Abstract: Given an edge weighted graph and a spanning tree, $T$, the {\em tree augmentation problem} is to pick a minimum weight set of edges, $F$, such that $T\cup F$ is 2-edge connected. Williamson et.al.
Abstract: Consider the following 2-respecting min-cut problem: Given a weighted graph G and its spanning tree T, find the minimum cut among the cuts that contain at most two edges in T.
Reed--Muller (RM) codes, a classical family of codes known for their elegant algebraic structure, have recently been shown to achieve capacity under maximum-likelihood (ML) decoding on the binary erasure channel and th