In this paper, we present an eecient algorithm to compute a tighter lower bound for the one-dimensional bin packing problem. The time complexity of the algorithm is O(n log n). We have simulated the algorithm on randomly generated bin packing problems with item sizes drawn uniformly from (a; b], 0 a < b B. If our lower bound is used, on average, the error of BFD is less than 2%. For a + b B, th...

The Minimum Energy Broadcast problem consists in finding the minimum-energy range assignment for a given set S of n stations of an ad hoc wireless network that allows a source station to perform broadcast operations over S. We prove a nearly tight asymptotical bound on the optimal cost for the Minimum Energy Broadcast problem on square grids. We emphasize that finding tight bounds for this prob...

We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3. In fact, for constant d our alg...

Given d > 2 and a set of n grid points Q in <, we design a randomized algorithm that finds a w-wide separator, which is determined by a hyper-plane, in O(n 2 d log n) sublinear time such that Q has at most ( d d+1 + o(1))n points one either side of the hyper-plane, and at most cdwn d−1 d points within w 2 distance to the hyper-plane, where cd is a constant for fixed d. In particular, c3 = 1.209...

We consider a generalized one-dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin. We show that for any given positive constant , there exists a polynomial-time approximation algorithmwith an asymptotic worst-case performance ratio of no more than 1 + .

We generalize work of Bourgain-Kontorovich [6] and Zhang [31], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group A ≤ PSL2(K) satisfying certain conditions, where K is an imaginary quadratic field, and sh...

We consider approximate PCPs for multidimensional bin-packing problems. In particular, we show how a verifier can be quickly convinced that a set of multidimensional blocks can be packed into a small number of bins. The running time of the verifier is bounded by , where is the time required to test for heaviness. We give heaviness testers that can test heaviness of an element in the domain in t...

This paper presents a generic scheme for fractional packing in ideal clutters. Consider an ideal clutter with a nonnegative capacity function on its vertices. It follows from ideality that for any nonnegative capacity the total multiplicity of an optimal fractional packing is equal to the minimum capacity of a vertex cover. Our scheme finds an optimal packing using at most n edges with positive...

For bin packing, the input consists of n items with sizes s1, . . . , sn ∈ [0,1] which have to be assigned to aminimum number of bins of size 1. Recently, the second author gave an LP-based polynomial time algorithm that employed techniques from discrepancy theory to find a solution using at mostOPT +O(logOPT · log logOPT ) bins. In this paper, we present an approximation algorithm that has an ...

This article shows some techniques based on simulated annealing and genetic alghoritms for the resolution of a filling problem of a container of two dimensions using rectangular pieces of sizes not congruent. This problem is quite related to problems like bin-packing or strip-packing. The comparative was made using several type problems and having into account parameters like run time, number o...