21 research outputs found
Euclidean TSP with few inner points in linear space
Given a set of points in the Euclidean plane, such that just points
are strictly inside the convex hull of the whole set, we want to find the
shortest tour visiting every point. The fastest known algorithm for the version
when is significantly smaller than , i.e., when there are just few inner
points, works in time [Knauer and Spillner,
WG 2006], but also requires space of order . The best
linear space algorithm takes time [Deineko, Hoffmann, Okamoto,
Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space
time algorithm. The new insight is extending the
known divide-and-conquer method based on planar separators with a
matching-based argument to shrink the instance in every recursive call. This
argument also shows that the problem admits a quadratic bikernel.Comment: under submissio
The multi-stripe travelling salesman problem
In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP
A solvable case of the quadratic assignment problem
This short note investigates a restricted version of the quadratic assignment problem (QAP), where one of the coefficient matrices is a Kalmanson matrix, and where the other coefficient matrix is a symmetric circulant matrix that is generated by a decreasing function. This restricted version is called the Kalmanson-circulant QAP. We prove that – in strong contrast to the general QAP – this version can be solved easily. Our result naturally generalizes a well-known theorem of Kalmanson on the travelling salesman problem
Linearizable special cases of the QAP
We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set problem
Exact algorithms for the Hamiltonian cycle problem in planar graphs
We construct an exact algorithm for the Hamiltonian cycle problem in planar graphs with worst case time complexity , where c is some fixed constant that does not depend on the instance. Furthermore, we show that under the exponential time hypothesis, the time complexity cannot be improved to
Well-solvable cases of the QAP with block-structured matrices
We investigate special cases of the quadratic assignment problem (QAP) where one of the two underlying matrices carries a simple block structure. For the special case where the second underlying matrix is a monotone anti-Monge matrix, we derive a polynomial time result for a certain class of cut problems. For the special case where the second underlying matrix is a product matrix, we identify two sets of conditions on the block structure that make this QAP polynomially solvable respectively NP-hard
Well-solvable special cases of the Traveling Salesman Problem : a survey
The traveling salesman problem (TSP) belongs to the most basic, most important, and most investigated problems in combinatorial optimization. Although it is an -hard problem, many of its special cases can be solved efficiently in polynomial time. We survey these special cases with emphasis on the results that have been obtained during the decade 1985--1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler, and Shmoys [The Traveling Salesman Problem---A Guided Tour of Combinatorial Optimization, Wiley, Chichester, pp. 87--143]