Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Spectral density of random graphs with topological constraints

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.Comment: 24 pages, 6 figure

Dynamic programming optimization over random data: The scaling exponent for near-optimal solutions

A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over A ⊆ {1, 2,⋯, n}, the objective function |A| - Σ i ξ i 1 (i ∈ A, i + 1 ∈ A) for given ξi > 0. This problem, with random (ξi), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions difiering from the optimal solution in a small proportion δ of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order δ 2 but not of smaller order. We conjecture this relationship holds widely in the context of dynamic prog amming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in [D. J. Aldous and A. G. Percus, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 11211-11215] of relating such scaling exponents to the algorithmic difficulty of optimization problems. © 2009 Society for Industrial and Applied Mathematics

Near-minimal spanning trees: A scaling exponent in probability models

We study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion 8 of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1 + 0 (δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model. © Association des Publications de l'Institut Henri Poincaré, 2008