Abstract
We consider a finite family of vectors in the plane with sum equal to zero and each vector having at most unit length. It is proved (by constructing an algorithm) that any such family of vectors can be nonstrictly summed within a given unbounded convex set in the plane which satisfies some conditions. The application of the algorithm of the nonstrict vector summation to three scheduling problems with three machines enables one to construct polynomial-time approximation algorithms for their solutions with worst-case absolute errors independent of the number of jobs.
This research was partially supported by the Russian Foundation for Fundamental Research (Grant 93-01-00489) and the International Science Foundation (Grant NQC000).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
I. S. Belov and Ya. N. Stolin (1974) An algorithm for the single-route scheduling problem (in Russian), in: Mathematical Economics and Functional Analysis, Nauka, Moscow, pp. 248–257.
S.V. Sevast’yanov (1974) Asymptotical approach to some scheduling problems (in Russian), in: Abstracts of the Third All-Union Conference on Problems of Theoretical Cybernetics (Novosibirsk), Akad. Nauk, Sibirsk. Otdel., Inst. Mat., Novosibirsk, pp. 67–69.
S.V. Sevast’yanov (1975) Asymptotical approach to some scheduling problems (in Russian), Upravlyaemye Sistemy 14, 40–51.
A. I. Babushkin, A. L. Bashta, I. S. Belov, and B. I. Dushin (1975) Minimization of a flow-line working time (in Russian), Avtomat. i Telemekh. No. 6, 161–167.
A. I. Babushkin, A. L. Bashta, and I. S. Belov (1976) Scheduling of a three machine job shop (in Russian), Avtomat. i Telemekh. No. 6, 154–158.
A. I. Babushkin, A. L. Bashta, and I. S. Belov (1977) Construction of a schedule for the problem of counter-routes (in Russian), Kibernetika (Kiev) No. 4, 130–135.
S. V. Sevast’yanov (1981) Some generalizations of the Johnson problem (in Russian), Upravlyaemye Sistemy 21, 45–61.
S. V. Sevast’yanov (1986) An algorithm with an estimate for a problem with routings of parts of arbitrary shape and alternative executors (in Russian), Kibernetika (Kiev) No. 6, 74–79.
S. V. Sevast’yanov (1988) Geometry in scheduling theory (in Russian), in: Modeli i Metody Optimizatsii, Trudy Inst. Mat. Vol. 10, Novosibirsk, pp. 226–261.
E. Steinitz (1913) Bedingt konvergente Reihen und convexe Systeme, J. Reine Angew. Math. 143, 128–175.
W. Gross (1917) Bedingt konvergente Reihen, Monatsh. Math, und Phys. 28, 221–237.
V. Bergström (1931) Ein neuer Beweis eines Satzes von E. Steinitz Abh. Math. Sem. Univ. Hamburg 8, 148–152.
V. Bergström (1931) Zwei Satze über ebene Vektor polygone Abh. Math. Sem. Univ. Hamburg 8, 206–214.
I. Damsteeg and I. Halperin (1950) The Steinitz-Gross theorem on sums of vectors, Trans. Roy. Soc. Canada 44, 31–35.
M. I. Kadets (1953) On a property of polygonal paths in n-dimensional space (in Russian), Uspekhi Mat. Nauk 8, No. 1, 139–143.
F. A. Behrend (1954) The Steinitz-Gross theorem on sums of vectors Canad. J. Math. 6, 108–124.
S.V. Sevast’yanov (1978) Approximate solution of some problems of scheduling theory (in Russian), Metody Diskret. Analiz. 32, 66–75.
V. S. Grinberg and S. V. Sevast’yanov (1980) Value of the Steinitz constant, Functional Anal. Appl. 14, 125–126.
W. Banaszczyk (1987) The Steinitz constant of the plane J. Reine Angew. Math. 373, 218–220.
S. V. Sevast’yanov (1991) On a compact vector summation (in Russian), Diskret. Mat, 3, No. 3, 66–72.
R. T. Rockafellar (1970) Convex Analysis, Princeton University Press, Princeton, New Jersey.
M. R. Garey and D. S. Johnson (1979) Computers and Intractability, Freeman, San Francisco.
W. T. Trotter (1992) Combinatorics and Partially Ordered Sets. D imension Theory, The Johns Hopkins University Press, Baltimore and London.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Sevast’yanov, S.V. (1996). Nonstrict Vector Summation in Scheduling Problems. In: Korshunov, A.D. (eds) Discrete Analysis and Operations Research. Mathematics and Its Applications, vol 355. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1606-7_18
Download citation
DOI: https://doi.org/10.1007/978-94-009-1606-7_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7217-5
Online ISBN: 978-94-009-1606-7
eBook Packages: Springer Book Archive