Complexity and computability of solutions to linear programming systems

Through key examples and constructs, exact and approximate, complexity, computability, and solution of linear programming systems are reexamined in the light of Khachian's new notion of (approximate) solution. Algorithms, basic theorems, and alternate representations are reviewed. It is shown that the Klee-Minty example hasnever been exponential for (exact) adjacent extreme point algorithms and that the Balinski-Gomory (exact) algorithm continues to be polynomial in cases where (approximate) ellipsoidal centered-cutoff algorithms (Levin, Shor, Khachian, Gacs-Lovasz) are exponential. By model approximation, both the Klee-Minty and the new J. Clausen examples are shown to be trivial (explicitly solvable) interval programming problems. A new notion of computable (approximate) solution is proposed together with ana priori regularization for linear programming systems. New polyhedral constraint contraction algorithms are proposed for approximate solution and the relevance of interval programming for good starts or exact solution is brought forth. It is concluded from all this that the imposed problem ignorance of past complexity research is deleterious to research progress on computability or efficiency of computation.This research was partly supported by Project NR047-071, ONR Contract N00014-80-C-0242, and Project NR047-021, ONR Contract N00014-75-C-0569, with the Center for Cybernetic Studies, The University of Texas at Austin.     

The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

The plane with parallel coordinates

By means ofParallel Coordinates planar graphs of multivariate relations are obtained. Certain properties of the relationship correspond tothe geometrical properties of its graph. On the plane a point line duality with several interesting properties is induced. A new duality betweenbounded and unbounded convex sets and hstars (a generalization of hyperbolas) and between Convex Unions and Intersections is found. This motivates some efficient Convexity algorithms and other results inComputational Geometry. There is also a suprising cusp inflection point duality. The narrative ends with a preview of the corresponding results inR ^N .     

Sound and Complete Qualitative Simulation Needs “Quantitative” Filtering

The AI methodology of qualitative reasoning furnishes useful tools to scientists and engineers who need to deal with incomplete system knowledge during design, analysis, or diagnosis tasks. Qualitative simulators have a theoretical soundness guarantee; they cannot overlook any concrete equation implied by their input. On the other hand, the basic qualitative simulation algorithms have been shown to suffer from the incompleteness problem; they may allow non-solutions of the input equation to appear in their output. The question of whether a simulator with purely qualitative input which never predicts spurious behaviors can ever be achieved by adding new filters to the existing algorithm has remained unanswered. In this paper, we show that, if such a sound and complete simulator exists, it will have to be able to handle numerical distinctions with such a high precision that it must contain a component that would better be called a quantitative, rather than qualitative reasoner. This is due to the ability of the pure qualitative format to allow the exact representation of the members of a rich set of numbers.     

Maintaining Longest Paths Incrementally

Modeling and programming tools for neighborhood search often support invariants, i.e., data structures specified declaratively and automatically maintained incrementally under changes. This paper considers invariants for longest paths in directed acyclic graphs, a fundamental abstraction for many applications. It presents bounded incremental algorithms for arc insertion and deletion which run in O( + || log||) time and O() time respectively, where || and are measures of the change in the input and output. The paper also shows how to generalize the algorithm to various classes of multiple insertions/deletions encountered in scheduling applications. Preliminary experimental results show that the algorithms behave well in practice.     

Sometime = always + recursion ≡ always on the equivalence of the intermittent and invariant assertions methods for proving inevitability properties of programs

Summary We propose and compare two induction principles called always and sometime for proving inevitability properties of programs. They are respective formalizations and generalizations of Floyd invariant assertions and Burstall intermittent assertions methods for proving total correctness of sequential programs whose methodological advantages or disadvantages have been discussed in a number of previous papers. Both principles are formalized in the abstract setting of arbitrary nondeterministic transition systems and illustrated by appropriate examples. The sometime method is interpreted as a recursive application of the always method. Hence always can be considered as a special case of sometime. These proof methods are strongly equivalent in the sense that a proof by one induction principle can be rewritten into a proof by the other one. The first two theorems of the paper show that an invariant for the always method can be translated into an invariant for the sometime method even if every recursive application of the later is required to be of finite length. The third and main theorem of the paper shows how to translate an invariant for the sometime method into an invariant for the always method. It is emphasized that this translation technique follows the idea of transforming recursive programs into iterative ones. Of course, a general translation technique does not imply that the original sometime invariant and the resulting always invariant are equally understandable. This is illustrated by an example.     

Optimal multilayer channel routing with overlap

This paper presents algorithms for multiterminal net channel routing where multiple interconnect layers are available. Major improvements are possible if wires are able to overlap, and our generalized main algorithm allows overlap, but only on everyKth (K 2) layer. Our algorithm will, for a problem with densityd onL layers,L K + 3,provably use at most three tracks more than optimal: (d + 1)/L/K + 2 tracks, compared with the lower bound of d/L/K. Our algorithm is simple, has few vias, tends to minimize wire length, and could be used if different layers have different grid sizes. Finally, we extend our algorithm in order to obtain improved results for adjacent (K = 1) overlap: (d + 2)/2L/3 + 5 forL 7.This work was supported by the Semiconductor Research Corporation under Contract 83-01-035, by a grant from the General Electric Corporation, and by a grant at the University of the Saarland.     

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A (upper approximation minus lower approximation), where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U-uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U(A) = 1/h(A) ₀ ^h(A) log [1+(A)]d, where h(A) is the largest membership value obtained by any element of A and (A) is the measure of the -cut of A defined by the Lebesge integral of its characteristic function.     

A Typology of Translation Problems for Eurotra Translation Machines

This paper presents a detailed study of Eurotra Machine Translation engines, namely the mainstream Eurotra software known as the E-Framework, and two unofficial spin-offs – the C,A,T and Relaxed Compositionality translator notations – with regard to how these systems handle hard cases, and in particular their ability to handle combinations of such problems. In the C,A,T translator notation, some cases of complex transfer are wild, meaning roughly that they interact badly when presented with other complex cases in the same sentence. The effect of this is that each combination of a wild case and another complex case needs ad hoc treatment. The E-Framework is the same as the C,A,T notation in this respect. In general, the E-Framework is equivalent to the C,A,T notation for the task of transfer. The Relaxed Compositionality translator notation is able to handle each wild case (bar one exception) with a single rule even where it appears in the same sentence as other complex cases.     

Directional differentiation of marginal functions in mathematical programming problems

We consider regular mathematical programming problems of the form f(x, y) inf, y F(x), x Rⁿ, where F(x) = {y R^m h_i| (x, y) 0, , h_i (x, y) = 0, . The directional derivatives offunctions (x) = inf{f(x, y)|y F(x)} are estimated.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 70–77, November–December, 1991.