8 research outputs found
Approximating pathwidth for graphs of small treewidth
We describe a polynomial-time algorithm which, given a graph with
treewidth , approximates the pathwidth of to within a ratio of
. This is the first algorithm to achieve an
-approximation for some function .
Our approach builds on the following key insight: every graph with large
pathwidth has large treewidth or contains a subdivision of a large complete
binary tree. Specifically, we show that every graph with pathwidth at least
has treewidth at least or contains a subdivision of a complete
binary tree of height . The bound is best possible up to a
multiplicative constant. This result was motivated by, and implies (with
), the following conjecture of Kawarabayashi and Rossman (SODA'18): there
exists a universal constant such that every graph with pathwidth
has treewidth at least or contains a subdivision of a
complete binary tree of height .
Our main technical algorithm takes a graph and some (not necessarily
optimal) tree decomposition of of width in the input, and it computes
in polynomial time an integer , a certificate that has pathwidth at
least , and a path decomposition of of width at most . The
certificate is closely related to (and implies) the existence of a subdivision
of a complete binary tree of height . The approximation algorithm for
pathwidth is then obtained by combining this algorithm with the approximation
algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth
A 3-approximation for the pathwidth of Halin graphs
We prove that the pathwidth of Halin graphs can be 3-approximated in linear time. Our approximation algorithms is based on a combinatorial result about respectful edge orderings of trees. Using this result we prove that the linear width of Halin graph is always at most three times the linear width of its skeleton