Minimizing Wallpaper Waste, Part 1: A Class of Traveling Salesman Problems

Abstract

We address the problem of wallpapering a room so as to minimize the paper wasted. We show that the problem is equivalent to finding a shortest hamiltonian chain in a highly structured graph. When the chain connects two equivalent nodes (traveling salesman problem), the “nearest-neighbor” technique yields an optimal solution.

The Symmetric Circulant Traveling Salesman Problem

“…It is still an open question whether the Directed Hamiltonian Circuit problem, restricted to circulant (directed) graphs, remains NP-hard, or not. A solution in some special cases has been found by Garfinkel (1977), Fan Yang, et al (1997, and Bogdanowicz (2005). The Hamiltonian Circuit problem admits, instead, a polynomial time algorithm on the circulant undirected graphs, as shown by Burkard, and Sandholzer (1991).…”

“…As observed by Garfinkel in (1977), the permutation T 1 : Z n → Z n , defined as T 1 (i) = 〈i + a 1 〉 n , for any i∈Z n , is a Hamiltonian tour for D if and only if gcd(n, a 1 ) = 1. In this case T 1 is, clearly, optimal.…”

Optimization of Operation Sequence in CNC Machine Tools Using Genetic Algorithm

“…TSP is one of the most widely studied combinatorial optimization problems (7), (8), (9) . The most common practical interpretation of the TSP is that of a salesman seeking the shortest tour through N cities, visiting each city exactly once and returning back to the starting city.…”

Well-Solvable Special Cases of the Traveling Salesman Problem: A Survey

“…Garfinkel [57] proved that the number of directed cycles associated with the kth stripe of an n×n circulant matrix is given by gcd(k, n). Boesch and Tindell [10] proved that the circulant graph C n (c 1 , c 2 and conjectured that all connected circulant graphs are Hamiltonian.…”