Claire Kenyon

Approximation schemes for minimizing average weighted completion time with release dates

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 /spl alpha/(1+/spl epsiv/) approximation is of the form f(1//spl epsiv/, m)poly(n).

A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classicalNP-hard cutting-stock problem. The algorithm, based on a new linear-programming relaxation, finds a packing ofn rectangles whose total height is within a factor of (1 +e) of optimal (up to an additive term), and has running time polynomial both in n and in 1/ e.

Approximation schemes for clustering problems

Let <i>k</i> be a fixed integer. We consider the problem ofpartitioning an input set of points endowed with a distancefunction into <i>k</i> clusters. We give polynomial timeapproximation schemes for the following three clustering problems:Metric <i>k</i>-Clustering, l ²₂<i>k</i>-Clustering, and l²₂ <i>k</i>-Median.In the <i>k</i>-Clustering problem, the objective is to minimizethe sum of all intra-cluster distances. In the <i>k</i>-Medianproblem, the goal is to minimize the sum of distances from pointsin a cluster to the (best choice of) cluster center. In metricinstances, the input distance function is a metric. In l²₂ instances, the points are in R^<i>d</i> and the distance between two points <i>x,y</i>is measured by <i>x−y</i> ²₂ (noticethat (R ^<i>d</i>, ṡ ²₂ is nota metric space). For the first two problems, our results are thefirst polynomial time approximation schemes. For the third problem,the running time of our algorithms is a vast improvement overprevious work.

Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes

We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless PNP. On the positive side, we give an APTAS for the special case of packing d-dimensional cubes into the minimum number of unit cubes. Second, we give a polynomial time algorithm for packing arbitrary two-dimensional rectangles into at most OPT square bins with sides of length 1 , where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result has no additive constant term, i.e., is not an asymptotic result. As a corollary, we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum-area encasing rectangle.

Dynamic TCP acknowledgement and other stories about e/(e-1)

We present the first optimal randomized online algorithms for the TCP acknowledgment problem [5] and the Bahncard problem [7]. These problems are well-known to be generalizations of the classical online ski rental problem, however, they appeared to be harder. In this paper, we demonstrate that a number of online algorithms which have optimal competitive ratios of e/(e-1), including these, are fundamentally no more complex than ski rental. Our results also suggest a clear paradigm for solving ski rental-like problems.

Scheduling Independent Multiprocessor Tasks

We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of length within a factor of (1 + c) of optimal in the non-preemptive model.

The data broadcast problem with non-uniform transmission times

AbstractThe Data Broadcast Problem consists of finding an infinite schedule to broadcast a given set of messages so as to minimize a linear combination of the average service time to clients requesting messages, and of the cost of the broadcast. This problem also models the Maintenance Scheduling Problem and the Multi-Item Replenishment Problem. Previous work concentrated on a discrete-time restriction where all messages have transmission time equal to 1. Here, we study a generalization of the model to a setting of continuous time and messages of non-uniform transmission times. We prove that the Data Broadcast Problem is strongly NP -hard, even if the broadcast costs are all zero, and give 3-approximation algorithms.

Tiling a polygon with rectangles

The authors study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). They present a linear time algorithm for deciding if a polygon can be tiled with 1 * m and k * 1 tiles (and giving a tiling when it exists), and a quadratic algorithm for the same problem when the tile types are m * k and k * m.<<ETX>>

On the Sum-of-Squares algorithm for bin packing

In this article we present a theoretical analysis of the online <i>Sum-of-Squares</i> algorithm (<i>SS</i>) for bin packing along with several new variants. <i>SS</i> is applicable to any instance of bin packing in which the bin capacity <i>B</i> and item sizes <i>s</i>(<i>a</i>) are integral (or can be scaled to be so), and runs in time <i>O</i>(<i>nB</i>). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, <i>SS</i> also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, <i>SS</i> has expected waste at most <i>O</i>(log <i>n</i>). We also discuss several interesting variants on <i>SS</i>, including a randomized <i>O</i>(<i>nB</i> log <i>B</i>)-time online algorithm <i>SS</i>* whose expected behavior is essentially optimal for all discrete distributions. Algorithm <i>SS</i>* depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution <i>F</i>, just what is the growth rate for the optimal expected waste.

Dynamic TCP Acknowledgment and Other Stories about e/(e - 1)

AbstractWe present the first optimal randomized online algorithms for the TCP acknowledgment problem [3] and the Bahncard problem [5]. These problems are well known to be generalizations of the classical online ski-rental problem, however, they appeared to be harder. In this paper we demonstrate that a number of online algorithms which have optimal competitive ratios of e/(e-1) , including these, are fundamentally no more complex than ski rental. Our results also suggest a clear paradigm for solving ski-rental-like problems.