16 research outputs found
Message passing for the coloring problem: Gallager meets Alon and Kahale
Message passing algorithms are popular in many combinatorial optimization
problems. For example, experimental results show that {\em survey propagation}
(a certain message passing algorithm) is effective in finding proper
-colorings of random graphs in the near-threshold regime. In 1962 Gallager
introduced the concept of Low Density Parity Check (LDPC) codes, and suggested
a simple decoding algorithm based on message passing. In 1994 Alon and Kahale
exhibited a coloring algorithm and proved its usefulness for finding a
-coloring of graphs drawn from a certain planted-solution distribution over
-colorable graphs. In this work we show an interpretation of Alon and
Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus
showing a connection between the two problems - coloring and decoding. This
also provides a rigorous evidence for the usefulness of the message passing
paradigm for the graph coloring problem. Our techniques can be applied to
several other combinatorial optimization problems and networking-related
issues.Comment: 11 page
Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs
For an increasing monotone graph property \mP the \emph{local resilience}
of a graph with respect to \mP is the minimal for which there exists
of a subgraph with all degrees at most such that the removal
of the edges of from creates a graph that does not possesses \mP.
This notion, which was implicitly studied for some ad-hoc properties, was
recently treated in a more systematic way in a paper by Sudakov and Vu. Most
research conducted with respect to this distance notion focused on the Binomial
random graph model \GNP and some families of pseudo-random graphs with
respect to several graph properties such as containing a perfect matching and
being Hamiltonian, to name a few. In this paper we continue to explore the
local resilience notion, but turn our attention to random and pseudo-random
\emph{regular} graphs of constant degree. We investigate the local resilience
of the typical random -regular graph with respect to edge and vertex
connectivity, containing a perfect matching, and being Hamiltonian. In
particular we prove that for every positive and large enough values
of with high probability the local resilience of the random -regular
graph, \GND, with respect to being Hamiltonian is at least .
We also prove that for the Binomial random graph model \GNP, for every
positive and large enough values of , if
then with high probability the local resilience of \GNP with respect to being
Hamiltonian is at least . Finally, we apply similar
techniques to Positional Games and prove that if is large enough then with
high probability a typical random -regular graph is such that in the
unbiased Maker-Breaker game played on the edges of , Maker has a winning
strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur
Vertex Percolation on Expander Graphs
We say that a graph G = (V, E) on n vertices is a β-expander for some constant β> 0 if every U ⊆ V of cardinality |U | ≤ n 2 satisfies |NG(U) | ≥ β|U | where NG(U) denotes the neighborhood of U. In this work we explore the process of deleting vertices of a β-expander independently at random with probability n −α for some constant α> 0, and study the properties of the resulting graph. Our main result states that as n tends to infinity, the deletion process performed on a β-expander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o(n) vertices that is in itself an expander graph, and constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of (n, d, λ)-graphs, that are such expanders, we compute the values of α, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of d-regular graphs with high probability is an expander and meets the additional constraints, this result strengthens a recent result due to Greenhill, Holt and Wormald about vertex percolation on random d-regular graphs. We conclude by showing that performing the above described deletion process on graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in a connected expander graph.