Report on “A dynamical programming solution for shortest path itineraries in
robotics” by M. Talbot
This paper concerns a problem in autonomous mobile robotics where one wants to find
the shortest path from one location to another under the condition that certain other
locations must be visited. As the author notes, this problem is similar to, but not exactly
the same as, the well-known traveling salesman problem. A two-phase approach is
proposed for this problem using integer programming techniques. Overall, the
application is very interesting, and the paper could be acceptable, but major changes need
to be made. As it is written, it should not be published. But, if the changes below are
made, it should be considered again.
There is a problem with this approach. In the second-phase of the algorithm, a modified
shortest path problem is solved (formulation C). The author claims that the solution to
this problem will never have subtours, which is incorrect. If, in the example used, the
arcs (11, 15) and (15, 11) each have time 4 instead of 6, an optimal solution is x1,8 =
x8,7 = x11,13 = x13,11 = 1, giving a subtour. Thus, subtours need to be considered in
this formulation.
The author also needs to be careful in how he solves his problems. First, in order to
generate formulation C, the author goes through a number of shortest path calculations.
One would not solve these problems using linear programming, because it is too slow.
One would either use a combinatorial shortest path algorithm like Dijkstra’s method
repeatedly, or would use an All-Pairs shortest path algorithm. In either case, the
complexity is polynomial and performs much better than any LP solver; such
combinatorial methods take advantage of the special structure of the problem while LP
solvers generally don’t. Second, formulation C cannot be solved in general by the
simplex method. This is because there is no guarantee that the LP solution would be
integral. Thus, some sort of IP algorithm such as branch-and-bound would be needed.
Some other minor points on the paper: (1) the author should forego the table of contents;
(2) the author notes the given problem is NP-complete. He should give a reference to
this. Because subtours are allowed, is it possible this problem can be solved in
polynomial time? (3) Since most undergraduates may not have seen any linear
programming, LP and IP should be introduced, and the simplex method mentioned as one
algorithm to solve IP. (4) There is no need to mention sensitivity analysis and duality.
(5) The MCNFP model needs summations in the first group of constraints. (6) Remove
the section on the mechanics of the simplex algorithm. (7) Sections 10 and 11 can be
combined into 1 section. This will make the paper flow better.