Related papers: Algorithms for Solving Rubik's Cubes

The maximal clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry,…

The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini,…

The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is…

Given pairwise distinct vertices $\{\alpha_i , \beta_i\}^k_{i=1}$ of the $n$-dimensional hypercube $Q_n$ such that the distance of $\alpha_i$ and $\beta_i$ is odd, are there paths $P_i$ between $\alpha_i$ and $\beta_i$ such that $\{V…

Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…

The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a $4 \times 4$ board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the…

A generalization of Rip\`a's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$…

We consider the quantum time complexity of the all pairs shortest paths (APSP) problem and some of its variants. The trivial classical algorithm for APSP and most all pairs path problems runs in $O(n^3)$ time, while the trivial algorithm in…

Consider a distribution of pebbles on a graph. A pebbling move removes two pebbles from a vertex and place one at an adjacent vertex. A vertex is reachable under a pebble distribution if it has a pebble after the application of a sequence…

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This…

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let $S_n$ denote the set of permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their Ulam distance…

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…

We study numerically the cluster structure of random ensembles of two NP-hard optimization problems originating in computational complexity, the vertex-cover problem and the number partitioning problem. We use branch-and-bound type…

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…

A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…

Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erd\"os-R\'enyi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of…

We study the communication complexity of a number of graph properties where the edges of the graph $G$ are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: * An Omega(n) lower bound…

Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local…

The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving…