Combinatorial characterization of the matching exponent

Determine a combinatorial characterization of the function s_*(n), defined as the minimum, over all orderings of the x-variables, of the maximum split-set cardinality over all induced partitions and all pairs a<b.

Background

The paper studies the OBDD width of a restricted middle-bit integer multiplication function in which the y-input has exactly two bits set to 1. For a partition (L,R) of the x-variables and indices a<b, the authors define a split set (L,R;a,b) that records residue classes containing adjacent variables separated between L and R. The function s_*(n) is then defined by optimizing the maximum cardinality of these split sets over x-variable orderings, partitions, and pairs (a,b).

The authors prove that s_(n) determines the OBDD-width exponent up to constant factors: every variable ordering has a level of width at least 2^{s_(n)}, while a suitable ordering achieves width at most a polynomial factor times 2^{s_*(n)}. They leave unresolved the task of giving a more direct combinatorial characterization of s_*(n), beyond its optimization-based definition.