44,628 research outputs found
A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors
In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring
number of a graph is bounded above by when is even and
when is odd. They gave a simple
construction which requires this many colors. The conjecture has been verified
for . For , the conjectured bound is 20. Previously,
the best known upper bound was 23 due to Horak. In this paper we give an
algorithm that uses at most 22 colors.Comment: 9 pages, 4 figure
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
Improved Analysis of Deterministic Load-Balancing Schemes
We consider the problem of deterministic load balancing of tokens in the
discrete model. A set of processors is connected into a -regular
undirected network. In every time step, each processor exchanges some of its
tokens with each of its neighbors in the network. The goal is to minimize the
discrepancy between the number of tokens on the most-loaded and the
least-loaded processor as quickly as possible.
Rabani et al. (1998) present a general technique for the analysis of a wide
class of discrete load balancing algorithms. Their approach is to characterize
the deviation between the actual loads of a discrete balancing algorithm with
the distribution generated by a related Markov chain. The Markov chain can also
be regarded as the underlying model of a continuous diffusion algorithm. Rabani
et al. showed that after time , any algorithm of their
class achieves a discrepancy of , where is the spectral
gap of the transition matrix of the graph, and is the initial load
discrepancy in the system.
In this work we identify some natural additional conditions on deterministic
balancing algorithms, resulting in a class of algorithms reaching a smaller
discrepancy. This class contains well-known algorithms, eg., the Rotor-Router.
Specifically, we introduce the notion of cumulatively fair load-balancing
algorithms where in any interval of consecutive time steps, the total number of
tokens sent out over an edge by a node is the same (up to constants) for all
adjacent edges. We prove that algorithms which are cumulatively fair and where
every node retains a sufficient part of its load in each step, achieve a
discrepancy of in time . We
also show that in general neither of these assumptions may be omitted without
increasing discrepancy. We then show by a combinatorial potential reduction
argument that any cumulatively fair scheme satisfying some additional
assumptions achieves a discrepancy of almost as quickly as the
continuous diffusion process. This positive result applies to some of the
simplest and most natural discrete load balancing schemes.Comment: minor corrections; updated literature overvie
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
Functional limit theorems for random regular graphs
Consider d uniformly random permutation matrices on n labels. Consider the
sum of these matrices along with their transposes. The total can be interpreted
as the adjacency matrix of a random regular graph of degree 2d on n vertices.
We consider limit theorems for various combinatorial and analytical properties
of this graph (or the matrix) as n grows to infinity, either when d is kept
fixed or grows slowly with n. In a suitable weak convergence framework, we
prove that the (finite but growing in length) sequences of the number of short
cycles and of cyclically non-backtracking walks converge to distributional
limits. We estimate the total variation distance from the limit using Stein's
method. As an application of these results we derive limits of linear
functionals of the eigenvalues of the adjacency matrix. A key step in this
latter derivation is an extension of the Kahn-Szemer\'edi argument for
estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and
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