space as the observed data. By relaxing this requirement, it is…
Consider the commonly known puzzle, given $k$ glass balls, find an optimal algorithm to determine the lowest floor of a building of $n$ floors from which a thrown glass ball will break. This puzzle was originally posed in its original form…
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…
Given a set (or multiset) S of n numbers and a target number t, the subset sum problem is to decide if there is a subset of S that sums up to t. There are several methods for solving this problem, including exhaustive search,…
The subset sum problem is known to be an NP-hard problem in the field of computer science with the fastest known approach having a run-time complexity of $O(2^{0.3113n})$. A modified version of this problem is known as the perfect sum…
The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we…
$k$-means clustering is a well-studied problem due to its wide applicability. Unfortunately, there exist strong theoretical limits on the performance of any algorithm for the $k$-means problem on worst-case inputs. To overcome this barrier,…
Given a k-dimensional subspace M\subseteq \R^n and a full rank integer lattice L\subseteq \R^n, the \emph{subspace avoiding problem} SAP is to find a shortest vector in L\setminus M. Treating k as a parameter, we obtain new parameterized…
We study the Subset Balancing problem: given $x \in \mathbb{Z}^n$ and a coefficient set $C \subseteq \mathbb{Z}$, find a nonzero vector $c \in C^n$ such that $c\cdot x = 0$. The standard meet-in-the-middle algorithm runs in time…