Number of order-relational different weighted $K_4$

Question

Weighted $K_4$ are the simplest non-trivial symmetric TSP instances that already exhibit a rich variety of properties, e.g. that the shortest edge is not contained in the optimal solution or that the longest edge is contained in the optimal solution and I was wondering how many essentially different order-relational properties weighted $K_4$ can exhibit.

Defintions:

$K_4:= G(V,E,\Omega)$
$V=\lbrace 1,\,2,\,3,\,4\rbrace$
$E=\big\lbrace\,\lbrace 1,2\rbrace,\,\lbrace 1,3\rbrace,\,\lbrace 1,4\rbrace,\,\lbrace 2,3\rbrace,\,\lbrace 2,4\rbrace,\,\lbrace 3,4\rbrace\,\big\rbrace$
$\Omega=\lbrace \omega_{12},\omega_{13},\omega_{14},\omega_{23},\omega_{24},\omega_{34}\rbrace⊂\mathbb{R}_+,\ \operatorname{card}⁡(Ω)=6,\\ \phantom{\Omega=}\ i\ne j\implies\omega_i\ne\omega_j$

$\mathrm{w}\in\mathbb{R}_+^6:\mathrm{w}\in\Omega,\ i\lt j\implies \mathrm{w}_i\lt\mathrm{w}_j$ is the sorted sequence of edge weights.

$G$'s set of weighted triangles is denoted by$\lbrace T_1, T_2, T_3, T_4\rbrace$ and defined via: $T_1:=\lbrace e_{12},e_{13},e_{23}\rbrace,\quad |T_1| := \omega_{12}+\omega_{13}+\omega_{23}=\mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(6)}$
$T_2:=\lbrace e_{12},e_{14},e_{24}\rbrace,\quad |T_2| := \omega_{12}+\omega_{14}+\omega_{24}=\mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(4)}$
$T_3:=\lbrace e_{13},e_{14},e_{34}\rbrace,\quad |T_3| := \omega_{13}+\omega_{14}+\omega_{34}=\mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(2)}$
$T_4:=\lbrace e_{34},e_{24},e_{23}\rbrace,\quad |T_4| := \omega_{34}+\omega_{24}+\omega_{23}=\mathrm{w}_{\pi(2)}+\mathrm{w}_{\pi(4)}+\mathrm{w}_{\pi(6)}$

$|T_1|\lt|T_4|,\quad |T_2|\lt|T_4|,\quad |T_3|\lt|T_4|$

$G$'s set of weighted perfect matchings is denoted by $\lbrace M_1, M_2, M_3\rbrace$ and defined via
$M_1:=\lbrace e_{12},e_{34} \rbrace,\quad|M_1| := \omega_{12}+\omega_{34}=\mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(2)}$
$M_2:=\lbrace e_{13},e_{24}\rbrace,\quad|M_2| := \omega_{13}+\omega_{24}=\mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(4)}$
$M_3:=\lbrace e_{14},e_{23}\rbrace,\quad|M_3| := \omega_{14}+\omega_{23}=\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(6)}$

$|M_1|\lt|M_2|\lt|M_3|$

If a canonical order is imposed on the edges, like e.g.
$\Big(\ e_{12},\, e_{34},\, e_{13},\, e_{24},\, e_{14},\, e_{23}\ \Big):=\\ \Big(\ M_1\setminus T_4,\ M_1\cap T_4,\ M_2\setminus T_4,\ M_2\cap T_4,\ M_3\setminus T_4,\ M_3\cap T_4\ \Big)$
each edge's weight is also unambiguously defined by its position in that order, i.e. that position serves as the index into the ascending order of edge-weights, interpreted in this question as an index permutation denoted by $\pi(i)$.

Questions:

what is the cardinality of the following set of permutations: \begin{align}\mathscr{P}= & \lbrace\\ & \pi(1,2,3,4,5,6)\,\big|\\ & \exists \mathrm{w}\in\mathbb{R}_+^6:\\ & i\lt j\implies\mathrm{w}_i\lt\mathrm{w}_j,\\ & \mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(2)}\lt\mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(4)}\lt\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(6)},\\ & \mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(6)}\lt\mathrm{w}_{\pi(2)}+\mathrm{w}_{\pi(4)}+\mathrm{w}_{\pi(6)},\\ & \mathrm{w}_{\pi(1)}+\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(4)}\lt\mathrm{w}_{\pi(2)}+\mathrm{w}_{\pi(4)}+\mathrm{w}_{\pi(6)},\\ & \mathrm{w}_{\pi(3)}+\mathrm{w}_{\pi(5)}+\mathrm{w}_{\pi(2)}\lt\mathrm{w}_{\pi(2)}+\mathrm{w}_{\pi(4)}+\mathrm{w}_{\pi(6)},\\ & \rbrace\end{align}

• without further constraints?
• what is the smallest set of real values that generate all possible valid permutation?

edit:
the notation I use is to be interpreted in the following way: $\mathrm{w}_{\pi(i)}$ is the index in the ordered sequence of weights, to which index $i$ in the canonical sequence will be mapped via the permutation. if e.g. $\pi(6)=5$ then edge $e_{23}$'s weight $\omega_{23}$ is the 5th largest of the six weights.

Answer 1

The first question may be answered, as you yourself implied in comments, by using a linear programming solver. Applying GNU's lpsolve to the question of which permutations have solutions, I find that there are 92 of them.

The solutions produced by lpsolve give an upper bound to the second question of 11, because they involve only the integers 0 to 10. A bit of brute-force testing shows that the smallest subset of the first 24 natural numbers which works has size 9: e.g. $\{0, 2, 3, 7, 9, 10, 11, 15, 18\}$.

For lower bounds, note that the permutations $142356$ and $231456$ impose opposite constraints on the sums $w_1 + w_4 <> w_2 + w_3$, so that at least 7 distinct values are required.

To exhaustively check potential solutions with 7 distinct values, I first ordered the permutations by the number of constraints they have. Of the five constraints following $i\lt j\implies\mathrm{w}_i\lt\mathrm{w}_j$, each permutation automatically satisfies between two and five of them just from $i\lt j\implies\mathrm{w}_i\lt\mathrm{w}_j$. E.g. the permutation $123456$ automatically satisfies $\mathrm{w}_{\pi(1)} + \mathrm{w}_{\pi(2)} < \mathrm{w}_{\pi(3)} + \mathrm{w}_{\pi(4)}$ because that reduces to $\mathrm{w}_1 + \mathrm{w}_2 < \mathrm{w}_3 + \mathrm{w}_4$ which follows from $\mathrm{w}_1 < \mathrm{w}_3$ and $\mathrm{w}_2 < \mathrm{w}_4$. If we label our seven values as $t_0$ to $t_6$ then we must assign a subset of six of those to each of the 92 valid permutations; by starting with the permutations which impose most additional constraints and using lpsolve to test for feasibility after each assignment, the cases to test reduce from $\binom{7}{6}^{92}$ to something manageable; in fact, I find that the first 10 permutations suffice to show that there is no solution with seven values. These permutations are $142365$, $152364$, $152463$, $213645$, $214536$, $231465$, $231564$, $231654$, $241563$, $241653$.

Thus the answer is restricted to 8 or 9. I am attempting a similar exhaustive search to eliminate 8, but I'm pessimistic that it will complete in a reasonable timeframe.