The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions [4], and is provably not hard under "local" reductions computable in TIME(n 0.49 ) [26]. The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of
The jeu-de-taquin-based Littlewood-Richardson rule of H. for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to K-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's 'd-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of d-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain K-theoretic Schubert structure constants in the Kac-Moody setting.
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