26809229
Packing Squares into a Square
Abstract The problem of determining whether a set of rectangles can be orthogonally packed into a larger rectangle has been studied as a geometric packing problem and as a floor planning problem. Recently, there is some renewed interest in this problem, as it is related to a job scheduling problem in a partitionable mesh connected system. In this paper we show that the problem of deciding whether a set of squares can be packed into a larger square is strongly NP-complete, answering an open question posed by Li and Cheng.
7180182
Minimizing mean flow time with error constraint
In a new model of task systems studied by J.W.S. Liu et al. (1987) each task is logically decomposed into two subtasks, mandatory and optional. The authors discuss preemptively scheduling this kind of task system on p>or=1 identical processors so as to minimize the mean flow time. Given a task system and an error threshold K, the goal is to find a preemptive schedule such that each task is executed in the interval of its release time and deadline, the average error is no more than K, and the mean flow time of the schedule is minimized. Such a schedule is called an optimal schedule. It is shown that the problem of finding an optimal schedule is NP-hard for each p>or=1, even if all tasks have the same ready time and deadline. For a single processor, a pseudo-polynomial time algorithm and polynomial time algorithms for various special cases of the problem are given.<<ETX>>
195859283
On fixed edges and edge-reconstruction of series-parallel networks
A graph configuration is a vertex labeled simple graph. We say a configuration is exclusive for graphs with a property in a class of graphs if every graph in the class containing the configuration has the property. We use the method of exclusive configurations to search for graphs without fixed edges and forced edges investigate the fixed edges. An edge e of a graph G is said to be a fixed edge if G-e+e'/spl cong/G implies that e'=e, and a forced edge if G-e+e' is an edge-reconstruction of G implies that e'=e. It is proved that all series-parallel networks contain fixed edges and forced edge except P/sub 2//spl times/K/sub 1/ and P/sub 3//spl times/K/sub 1/. A similar method is used to investigate the alternative wire problem in circuit design. Let G be a Boolean network and w a wire in G. Wire w' is said to be an alternative wire of w if w'/spl ne/w and G-w+w' has the same functionality as that of G. A wire is said to be fixed if it has no alternative wire. We present a set of exclusive configurations for Boolean networks in which every wire is fixed.
12009161
Dynamic Placement of Records in Linear Storage
This paper considers allocation of space in a hnear storage medium when space must be allocated dynamically as customers arrive A heuristic is proposed for this problem and for a simple model of the resultmg reference sequence, we show that the average distance between consecutive references is asymptotically 7n/30, where n is the size of the storage For optimal static placement where one waits for all arrivals before any space allocation, the average distance is shown to be asymptotically 7n/30 For random placement, the average distance is asymptotically n/3 Thus, the heuristic is asymptotically optimal in a strong sense For reasonable values of n, we demonstrate that the heuristic is nearly as good as optimal static placement and much better than random placement
21236849
Near-Optimal Solutions to a 2-Dimensional Placement Problem
We consider the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized. A simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric. For the rectilinear and maximum metrics, we show that there is no such heuristic. For the special case in which all access probabilities are equal, however, heuristics within an additive constant of optimal do exist, and their implementation requires solution of differential equations for which we give numerical solutions.
12439298
The Steiner Tree Problem in Orientation Metrics
Given a set?of?i(i=1,2,?,k) orientations (angles) in the plane, one can define a distance function which induces a metric in the plane, called the orientation metric 3. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2. Specifically, if there are?orientations, forming anglesi?/?, 0?i???1, with thex-axis, where??2 is an integer, we call the metric a??-metric. Note that the?2-metric is the well-known rectilinear metric and the?∞corresponds to the Euclidean metric. In this paper, we will concentrate on the?3-metric. In the?2-metric, Hanan has shown that there exists a solution of the Steiner tree problem such that all Steiner points are on the intersections of grid lines formed by passing lines at directionsi?/2,i=0,1, through all demand points. But this is not true in the?3-metric. In this paper, we mainly prove the following theorem: LetP,Q, andOi(i=1,2,?,k) be the set ofndemand points, the set of Steiner points, and the set of theith generation intersection points, respectively. Then there exists a solutionGof the Steiner problemSnsuch that all Steiner points are in ?ki=1Oi, wherek?? (n?2)/2?.
15615911
Time-varying shortest path problems with constraints
The shortest path problem is considered with yet another extension. The transit times b(x,y,u) and the arc costs c(x,y,u) on arc (x,y) vary as a function of the departure time u at the beginning vertex of the arc. The problem considered is to find the shortest path between a pair of nodes such that the total traverse time is lesser than some specified value T. Waiting times at nodes are considered as decision variables in this paper as well. Three variations of the problem are analyzed in which the waiting times are allowed to be arbitrary, the waiting times are zero and the waiting time has an upper bound at each vertex. Algorithms are also discussed to solve the problem. Some of the applications of the problem are encountered in communication networks and freight delivery problems. Suppose that some freight is to be sent from one city to another in a network before a deadline T. Different types of freight services with different costs and transit times are available between the two cities. An optimal solution would have to specify the route as well as the waiting times of the freight at each city so that the overall cost is minimum. A dynamic programming approach is used to compute the shortest path from the origin node to the destination node. Three cases are considered for the waiting times and an algorithm is suggested for each of those cases. This problem is obviously an NP-Complete problem as it is more difficult to solve than a problem with constant transit times and arc costs such as the previous problem which is NP-Complete. Some definitions: Waiting time w(x i) at a vertex x i is a nonnegative integer with an upper bound U x >= 0. Let P be a path from vertex x 1 to x r. The departure time T(x i) at a vertex x i can be expressed as below: T(x 1) = w(x 1) The departure time for all other vertices on the path can be expressed recursively as : The time of path P is defined as T(x r) where x r is the final node on the path..