20,083 research outputs found
Approximability of the Unsplittable Flow Problem on Trees
We consider the approximability of the Unsplittable Flow Problem (UFP) on tree graphs, and give a deterministic quasi-polynomial time approximation scheme for the problem when the number of leaves in the tree graph is at most poly-logarithmic in (the number of demands), and when all edge capacities and resource requirements are suitably bounded. Our algorithm generalizes a recent technique that obtained the first such approximation scheme for line graphs. Our results show that the problem is not APX-hard for such graphs unless NP \subseteq DTIME(2^{polylog(n)}). Further, a reduction from the Demand Matching Problem shows that UFP is APX-hard when the number of leaves is Omega(n^\epsilon) for any constant \epsilon \u3e 0. Together, the two results give a nearly tight characterization of the approximability of the problem on tree graphs in terms of the number of leaves, and show the structure of the graph that results in hardness of approximation
An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
We provide a new algorithm for generating the Baker--Campbell--Hausdorff
(BCH) series Z = \log(\e^X \e^Y) in an arbitrary generalized Hall basis of
the free Lie algebra generated by and . It is based
on the close relationship of with a Lie algebraic structure
of labeled rooted trees. With this algorithm, the computation of the BCH series
up to degree 20 (111013 independent elements in ) takes less
than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We
also address the issue of the convergence of the series, providing an optimal
convergence domain when and are real or complex matrices.Comment: 30 page
On the proper intervalization of colored caterpillar trees
This paper studies the computational complexity of the Proper interval colored graph problem (picg), when the input graph is a colored caterpillar, parameterized by hair length. In order prove our result we establish a close relationship between the picg and a graph layout problem the Proper colored layout problem (pclp). We show a dichotomy: the picg and the pclp are NP-complete for colored caterpillars of hair length ≥ 2, while both problems are in P for colored caterpillars of hair length < 2. For the hardness results we provide a reduction from the Multiprocessor Scheduling problem, while the polynomial time results follow from a characterization in terms of forbidden subgraphs.Preprin
Combinatorics of Hard Particles on Planar Graphs
We revisit the problem of hard particles on planar random tetravalent graphs
in view of recent combinatorial techniques relating planar diagrams to
decorated trees. We show how to recover the two-matrix model solution to this
problem in this purely combinatorial language.Comment: 35 pages, 20 figures, tex, harvmac, eps
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