Bernhard Korte

An Analysis of the Greedy Heuristic for Independence Systems

The worst case behaviour of the greedy heuristic for independence systems is analyzed by deriving lower bounds for the ratio of the greedy solution value to the optimal value. For two special independence systems, this ratio can be bounded by 1/2, for two other independence systems, it converges with increasing problem size to zero. The main theorem states that for every independence system ( E , F ) the ratio is bounded by I/k, k such that ( E , F ) can be represented as the intersection of k matroids.

ON THE EXISTENCE OF FAST APPROXIMATION SCHEMES

We characterize those combinatorial optimization problems which can be solved approximately by polynomially bounded algorithms. Using slight modifications of the Sahni and Ibarra and Kim algorithms for the knapsack problem we prove that there is no fast approximation scheme unless their algorithmic ideas apply. Hence we show that these algorithms are not only the origin but also prototypes for all polynomial or fully polynomial approximation schemes.

Cycle time and slack optimization for VLSI-chips

We consider the problem of finding an optimal clock schedule, i.e. optimal arrival times for clock signals at latches of a VLSI chip. We describe a general model which includes all previously considered models. Then we show how to optimize the cycle time and optimally balance slacks on data paths and on clocktree paths. The problem of finding a clock schedule with the optimum cycle time was solved before, either by linear programming or by binary search, using a test for negative circuits in a digraph as a subroutine. We show for the first time that a direct combinatorial algorithm solves this problem optimally. Incidentally, this yields a new efficient method for timing analysis with transparent latches. Moreover, we extend this algorithm to the slack balancing problem: To make the chip less sensitive to routing detours, process variations and manufacturing skew it is desirable to have as few critical paths as possible. We show how to find the clock schedule with minimum number of critical paths (optimum slack distribution) in a well-defined sense. Rather than fixed dock arrival times we show how to obtain as large as possible intervals for the clock arrival times. This can be considered as slack on clocktree paths. Indeed, we can find the global optimum of simultaneous optimization of slacks on all data paths and clocktree paths. All the above is done by very efficient network optimization algorithms, based on parametric shortest paths. Our computational results with recent IBM processor chips show that the number of critical paths decreases dramatically, in addition to a considerable improvement of the cycle time. The running times are reasonable even for the largest designs.

Greedoids - A Structural Framework for the Greedy Algorithm

In this paper we introduce combinatorial structures called “greedoids” which are more general than matroids and which can be considered as relaxations of them. These structures are characterized by the optimality of the greedy solution for a broad class of objective functions of which breadth first search, shortest path, scheduling under precedence constraints are special cases. Besides some basic structural and algorithmic facts about greedoids we mainly discuss many different examples of greedoids. A subsequent paper will deal with more structural properties of greedoids.

Clock Scheduling and Clocktree Construction for High Performance ASICS

In this paper we present a new method for clock schedulingand clocktree construction that improves the performance ofhigh-end ASICs significantly.First, we compute a clock schedule that yields the optimumcycle time and the best possible clock distribution with respectto early and late mode; in particular the number of criticaltests is minimized. Second, individual arrival time intervalsare computed for all endpoints of the clocktree. Finally, weconstruct a clocktree that realizes arrival times within theseintervals and exploits positive slacks to save power consumption.We demonstrate the superiority of our method to previousapproaches by experimental results on industrial ASICs withup to 194 000 registers and more than 160 clock domains. Weimproved the clock frequencies by 5-28% up to 1.033 GHz (in hardware).

Structural properties of greedoids

Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. In this paper they are studied from a structural aspect. Definitions of basic matroid-theoretical concepts such as rank and closure can be generalized to greedoids, even though they loose some of their fundamental properties. The rank function of a greedoid is only “locally” submodular. The closure operator is not monotone but possesses a (relaxed) Steinitz—McLane exchange property. We define two classes of subsets, called rank-feasible and closure-feasible, so that the rank and closure behave nicely for them. In particular, restricted to rank-feasible sets the rank function is submodular. Finally we show that Rado’s theorem on independent transversals of subsets of matroids remains valid for feasible transversals of certain sets of greedoids.

On the RAS-algorithm

Given a nonnegative real (m, n) matrixA and positive vectorsu, v, then the biproportional constrained matrix problem is to find a nonnegative (m, n) matrixB such thatB=diag (x) A diag (y) holds for some vectorsx ∈ ℝm andy ∈ ℝn and the row (column) sums ofB equalui (vj)i=1,...,m(j=1,..., n). A solution procedure (called the RAS-method) was proposed by Bacharach [1] to solve this problem. The main disadvantage of this algorithm is, that round-off errors slow down the convergence. Here we present a modified RAS-method which together with several other improvements overcomes this disadvantage.ZusammenfassungSeiA eine reelle (m, n) Matrix undu, v positive Vektoren. Das nichtnegative Matrixproblem besteht nun in der Aufgabe, eine nichtnegative (m, n) Matrix zu bestimmen, so daßB=diag(x) A diag (y) für Vektorenx ∈ ℝm undy ∈ ℝn gilt undui(vj)i=1, ...,m (j=1,...,n) die Zeilen- und Spaltensummen vonB darstellen. Eine Lösungsmethode (RAS-Verfahren) wurde von Bacharach [1] vorgeschlagen. Ein wesentlicher Nachteil dieses Algorithmus ist die Verlangsamung der Konvergenzgeschwindigkeit durch Rundungsfehler. Hier schlagen wir einen modifizierten RAS-Algorithmus vor, der zusammen mit anderen Verbesserungen diesen Nachteil überwindet.

Lower bounds on the worst-case complexity of some oracle algorithms

Abstract To give a proper definition of the complexity of very general computational problems such as optimization problems over arbitrary independence systems or fixed-point problems for continuous functions, it is useful to consider the input for these problems as “oracles” R which can be called by the algorithms for some values x ∈ X and which then give back some information R ( x ) about x , e.g. whether x belongs to the independence system or the point into which x is mapped by the continuous function. A lower bound on the complexity of an algorithm using an oracle R is the number of calls on R in the worst case. Using this technique it is shown that there is no polynomial approximative algorithm for the maximization problem over a general independence system which has a better worst-case behaviour than the greedy algorithm. Moreover several formalizations of the problem of approximating a fixed point of a continuous function are considered, and it is shown that none of these problems can be solved by a bounded algorithm.

Greedoids and Linear Objective Functions

Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. They can be characterized algorithmically via the optimality of the greedy algorithm for a class of objective functions, which are in general not linear and do not include all linear functions. It is therefore natural to ask the following questions: (1) What are those linear objective functions which can be optimized over any greedoid by the greedy algorithm; (2) what are those greedoids over which the linear objective function can be optimized by the greedy algorithm. This paper gives an answer to both questions. Moreover, it gives slimming procedures for obtaining such greedoids from matroids and it gives briefly some (negative) oracle results about greedoid optimization and greedoid recognition.

Maximum mean weight cycle in a digraph and minimizing cycle time of a logic chip

The maximum mean weight cycle problem is well-known: given a digraph G with weights c:E(G) → R, find a directed circuit in G whose mean weight is maximum. Closely related is the minimum balance problem: Find a potential π: V(G) → R such that the numbers slack(e):=π(w)-π(v)-c((v, w))(e=(v, w)∈E(G)) are optimally balanced: for any subset of vertices, the minimum slack on an entering edge should equal the minimum slack on a leaving edge. Both problems can be solved by a parametric shortest path algorithm.We describe an application of these problems to the design of logic chips. In the simplest model, optimizing the clock schedule of a chip to minimize the cycle time is equivalent to a maximum mean weight cycle problem. It is very important to find a solution with well-balanced slacks; this problem, in the simple model, is a minimum balance problem.However, in practical situations many constraints have to be taken into account. Therefore minimizing the cycle time and finding the optimum slack distribution are more general problems. We show how a parametric shortest path algorithm can be extended to solve these problems efficiently.Computational results with recent IBM processor chips show that the cycle time reduces considerably. Moreover, the number of critical paths (with small slack) decreases dramatically. As a result we obtain significantly faster chips. The running time of our algorithm is reasonable even for very large designs.