New special cases of the quadratic assignment problem with diagonally structured coefficient matrices

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time

Optimizing the Incidences between Points and Arcs on a Circle

Projet PRAXITELEGiven a set P of 2n+1 points regularly spaced on a circle, a number pi for pairwise distinct points and a number alpha for pairwise distinct and fixed length arcs incident to points in P, the sum of incidences between alpha arcs and pi points, is optimized by contiguously assigning both arcs and points. An extension to negative incidences by considering $\pm 1$ weights on points is provided. Optimizing a special case of a bilinear form (Hardy, Littlewood and Pólya' theorem) as well as Circulant $\times$ anti-Monge QAP directly follow

Travelling salesman paths on Demidenko matrices

In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t. In this paper we show that a special case of the Path-TSP with a Demidenko distance matrix is solvable in polynomial time. Demidenko distance matrices fulfill a particular condition abstracted from the convex Euclidian special case by Demidenko (1979) as an extension of an earlier work of Kalmanson (1975). We identify a number of crucial combinatorial properties of the optimal solution and design a dynamic programming approach with time complexity O(n6)

A Note on the Maximum of a certain Bilinear Form

In this note a generalization of a result by Hardy, Littlewood and P'olya (1926) is derived on computing the maximum of a certain bilinear form. The proof is elementary