Extremal optimization for sensor report pre-processing

We describe the recently introduced extremal optimization algorithm and apply it to target detection and association problems arising in pre-processing for multi-target tracking. Here we consider the problem of pre-processing for multiple target tracking when the number of sensor reports received is very large and arrives in large bursts. In this case, it is sometimes necessary to pre-process reports before sending them to tracking modules in the fusion system. The pre-processing step associates reports to known tracks (or initializes new tracks for reports on objects that have not been seen before). It could also be used as a pre-process step before clustering, e.g., in order to test how many clusters to use. The pre-processing is done by solving an approximate version of the original problem. In this approximation, not all pair-wise conflicts are calculated. The approximation relies on knowing how many such pair-wise conflicts that are necessary to compute. To determine this, results on phase-transitions occurring when coloring (or clustering) large random instances of a particular graph ensemble are used.Comment: 10 page

Extremal \u3cem\u3eH\u3c/em\u3e-Colorings of Trees and 2-connected Graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H -colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n -vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric