Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions

A $K_4$-decomposition of a graph is a partition of its edges into $K_4$s. A fractional $K_4$-decomposition is an assignment of a nonnegative weight to each $K_4$ in a graph such that the sum of the weights of the $K_4$s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on $n$ vertices with minimum degree at least $\frac{31}{33}n$ has a fractional $K_4$-decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least $\frac{399}{400}n$. Together with a result of Barber, K\"uhn, Lo, and Osthus, this result implies that for all $\varepsilon> 0$, every large enough $K_4$-divisible graph on $n$ vertices with minimum degree at least $(\frac{31}{33}+\varepsilon)n$ admits a $K_4$-decomposition.