Question

I am looking for a name for this problem or any leads on an algorithm or source code:

Example: You want to find the best route to visit the 100 largest cities in the US (classic TSP) but before you can visit any given city you must visit the capital of the state it is in.

Example: You are collecting permission slips from the students of several professors. You need to visit every student and every professor but you can’t visit a professor until you have seen all of his students.

Some Googling turns up the sequential ordering problem or “SOP” but there is not so much literature that I am convinced that this is a widely accepted name.

I don’t think these partial orderings can be represented within the classic TSP simply by choosing which edges to use in the graph (e.g. you can’t initially go from New York to Chicago, but once you visit Springfield you can) or weights, but I may be wrong.

Answer 1

The Sequential Ordering Problem was first introduced by Escudero in 1988 in a paper entitled “An Inexact Algorithm for the Sequential Ordering Problem” (this appeared in the European Journal of Operational Research), so this is the original name for the problem. The abstract of the paper reads:

Given the directed G= (N, A) and the
penalty matrix C, the Sequential
Ordering Problem (hereafter, SOP)
consists of finding the permutation of
the nodes from the set N, such that it
minimizes a C-based function and does
not violate the precedence
relationships given by the set A.
Strong sufficient conditions for the
infeasibility of a SOP’s instance are
imbedded in a procedure for the SOP’s
preprocessing. Note that it is one of
the key steps in any algorithm that
attempts to solve SOP. By dropping the
constraints related to the precedence
relationships, SOP can be converted in
the classical Asymmetric Traveling
Salesman Problem (hereafter, ATSP).
The algorithm obtains (hopefully)
satisfactory solutions by modifying
the optimal solution to the related
Assignment Problem (hereafter, AP) if
it is not a Feasible Sequential
Ordering (hereafter, FSO). The new
solution ‘patches’ the subtours (if
any) giving preference to the patches
with zero reduced cost in the linking
arcs. The AP-based lower bound on the
optimal solution to ATSP is tightened
by using some of the procedures given
in [1]. In any case, a local search
for improving the initial FSO is
performed; it uses 3- and 4-change
based procedures. Computational
results on a broad set of cases are
reported.

Escudero and his collaborators have a number of papers on the subject, with references to even more. Searching for papers by him or that reference this paper may help you if you’re looking through the literature.

SOP is a well-studied constrained version of the Asymmetric Travelling Salesman Problem, so much of the literature on ATSP may be related.