Three disproofs of the Gilbert-Pollak Steiner ratio conjecture in more than
two dimensions
W.D. Smith, NECI
TR was "two disproofs" but now coauthored with D-Z Du and "three"
10/13/92
The Gilbert-Pollak conjecture, posed in 1968, was the most important
conjecture in the area of ``Steiner trees.'' The ``Steiner minimal
tree'' (SMT) of a point set $P$ is the shortest network of ``wires''
which will suffice to ``electrically'' interconnect $P$. The ``minimum
spanning tree'' (MST) is the shortest such network when only {\it
intersite line segments} are permitted. The GP conjecture stated that
$$\rho_d = \inf_{P \subset {\bf R}^d}
\frac{\ell \ro{SMT} ( P )}{\ell \ro{MST} ( P )}$$
was achieved when $P$ was the vertices
of a regular $d$-simplex. This conjecture is now settled: it is true
when $d=2$ and false when $d \ge 3$.
Indeed we give bounds showing that the point
sets minimizing the Steiner ratio necessarily
have an exponentially large number of sites, in high dimensions.
The real question now is: What
are the true minimal-$\rho$ point sets? The paper introduces the
``$d$-dimensional sausage'' point sets, which may have a lot to do
with the answer.
KEYWORDS:
GILBERT-POLLAK CONJECTURE, STEINER TREES, MINIMUM
SPANNING TREES, INEQUALITIES, NETWORKS, SAUSAGE