13,988 research outputs found
Sampling Balanced Forests of Grids in Polynomial Time
We prove that a polynomial fraction of the set of -component forests in
the grid graph have equal numbers of vertices in each component,
for any constant . This resolves a conjecture of Charikar, Liu, Liu, and
Vuong, and establishes the first provably polynomial-time algorithm for
(exactly or approximately) sampling balanced grid graph partitions according to
the spanning tree distribution, which weights each -partition according to
the product, across its pieces, of the number of spanning trees of each
piece. Our result follows from a careful analysis of the probability a
uniformly random spanning tree of the grid can be cut into balanced pieces.
Beyond grids, we show that for a broad family of lattice-like graphs, we
achieve balance up to any multiplicative constant with
constant probability, and up to an additive constant with polynomial
probability. More generally, we show that, with constant probability,
components derived from uniform spanning trees can approximate any given
partition of a planar region specified by Jordan curves. These results imply
polynomial time algorithms for sampling approximately balanced tree-weighted
partitions for lattice-like graphs.
Our results have applications to understanding political districtings, where
there is an underlying graph of indivisible geographic units that must be
partitioned into population-balanced connected subgraphs. In this setting,
tree-weighted partitions have interesting geometric properties, and this has
stimulated significant effort to develop methods to sample them
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
Let be a graph of order and let be the characteristic polynomial of its
Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c},
M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves or
vertices of degree two, Linear Algebra and its Applications
431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian
coefficients in the set of all -vertex unicyclic graphs
with the number of leaves , we investigate properties of the minimal
elements in the partial set of the Laplacian
coefficients, where denote the set of -vertex
unicyclic graphs with the number of leaves and girth . These results are
used to disprove their conjecture. Moreover, the graphs with minimum
Laplacian-like energy in are also studied.Comment: 19 page, 4figure
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
Failure Localization in Power Systems via Tree Partitions
Cascading failures in power systems propagate non-locally, making the control
and mitigation of outages extremely hard. In this work, we use the emerging
concept of the tree partition of transmission networks to provide an analytical
characterization of line failure localizability in transmission systems. Our
results rigorously establish the well perceived intuition in power community
that failures cannot cross bridges, and reveal a finer-grained concept that
encodes more precise information on failure propagations within tree-partition
regions. Specifically, when a non-bridge line is tripped, the impact of this
failure only propagates within well-defined components, which we refer to as
cells, of the tree partition defined by the bridges. In contrast, when a bridge
line is tripped, the impact of this failure propagates globally across the
network, affecting the power flow on all remaining transmission lines. This
characterization suggests that it is possible to improve the system robustness
by temporarily switching off certain transmission lines, so as to create more,
smaller components in the tree partition; thus spatially localizing line
failures and making the grid less vulnerable to large-scale outages. We
illustrate this approach using the IEEE 118-bus test system and demonstrate
that switching off a negligible portion of transmission lines allows the impact
of line failures to be significantly more localized without substantial changes
in line congestion
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Linial arrangements and local binary search trees
We study the set of NBC sets (no broken circuit sets) of the Linial
arrangement and deduce a constructive bijection to the set of local binary
search trees. We then generalize this construction to two families of Linial
type arrangements for which the bijections are with some -ary labelled trees
that we introduce for this purpose.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1403.257
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