A 6-Approximation Algorithm for Computing Smallest Common AoN-Supertree with Application to the Reconstruction of Glycan Trees

A node-labeled rooted tree T (with root r) is an all-or-nothing subtree (called AoN-subtree) of a node-labeled rooted tree T′ if (1) T is a subtree of the tree rooted at some node u (with the same label as r) of T′, (2) for each internal node v of T, all the neighbors of v in T′ are the neighbors of v in T. Tree T′ is then called an AoN-supertree of T. Given a set ${\mathcal {T}}=\{{T}_1,{T}_2,\cdots, {T}_n\}$ of nnode-labeled rooted trees, smallest common AoN-supertree problem seeks the smallest possible node-labeled rooted tree (denoted as ${\textbf{LCST}}$) such that every tree Ti in ${\mathcal {T}}$ is an AoN-subtree of ${\textbf{LCST}}$. It generalizes the smallest superstring problem and it has applications in glycobiology. We present a polynomial-time greedy algorithm with approximation ratio 6.