The ζ(2) limit in the random assignment problem

Abstract

The random assignment (or bipartite matching) problem asks about A n = min π n i=1 c i π i , where c i j is a n × n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. used the replica method from statistical physics to argue nonrigorously that EA n → ζ 2 = π 2 /6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ 2 limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost-optimal matching coincides with the optimal matching except on a small proportion of edges.

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

“…Nevertheless, progress on the limit value for E[A n ] has been more definitive. In 2001 -by means of the objective method -Aldous [4] finally proved the ζ(2) limit formula that Mézard and Parisi [50] first brought to light in 1987.…”

“…The proof of Lemma 11 is unfortunately too lengthy to be included here, and even a convincing sketch would run beyond our boundaries, so for most details we must refer the reader to Proposition 18 of Aldous [4], which additionally proves that inequality (5.63) is strict except when the matching M is almost surely identical to the matching defined by ϕ.…”

Phase transitions in the coloring of random graphs

“…The only success so far was obtained by proving a far too strong condition: the Gibbs measure uniqueness [49,50,51,52]. Roughly speaking: the Gibbs measure (2) is unique if the behavior of a spin i is totally independent from the boundary conditions (i.e.…”

“…To see if a solution {s i } belongs to a cluster with frozen variables or not we initialize warning propagation with u i→j s = δ s,si , and update iteratively according to (49) until a fixed point is reached (the update every time converge, because starting from a solution we are only adding white edges). In the fixed point or all edges are white, then the solution {s i } does not belong to a frozen cluster, or some of the edges stay colored (non-white), then the solution {s i } belongs to a frozen cluster.…”

Average-Case Analyses of Vickrey Costs

“…exponential(1) random variables, Mézard and Parisi [MP85, MP86, MP87] gave a sophisticated mathematical physics argument, using the "replica method" (related to the "cavity method"), that the minimum cost AP satisfies lim n→∞ E(AP) = ζ(2) = π 2 /6. Aldous [Ald92,Ald01] made this mathematically rigorous through reasoning about a "Poisson weighted infinite tree". For finite values of n, Parisi [Par98] conjectured the expected cost to be n i=1 i −2 , Coppersmith and Sorkin [CS99] extended the conjecture to cheapest cardinality-k assignments in K m,n , and these results were proved simultaneously, by different methods, by Linusson and Wästlund [LW04] and Nair, Prabhakar and Sharma [NPS05].…”