2
$\begingroup$

Question:

have there been any serious (meaning by a reputated mathematician) attempts to solve the euclidean TSP in the complex plane by interpreting the $(x,y)$ coordinates of the real plane as complex numbers and then applying function-theoretic methods?

To be more concrete:
it would be an obvious idea to interpret the $(x,y)$ coordinates of the real instance as the complex zeroes or poles of some complex function and then try to express simple closed tours e.g. as certain conformal mappings of the unit circle.
As it is often the case, that a real-analytic problem is easier solved in the complex plane, it seems justified to put some hope into an easier solution of the ETSP by transferring it to the complex plane or, maybe even to quaternian space.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Can you explain what TSP and ETSP are?? $\endgroup$
    – Alexandre Eremenko
    Commented May 14, 2017 at 9:49
  • 1
    $\begingroup$ Travelling Salesman Problem (TSP), and Euclidean TSP presumably $\endgroup$
    – kodlu
    Commented May 14, 2017 at 9:56

1 Answer 1

Reset to default
1
$\begingroup$

If you map the boundary of the unit circle conformally, then the image will not have kinks, but an optimal tour should have kinks. Probably I just did not get what you had in mind...

Improve this answer
$\endgroup$
3
  • $\begingroup$ Wait, what?! The Riemann mapping theorem allows you to conformally map the interior of any Jordan curve (including any polygon; cf. Schwarz-Christoffel transformation) to the interior of the unit disk. $\endgroup$
    – Adam P. Goucher
    Commented May 14, 2017 at 12:32
  • $\begingroup$ Oops, this was actually not meant as an answer... $\endgroup$
    – Dirk
    Commented May 14, 2017 at 12:35
  • 1
    $\begingroup$ @AdamP.Goucher that was exactly what I had in mind, but "forgot" to restrict conformity to the open disk; the tour would then be the the closure of the open disk's image. $\endgroup$
    – Manfred Weis
    Commented May 14, 2017 at 18:27

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .