Counting Linear Extensions of Modular Partial Orders

The counting of linear extensions is a prominent problem about partial orders. Unfortunately, the problem is computationally hard and in fact, relatively few counting procedures have been proposed in the literature. In this paper, we present a new counting algorithm based on the modular decomposition of posets. To handle the prime substructures of posets, we adopt a decomposition scheme we introduced in a previous work. This scheme is based on two complementary rules: (1) the BIT-rule that can “consume” individual elements in a chain, and (2) the SPLIT-rule that can “break” antichains. The symbolic counting algorithm we propose is based on a multivariate integration problem that can be built along the decomposition. We show that for a poset of size n the parametric complexity is $\mathcal{O}\left(n^{k+1}\right)$ with k the modular-width of the poset. This algorithm provides polynomial solutions for a large class of sparse posets, including the N-sparse and the N-extensible posets. We also introduce a new and finer parameter, the BIT-width, and show that the counting complexity becomes $\mathcal{O}\left(n^{w+1}\right)$ for the class of so-called BIT-modular posets. This is an interesting bound since $w \leq k$, and moreover large modular-width posets can have low BIT-width. An illustrative example is that of caterpillar posets for which k can be arbitrarily large and $w=1$.