Related papers: Combinatorial Explorations in Su-Doku
The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject…
The Sequential Multiple Knapsack Problem is a special case of Multiple knapsack problem in which the items sizes are divisible. A characterization of the optimal solutions of the problem and a description of the convex hull of all the…
The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of…
We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1 x 2 pieces on a hexagonal grid board of 2n + 1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single…
A Constraint Satisfaction Problem (CSP) is a computational problem where we are given variables and constraints about them; the question is whether the variables can be assigned values such that all constraints are satisfied. We give an…
Nonogram is a popular combinatorial puzzle (similar in nature to Sudoku or Minesweeper) in which a puzzle solver must determine if there exists a setting of the puzzle parameters that satisfy a given set of constraints. It has long been…
Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with…
Seriation methods order a set of descriptions given some criterion (e.g., unimodality or minimum distance between similarity scores). Seriation is thus inherently a problem of finding the optimal solution among a set of permutations of…
The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of…
We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with…
Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then…
We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of…
Many puzzle video games, like Sokoban, involve moving some agent in a maze. The reachable locations are usually apparent for a human player, and the difficulty of the game is mainly related to performing actions on objects, such as pushing…
We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and…
The Rubix Cube is a 3-dimensional single-player combination puzzle attracting attention in the reinforcement learning community. A Rubix Cube has six faces and twelve possible actions, leading to a small and unconstrained action space and a…
Many popular puzzle and matching games have been analyzed through the lens of computational complexity. Prominent examples include Sudoku, Candy Crush, and Flood-It. A common theme among these widely played games is that their generalized…
In this article we propose a heuristic algorithm to explore search space trees associated with instances of combinatorial optimization problems. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is…
Solving avoidability problems in the area of string combinatorics often requires, in an initial step, the construction, via a computer program, of a very long word that does not contain any word that matches a given pattern. It is well…
Global constraints proved themselves to be an efficient tool for modelling and solving large-scale real-life combinatorial problems. They encapsulate a set of binary constraints and using global reasoning about this set they filter the…
Based on a previous generalization by the author of Latin squares to Latin boards, this paper generalizes partial Latin squares and related objects like partial Latin squares, completable partial Latin squares and Latin square puzzles. The…