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.

In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let D = (V, A) be a digraph, and let w ∈ Z A ≥0 . A dicut is a cut δ + (U ) ⊆ A for some nonempty proper vertex subset U such that δ − (U ) = ∅, a dijoin is an arc subset that intersects every dicut at least once, and more generally a k-dijoin is an arc subset that intersects every dicut at least k times.Suppose every dicut has weight at least τ , for some integer τ ≥ 2. Let ρ(τ, D, w) := 1 τ v∈V m v , where each m v is the integer in {0, 1, . . . , τ − 1} equal to w(δ + (v)) − w(δ − (v)) mod τ . In this paper, we prove the following results, amongst others:1. If w = 1, then A can be partitioned into a k-dijoin and a (τ − k)-dijoin, for every k ∈ [τ ].2. If ρ(τ, D, w) ∈ {0, 1}, then there is an equitable w-weighted packing of dijoins of size τ .3. If ρ(τ, D, w) = 2, then there is a w-weighted packing of dijoins of size τ . 4. If w = 1 and ρ(τ, D, w) = 3, then A can be partitioned into two dijoins and a (τ − 2)-dijoin.Each result is best possible: (1) and ( 4) do not hold for general w, (2) does not hold for ρ(τ, D, w) = 2 even if w = 1, and (3) does not hold for ρ(τ, D, w) = 3.The results are rendered possible by a Decompose, Lift, and Reduce procedure, which turns (D, w) into a set of sink-regular weighted (τ, τ +1)-bipartite digraphs, each of which is a weighted digraph where every vertex is a sink of weighted degree τ or a source of weighted degree τ, τ + 1, and every dicut has weight at least τ . From there, an application of the classical alternating path technique proves (2). For each of the weighted digraphs, we define two matroids M 0 , M 1 of rank ρ(τ, D, w) whose common ground set is the set of sources of weighted degree τ + 1, where M 0

In the Steiner Tree Augmentation Problem (STAP), we are given a graph G = (V, E), a set of terminals R ⊆ V , and a Steiner tree T spanning R. The edges L := E \ E(T ) are called links and have non-negative costs. The goal is to augment T by adding a minimum cost set of links, so that there are 2 edge-disjoint paths between each pair of vertices in R. This problem is a special case of the Survivable Network Design Problem which can be approximated to within a factor of 2 using iterative rounding [14].We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular we achieve a ratio of (1+ln 2+ε) ≈ 1.69+ε. To do this, we use the Local Greedy approach of [22] for the Tree Augmentation Problem and generalize their main decomposition theorem from links (of size two) to hyper-links.We also consider the Node-Weighted Steiner Tree Augmentation Problem (NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a cheapest subset S ⊆ V \ R so that G[R ∪ S] is 2-edge-connected. We provide a O(log 2 (|R|))-approximation algorithm for NW-STAP. To do this, we use a greedy algorithm leveraging the spider decomposition of optimal solutions.

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We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the ODD-LP relaxation for the (weighted) Tree Augmentation Problem for a k-level tree instance is at most 2 − 1 2 k−1 . For 2-and 3-level trees, these ratios are 3 2 and 7 4 respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.