1,251 research outputs found
On bounding the bandwidth of graphs with symmetry
We derive a new lower bound for the bandwidth of a graph that is based on a
new lower bound for the minimum cut problem. Our new semidefinite programming
relaxation of the minimum cut problem is obtained by strengthening the known
semidefinite programming relaxation for the quadratic assignment problem (or
for the graph partition problem) by fixing two vertices in the graph; one on
each side of the cut. This fixing results in several smaller subproblems that
need to be solved to obtain the new bound. In order to efficiently solve these
subproblems we exploit symmetry in the data; that is, both symmetry in the
min-cut problem and symmetry in the graphs. To obtain upper bounds for the
bandwidth of graphs with symmetry, we develop a heuristic approach based on the
well-known reverse Cuthill-McKee algorithm, and that improves significantly its
performance on the tested graphs. Our approaches result in the best known lower
and upper bounds for the bandwidth of all graphs under consideration, i.e.,
Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and
Kneser graphs, with up to 216 vertices
Semidefinite programming and eigenvalue bounds for the graph partition problem
The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds
Bounds on separated pairs of subgraphs, eigenvalues and related polynomials
We give a bound on the sizes of two sets of vertices at a given minimum distance (a separated pair of subgraphs) in a graph in terms of polynomials and the spectrum of the graph. We find properties of the polynomial optimizing the bound. Explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance are given, and we find graphs for which the bounds are tight.Graphs;Eigenvalues;Polynomials;mathematics
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
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