Linear orderings of random geometric graphs (extended abstract)

In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.Postprint (published version

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Factors of IID on Trees

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur

Convergence theorems for some layout measures on random lattice and random geometric graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. For full square lattices, we give optimal layouts for the problems still open. Our convergence theorems can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidian TSP on random points in the $d$-dimensional cube. As the considered layout measures are non-subadditive, we use percolation theory to obtain our results on random lattices and random geometric graphs. In particular, we deal with the subcritical regimes on these class of graphs.Postprint (published version