Decomposition of Trees and Paths via Correlation [article]

Jan-Hendrik Lange, Bjoern Andres
2017 arXiv   pre-print
We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for the special case of the tree being a star, we show that the problem is NP-hard. For the special case of the tree being a path, this problem is known to be polynomial time solvable. We characterize several classes of facets of the combinatorial polytope
more » ... ed with a formulation of this clustering problem in terms of lifted multicuts. In particular, our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths, which establishes a connection to the combinatorial properties of alternative formulations such as set partitioning.
arXiv:1706.06822v2 fatcat:nsm2yilpkvefzkzxyn7ctq665m