an:04027206
Zbl 0631.90081
Gilmore, P. C.; Lawler, E. L.; Shmoys, D. B.
Well-solved special cases
EN
The traveling salesman problem, a guided tour of combinatorial optimization, 87-143 (1985).
1985
a
90C35 90-02 68Q25 05C35
survey; travelling salesman; polynomial algorithms; pyramidal tours; subtour patching; circulants; open Hamiltonian path; limited bound widths; Halin graphs
[For the entire collection see Zbl 0562.00014.]
The authors give an excellent survey on special travelling salesman problems which can be solved by polynomial algorithms. There are two kinds of conditions possible: algebraic conditions for the cost matrix or graph theoretic conditions on the underlying network. In particular the following problem classes are treated: the ``constant'' TSP yields the same objective function value for all tours. Its cost elements have the form \(c_{ij}=ai+b_ j\) for given vectors \(a=(a_ i)\) and \(b=(b_ j)\). Also ``small TSP's'' with \(c_{ij}=\min \{a_ i,b_ j\}\) can be solved polynomially, whereas Sarvanov showed that the TSP with product matrices \(c_{ij}=a_ ib_ j\) is NP-hard. A proof of this famous theorem is given. Moreover, an extensive survey on the general Demidenko- conditions, the theory of pyramidal tours and subtour patching is given (inclusive proofs). Here also the Gilmore-Gomory case is treated.
Moreover it is mentioned that circulants as cost-matrices allow to find polynomially an open Hamiltonian path whereas it is an open question whether the Hamiltonian tour problem is NP-hard. TSPs with graded matrices \((c_{ij}\leq c_{ij+1}\) for all i,j) are NP-hard for sum objectives, but polynomially tractible for bottleneck objectives. TSPs with upper triangular cost-matrices can be solved via assignment problems. This leads to graphtheoretic conditions: TSPs on networks with limited bound widths and on Halin graphs can be solved by efficient algorithms.
R.E.Burkard
Zbl 0562.00014