80,133 research outputs found
Shortest Distances as Enumeration Problem
We investigate the single source shortest distance (SSSD) and all pairs
shortest distance (APSD) problems as enumeration problems (on unweighted and
integer weighted graphs), meaning that the elements -- where
and are vertices with shortest distance -- are produced and
listed one by one without repetition. The performance is measured in the RAM
model of computation with respect to preprocessing time and delay, i.e., the
maximum time that elapses between two consecutive outputs. This point of view
reveals that specific types of output (e.g., excluding the non-reachable pairs
, or excluding the self-distances ) and the order of
enumeration (e.g., sorted by distance, sorted row-wise with respect to the
distance matrix) have a huge impact on the complexity of APSD while they appear
to have no effect on SSSD.
In particular, we show for APSD that enumeration without output restrictions
is possible with delay in the order of the average degree. Excluding
non-reachable pairs, or requesting the output to be sorted by distance,
increases this delay to the order of the maximum degree. Further, for weighted
graphs, a delay in the order of the average degree is also not possible without
preprocessing or considering self-distances as output. In contrast, for SSSD we
find that a delay in the order of the maximum degree without preprocessing is
attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit
Structure of conflict graphs in constraint alignment problems and algorithms
We consider the constrained graph alignment problem which has applications in
biological network analysis. Given two input graphs , a pair of vertex mappings induces an {\it edge conservation} if
the vertex pairs are adjacent in their respective graphs. %In general terms The
goal is to provide a one-to-one mapping between the vertices of the input
graphs in order to maximize edge conservation. However the allowed mappings are
restricted since each vertex from (resp. ) is allowed to be mapped
to at most (resp. ) specified vertices in (resp. ). Most
of results in this paper deal with the case which attracted most
attention in the related literature. We formulate the problem as a maximum
independent set problem in a related {\em conflict graph} and investigate
structural properties of this graph in terms of forbidden subgraphs. We are
interested, in particular, in excluding certain wheals, fans, cliques or claws
(all terms are defined in the paper), which corresponds in excluding certain
cycles, paths, cliques or independent sets in the neighborhood of each vertex.
Then, we investigate algorithmic consequences of some of these properties,
which illustrates the potential of this approach and raises new horizons for
further works. In particular this approach allows us to reinterpret a known
polynomial case in terms of conflict graph and to improve known approximation
and fixed-parameter tractability results through efficiently solving the
maximum independent set problem in conflict graphs. Some of our new
approximation results involve approximation ratios that are function of the
optimal value, in particular its square root; this kind of results cannot be
achieved for maximum independent set in general graphs.Comment: 22 pages, 6 figure
Dimension of posets with planar cover graphs excluding two long incomparable chains
It has been known for more than 40 years that there are posets with planar
cover graphs and arbitrarily large dimension. Recently, Streib and Trotter
proved that such posets must have large height. In fact, all known
constructions of such posets have two large disjoint chains with all points in
one chain incomparable with all points in the other. Gutowski and Krawczyk
conjectured that this feature is necessary. More formally, they conjectured
that for every , there is a constant such that if is a poset
with a planar cover graph and excludes , then
. We settle their conjecture in the affirmative. We also discuss
possibilities of generalizing the result by relaxing the condition that the
cover graph is planar.Comment: New section on connections with graph minors, small correction
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
Let be a Poisson structure on a finite-dimensional affine real manifold.
Can be deformed in such a way that it stays Poisson? The language of
Kontsevich graphs provides a universal approach -- with respect to all affine
Poisson manifolds -- to finding a class of solutions to this deformation
problem. For that reasoning, several types of graphs are needed. In this paper
we outline the algorithms to generate those graphs. The graphs that encode
deformations are classified by the number of internal vertices ; for we present all solutions of the deformation problem. For , first reproducing the pentagon-wheel picture suggested at
by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that
yields a new unique solution without -loops and tadpoles at .Comment: International conference ISQS'25 on integrable systems and quantum
symmetries (6-10 June 2017 in CVUT Prague, Czech Republic). Introductory
paragraph I.1 follows p.3 in arXiv:1710.00658 [math.CO]; 13 pages, 3 figures,
2 table
A characterization of consistent marked graphs
A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterzing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis
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