Pathwidth, trees, and random embeddings

Pathwidth, trees, and random embeddings

We prove that, for every integer k ≥ 1, every shortest-path metric on a graph of pathwidth k embeds into a distribution over random trees with distortion at most c = c(k), independent of the graph size. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair [GNRS04] states that for every minor-closed family of graphs F , there is a constant c(F) such that the multi-commodity max-flo...

CSP duality and trees of bounded pathwidth

We study non-uniform constraint satisfaction problems definable in monadic Datalog stratified by the use of non-linearity. We show how such problems can be described in terms of homomorphism dualities involving trees of bounded pathwidth and in algebraic terms. For this, we introduce a new parameter for trees that closely approximates pathwidth and can be characterised via a hypergraph searchin...

Pathwidth And Layered Drawings Of Trees

An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-...

Randomized Algorithms 2017A - Lecture 10 Metric Embeddings into Random Trees

Pathwidth is NP-Hard for Weighted Trees

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. We prove in this paper that the PATHWIDTH problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is polynomial.