10 research outputs found
Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs
In this paper, we present a simple factor 6 algorithm for approximating the
optimal multiplicative distortion of embedding a graph metric into a tree
metric (thus improving and simplifying the factor 100 and 27 algorithms of
B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi,
Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor
algorithm for approximating the optimal distortion of embedding a graph metric
into an outerplanar metric. For this, we introduce a general notion of metric
relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then
the distortion of any embedding of G into any metric induced by a H-minor free
graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which
either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an
outerplanar metric.Comment: 27 pages, 4 figires, extended abstract to appear in the proceedings
of APPROX-RANDOM 201
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
Spanners for Bounded Tree-Length Graphs
International audienc
Sparse Additive Spanners for Bounded Tree-Length Graphs
This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most , i.e., the tree-length graphs. For such graphs we construct additive 2-spanners with O(n log n) edges, and additive 6-spanners with O(n) edges. We also show a lower bound, and prove that there are graphs of tree-length for which every multiplicative -spanner (and thus every additive ( 1)-spanner) requires 1+1=() ) edges