Related papers: On the Complexity of the Skolem Problem at Low Ord…

An Ulam sequence U(1,n) is defined as the sequence starting with integers 1,n such that n > 1, and such that every subsequent term is the smallest integer that can be written as the sum of distinct previous terms in exactly one way. This…

We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology…

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This…

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$\Phi$(N)$?…

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to…

Given an integer array $A[1..n]$, the Range Minimum Query problem (RMQ) asks to preprocess $A$ into a data structure, supporting RMQ queries: given $a,b\in [1,n]$, return the index $i\in[a,b]$ that minimizes $A[i]$, i.e.,…

We consider the longest common subsequence (LCS) problem with the restriction that the common subsequence is required to consist of at least $k$ length substrings. First, we show an $O(mn)$ time algorithm for the problem which gives a…

Nearly linear recurrences are a generalisation of linear recurrences and are instances of linear time-invariant systems in control theory and linear constraint loops in program analysis. In this paper we formulate the Positivity Problem for…

Let ${\mathbf P}$ be the class of polynomial-time decision problems and $\mathbf{NP}$ be the class of nondeterministic polynomial time decision problems. We prove the following: Theorem 3. The classes ${\mathbf P}$ and $\mathbf{NP}$ are…

Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: "Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with…

This paper focuses on the study of the order of power series that are linear combinations of a given finite set of power series. The order of a formal power series, known as $\textrm{ord}(f)$, is defined as the minimum exponent of $x$ that…

The Sum of Square Roots (SSR) problem is the following computational problem: Given positive integers $a_1, \dots, a_k$, and signs $\delta_1, \dots, \delta_k \in \{-1, 1\}$, check if $\sum_{i=1}^k \delta_i \sqrt{a_i} > 0$. The problem is…

The linear ordering problem (LOP), which consists in ordering M objects from their pairwise comparisons, is commonly applied in many areas of research. While efforts have been made to devise efficient LOP algorithms, verification of whether…

Over the past decade, the low-degree heuristic has been used to estimate the algorithmic thresholds for a wide range of average-case planted vs null distinguishing problems. Such results rely on the hypothesis that if the low-degree moments…

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors…

In this paper, we address the Skolem problem for the $k$-generalized Pell sequence $(P_n^{(k)})_{n\geq2-k}$ extended to negative indices. We focus on identifying and bounding the indices $n<0$ for which $P_n^{(k)}=0.$ In particular, we…

We introduce the algorithm ExpoSort, a groundbreaking method that sorts an array of $n$ numbers in a spectacularly inefficient $\Theta(2^n)$ time. ExpoSort proudly claims the title of the first reluctant algorithm to decisively surpass the…

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it,…

Leveraging algorithmic stability to derive sharp generalization bounds is a classic and powerful approach in learning theory. Since Vapnik and Chervonenkis [1974] first formalized the idea for analyzing SVMs, it has been utilized to study…

We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items…