I found the following problem in a Computational Topology course that I am following: Write an algorithm that given a sequence $(\beta_0,\ldots,\beta_d)$ of integers builds a simplicial complex whose $k$'th Betti number is exactly $\beta_k$.
To find some solution to this problem is not particularly hard, for example, one can create $\beta_0$ vertices. Then use the last created vertex as one of the vertices of an "empty triangle" formed by adding two more vertices (and the corresponding 3 facets), use the last added vertex to keep adding empty triangles in a similar way until you have $\beta_1$ empty triangles. Then use the last added vertex as one of the vertices of an "empty tetrahedron" formed by adding three more vertices (and the corresponding facets) and so on and so forth.
I was wondering, if given the same problem statement is there any "efficient" algorithm that creates the simplicial complex which the minimum number of simiplices or with the minimum number of vertices. Or if there is any nice way to compute what is the minimum number of simiplices/vertices needed in a simplicial complex to match a given sequence of Betty numbers.