403 research outputs found
On computing tree and path decompositions with metric constraints on the bags
We here investigate on the complexity of computing the \emph{tree-length} and
the \emph{tree-breadth} of any graph , that are respectively the best
possible upper-bounds on the diameter and the radius of the bags in a tree
decomposition of . \emph{Path-length} and \emph{path-breadth} are similarly
defined and studied for path decompositions. So far, it was already known that
tree-length is NP-hard to compute. We here prove it is also the case for
tree-breadth, path-length and path-breadth. Furthermore, we provide a more
detailed analysis on the complexity of computing the tree-breadth. In
particular, we show that graphs with tree-breadth one are in some sense the
hardest instances for the problem of computing the tree-breadth. We give new
properties of graphs with tree-breadth one. Then we use these properties in
order to recognize in polynomial-time all graphs with tree-breadth one that are
planar or bipartite graphs. On the way, we relate tree-breadth with the notion
of \emph{-good} tree decompositions (for ), that have been introduced
in former work for routing. As a byproduct of the above relation, we prove that
deciding on the existence of a -good tree decomposition is NP-complete (even
if ). All this answers open questions from the literature.Comment: 50 pages, 39 figure
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
Treewidth distance on phylogenetic trees
In this article we study the treewidth of the display graph, an auxiliary graph structure obtained from the fusion of phylogenetic (i.e., evolutionary) trees at their leaves. Earlier work has shown that the treewidth of the display graph is bounded if the trees are in some formal sense topologically similar. Here we further expand upon this relationship. We analyse a number of reduction rules, commonly used in the phylogenetics literature to obtain fixed parameter tractable algorithms. In some cases (the subtree reduction) the reduction rules behave similarly with respect to treewidth, while others (the cluster reduction) behave very differently, and the behaviour of the chain reduction is particularly intriguing because of its link with graph separators and forbidden minors. We also show that the gap between treewidth and Tree Bisection and Reconnect (TBR) distance can be infinitely large, and that unlike, for example, planar graphs the treewidth of the display graph can be as much as linear in its number of vertices. A number of other auxiliary results are given. We conclude with a discussion and list a number of open problems
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