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

Error Graphs and the Reconstruction of Elements in Groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions. In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200