4,467 research outputs found
Relations between cumulants in noncommutative probability
We express classical, free, Boolean and monotone cumulants in terms of each
other, using combinatorics of heaps, pyramids, Tutte polynomials and
permutations. We completely determine the coefficients of these formulas with
the exception of the formula for classical cumulants in terms of monotone
cumulants whose coefficients are only partially computed.Comment: 27 pages, 7 figures, AMS LaTe
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
Boundary Partitions in Trees and Dimers
Given a finite planar graph, a grove is a spanning forest in which every
component tree contains one or more of a specified set of vertices (called
nodes) on the outer face. For the uniform measure on groves, we compute the
probabilities of the different possible node connections in a grove. These
probabilities only depend on boundary measurements of the graph and not on the
actual graph structure, i.e., the probabilities can be expressed as functions
of the pairwise electrical resistances between the nodes, or equivalently, as
functions of the Dirichlet-to-Neumann operator (or response matrix) on the
nodes. These formulae can be likened to generalizations (for spanning forests)
of Cardy's percolation crossing probabilities, and generalize Kirchhoff's
formula for the electrical resistance. Remarkably, when appropriately
normalized, the connection probabilities are in fact integer-coefficient
polynomials in the matrix entries, where the coefficients have a natural
algebraic interpretation and can be computed combinatorially. A similar
phenomenon holds in the so-called double-dimer model: connection probabilities
of boundary nodes are polynomial functions of certain boundary measurements,
and as formal polynomials, they are specializations of the grove polynomials.
Upon taking scaling limits, we show that the double-dimer connection
probabilities coincide with those of the contour lines in the Gaussian free
field with certain natural boundary conditions. These results have direct
application to connection probabilities for multiple-strand SLE_2, SLE_8, and
SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor
change
Approximating Minimum Cost Connectivity Orientation and Augmentation
We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph that admits an
orientation covering a nonnegative crossing -supermodular demand function,
as defined by Frank. An important example is -edge-connectivity, a
common generalization of global and rooted edge-connectivity.
Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of -edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most when and conjectured that it is constant
for all fixed . We disprove this conjecture by showing an
integrality gap even when
Gibbs distributions for random partitions generated by a fragmentation process
In this paper we study random partitions of 1,...n, where every cluster of
size j can be in any of w\_j possible internal states. The Gibbs (n,k,w)
distribution is obtained by sampling uniformly among such partitions with k
clusters. We provide conditions on the weight sequence w allowing construction
of a partition valued random process where at step k the state has the Gibbs
(n,k,w) distribution, so the partition is subject to irreversible fragmentation
as time evolves. For a particular one-parameter family of weight sequences
w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent
process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a
and b. Under further restrictions on a and b, the fragmentation process can be
realized by conditioning a Galton-Watson tree with suitable offspring
distribution to have n nodes, and cutting the edges of this tree by random
sampling of edges without replacement, to partition the tree into a collection
of subtrees. Suitable offspring distributions include the binomial, negative
binomial and Poisson distributions.Comment: 38 pages, 2 figures, version considerably modified. To appear in the
Journal of Statistical Physic
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