Consider a process in which information is transmitted from a given root node on a noisy tree network T. We start with an unbiased random bit R at the root of the tree and send it down the edges of T. On every edge the bit can be reversed with probability ε, and these errors occur independently. The goal is to reconstruct R from the values which arrive at the nth level of the tree. This model has been studied in information theory, genetics and statistical mechanics. We bound the reconstruction probability from above, using the maximum flow on T viewed as a capacitated network, and from below using the electrical conductance of T. For general infinite trees, we establish a sharp threshold: the probability of correct reconstruction tends to 1/2 as n → ∞ if 1 − 2ε 2 < p c T , but the reconstruction probability stays bounded away from 1/2 if the opposite inequality holds. Here p c T is the critical probability for percolation on T; in particular p c T = 1/b for the b + 1-regular tree. The asymptotic reconstruction problem is equivalent to purity of the "free boundary" Gibbs state for the Ising model on a tree. The special case of regular trees was solved in 1995 by Bleher, Ruiz and Zagrebnov; our extension to general trees depends on a coupling argument and on a reconstruction algorithm that weights the input bits by the electrical current flow from the root to the leaves.

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap |λ 1 − λ 2 |) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that for general graphs, if the relaxation time τ 2 satisfies τ 2 = O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.

We consider the problem of scheduling n jobs with release dates on m machines so as to minimize their average weighted completion time. We present the first known polynomial time approximation schemes for several variants of this problem. Our results include PTASs for the case of identical parallel machines and a constant number of unrelated machines with and without preemption allowed. Our schemes are efficient: for all variants the running time for a 1 + approximation is of the form f1= ; mpolyn.

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of xed width and minimum height, a classical N P-hard cutting-stock problem. The algorithm nds a packing of n rectangles whose total height is within a factor of (1 +) of optimal (up to an additive term), and has running time polynomial both in n and in 1=. It is based on a reduction to fractional bin-packing.

Abstract. We study the multidimensional generalization of the classical Bin Packing problem: Given a collection of d-dimensional rectangles of specified sizes, the goal is to pack them into the minimum number of unit cubes.A long history of results exists for this problem and its special cases. Currently, the best known approximation algorithm for packing two-dimensional rectangles achieves a guarantee of 1.69 in the asymptotic case (i.e., when the optimum uses a large number of bins) [3]. An important open question has been whether 2−dimensional bin packing is essentially similar to the 1−dimensional case in that it admits an asymptotic polynomial time approximation scheme (APTAS) [12,17] or not. We answer the question in the negative and show that the problem is APX hard in the asymptotic sense.On the positive side, we give the following results: First, we consider the special case where we have to pack d-dimensional cubes into the minimum number of unit cubes. We give an asymptotic polynomial time approximation scheme for this problem. This represents a significant improvement over the previous best known asymptotic approximation factor of 2 − (2/3), and settles the approximability of the problem. Second, we give a polynomial time algorithm for packing arbitrary rectangles into at most OPT square bins with sides of length 1 + ε, where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result does not have an additive constant term i.e., is not an asymptotic result. As a corollary, we obtain a polynomial time approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle, settling also the approximability of this problem.

In this paper we present a theoretical analysis of the deterministic on-line Sum of Squares algorithm (SS) for bin packing, introduced and studied experimentally in [8], along with several new variants.

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