461 research outputs found
An explicit formula for the Berezin star product
We prove an explicit formula of the Berezin star product on Kaehler
manifolds. The formula is expressed as a summation over certain strongly
connected digraphs. The proof relies on a combinatorial interpretation of
Englis' work on the asymptotic expansion of the Laplace integral.Comment: 19 pages, to appear in Lett. Math. Phy
Which Digraphs with Ring Structure are Essentially Cyclic?
We say that a digraph is essentially cyclic if its Laplacian spectrum is not
completely real. The essential cyclicity implies the presence of directed
cycles, but not vice versa. The problem of characterizing essential cyclicity
in terms of graph topology is difficult and yet unsolved. Its solution is
important for some applications of graph theory, including that in
decentralized control. In the present paper, this problem is solved with
respect to the class of digraphs with ring structure, which models some typical
communication networks. It is shown that the digraphs in this class are
essentially cyclic, except for certain specified digraphs. The main technical
tool we employ is the Chebyshev polynomials of the second kind. A by-product of
this study is a theorem on the zeros of polynomials that differ by one from the
products of Chebyshev polynomials of the second kind. We also consider the
problem of essential cyclicity for weighted digraphs and enumerate the spanning
trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for
publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00
Polynomial Kernels for Deletion to Classes of Acyclic Digraphs
We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made.
In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time
On the Kernel and Related Problems in Interval Digraphs
Given a digraph , a set is said to be absorbing set
(resp. dominating set) if every vertex in the graph is either in or is an
in-neighbour (resp. out-neighbour) of a vertex in . A set
is said to be an independent set if no two vertices in are adjacent in .
A kernel (resp. solution) of is an independent and absorbing (resp.
dominating) set in . We explore the algorithmic complexity of these problems
in the well known class of interval digraphs. A digraph is an interval
digraph if a pair of intervals can be assigned to each vertex
of such that if and only if .
Many different subclasses of interval digraphs have been defined and studied in
the literature by restricting the kinds of pairs of intervals that can be
assigned to the vertices. We observe that several of these classes, like
interval catch digraphs, interval nest digraphs, adjusted interval digraphs and
chronological interval digraphs, are subclasses of the more general class of
reflexive interval digraphs -- which arise when we require that the two
intervals assigned to a vertex have to intersect. We show that all the problems
mentioned above are efficiently solvable, in most of the cases even linear-time
solvable, in the class of reflexive interval digraphs, but are APX-hard on even
the very restricted class of interval digraphs called point-point digraphs,
where the two intervals assigned to each vertex are required to be degenerate,
i.e. they consist of a single point each. The results we obtain improve and
generalize several existing algorithms and structural results for subclasses of
reflexive interval digraphs.Comment: 26 pages, 3 figure
A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs
Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k=0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixed-parameter tractable algorithms. That is, we show that the problem admits a 2^{O(sqrt{k} log k)}n^{O(1)} time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n,k,k^2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments
Diffusion and consensus on weakly connected directed graphs
Let be a weakly connected directed graph with asymmetric graph Laplacian
. Consensus and diffusion are dual dynamical processes defined on
by for consensus and for diffusion. We
consider both these processes as well their discrete time analogues. We define
a basis of row vectors of the left null-space of
and a basis of column vectors of the right
null-space of in terms of the partition of into strongly
connected components. This allows for complete characterization of the
asymptotic behavior of both diffusion and consensus --- discrete and continuous
--- in terms of these eigenvectors.
As an application of these ideas, we present a treatment of the pagerank
algorithm that is dual to the usual one. We further show that the teleporting
feature usually included in the algorithm is not strictly necessary.
This is a complete and self-contained treatment of the asymptotics of
consensus and diffusion on digraphs. Many of the ideas presented here can be
found scattered in the literature, though mostly outside mainstream mathematics
and not always with complete proofs. This paper seeks to remedy this by
providing a compact and accessible survey.Comment: 19 pages, Survey Article, 1 figur
Similarities on Graphs: Kernels versus Proximity Measures
We analytically study proximity and distance properties of various kernels
and similarity measures on graphs. This helps to understand the mathematical
nature of such measures and can potentially be useful for recommending the
adoption of specific similarity measures in data analysis.Comment: 16 page
- …