1,115 research outputs found
Normal hypergraphs and the perfect graph conjecture
AbstractA hypergraph is called normal if the chromatic index of any partial hypergraph HⲠof it coincides with the maximum valency in Hâ˛. It is proved that a hypergraph is normal iff the maximum number of disjoint hyperedges coincides with the minimum number of vertices representing the hyperedges in each partial hypergraph of it. This theorem implies the following conjecture of Berge: The complement of a perfect graph is perfect. A new proof is given for a related theorem of Berge and Las Vergnas. Finally, the results are applied on a problem of integer valued linear programming, slightly sharpening some results of Fulkerson
Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation
We give the first rigorous proof of the convergence of Riemannian Hamiltonian
Monte Carlo, a general (and practical) method for sampling Gibbs distributions.
Our analysis shows that the rate of convergence is bounded in terms of natural
smoothness parameters of an associated Riemannian manifold. We then apply the
method with the manifold defined by the log barrier function to the problems of
(1) uniformly sampling a polytope and (2) computing its volume, the latter by
extending Gaussian cooling to the manifold setting. In both cases, the total
number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the
art. A key ingredient of our analysis is a proof of an analog of the KLS
conjecture for Gibbs distributions over manifolds
Non-degenerate colorings in the Brook's Theorem
Let and be two integers. We will call a proper coloring
of the graph a \textit{-nondegenerate}, if for any vertex of
with degree at least there are at least vertices of different colors
adjacent to it. In our work we prove the following result, which generalizes
Brook's Theorem. Let and be a graph without cliques on
vertices and the degree of any vertex in this graph is not greater than .
Then for every integer there is a proper -nondegenerate vertex
-coloring of , where During the primary proof,
some interesting corollaries are derived.Comment: 18 pages, 10 figure
Community detection in multiplex networks using locally adaptive random walks
Multiplex networks, a special type of multilayer networks, are increasingly
applied in many domains ranging from social media analytics to biology. A
common task in these applications concerns the detection of community
structures. Many existing algorithms for community detection in multiplexes
attempt to detect communities which are shared by all layers. In this article
we propose a community detection algorithm, LART (Locally Adaptive Random
Transitions), for the detection of communities that are shared by either some
or all the layers in the multiplex. The algorithm is based on a random walk on
the multiplex, and the transition probabilities defining the random walk are
allowed to depend on the local topological similarity between layers at any
given node so as to facilitate the exploration of communities across layers.
Based on this random walk, a node dissimilarity measure is derived and nodes
are clustered based on this distance in a hierarchical fashion. We present
experimental results using networks simulated under various scenarios to
showcase the performance of LART in comparison to related community detection
algorithms
Finitely forcible graphons
AbstractWe investigate families of graphs and graphons (graph limits) that are determined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of TurĂĄn, ErdĹsâSimonovits and ChungâGrahamâWilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractal-like) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are ârareâ, and exhibit simple and explicit non-forcible graphons
Quasirandom Load Balancing
We propose a simple distributed algorithm for balancing indivisible tokens on
graphs. The algorithm is completely deterministic, though it tries to imitate
(and enhance) a random algorithm by keeping the accumulated rounding errors as
small as possible.
Our new algorithm surprisingly closely approximates the idealized process
(where the tokens are divisible) on important network topologies. On
d-dimensional torus graphs with n nodes it deviates from the idealized process
only by an additive constant. In contrast to that, the randomized rounding
approach of Friedrich and Sauerwald (2009) can deviate up to Omega(polylog(n))
and the deterministic algorithm of Rabani, Sinclair and Wanka (1998) has a
deviation of Omega(n^{1/d}). This makes our quasirandom algorithm the first
known algorithm for this setting which is optimal both in time and achieved
smoothness. We further show that also on the hypercube our algorithm has a
smaller deviation from the idealized process than the previous algorithms.Comment: 25 page
Quasirandom permutations are characterized by 4-point densities
For permutations Ď and Ď of lengths |Ď|â¤|Ď| , let t(Ď,Ď) be the probability that the restriction of Ď to a random |Ď| -point set is (order) isomorphic to Ď . We show that every sequence {Ďj} of permutations such that |Ďj|ââ and t(Ď,Ďj)â1/4! for every 4-point permutation Ď is quasirandom (that is, t(Ď,Ďj)â1/|Ď|! for every Ď ). This answers a question posed by Graham
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