Document Zbl 0715.90063

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Program., Ser. A</a> <a href="/?q=in%3A155624" title="Articles in this Issue">46, No. 3, 259-271 (1990)</a>. </div> <div class="abstract">Summary: We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time $p_{ij}$. The objective is to find a schedule that minimizes the makespan. <br class="zbmathjax-paragraph">Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. <br class="zbmathjax-paragraph">In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than $\frac{3}{2}$ unless $P=NP$. We finally obtain a complexity classification for all special cases with a fixed number of processing times.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A4179126">Cited in <strong>12</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A4179126">Cited in <strong>262</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B35" title="MSC2020">90B35</a> </td> <td class="space"> Deterministic scheduling theory in operations research </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C10" title="MSC2020">90C10</a> </td> <td class="space"> Integer programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C60" title="MSC2020">90C60</a> </td> <td class="space"> Abstract computational complexity for mathematical programming problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C05" title="MSC2020">90C05</a> </td> <td class="space"> Linear programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90-08" title="MSC2020">90-08</a> </td> <td class="space"> Computational methods for problems pertaining to operations research and mathematical programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K05" title="MSC2020">65K05</a> </td> <td class="space"> Numerical mathematical programming methods </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aworst+case+analysis">worst case analysis</a>; <a href="/?q=ut%3Aparallel+machines">parallel machines</a>; <a href="/?q=ut%3Aindependent+jobs">independent jobs</a>; <a href="/?q=ut%3Amakespan">makespan</a>; <a href="/?q=ut%3Apolynomial+algorithm">polynomial algorithm</a>; <a href="/?q=ut%3Apolynomial+approximation+scheme">polynomial approximation scheme</a>; <a href="/?q=ut%3Arelaxations">relaxations</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0715.90063" data-ciurl="/ci/04179126" data-biburl="/bibtex/04179126.bib" data-amsurl="/amsrefs/04179126.bib" data-xmlurl="/xml/04179126.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04179126.pdf" title="Zbl 0715.90063 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF01585745" aria-label="DOI for “Approximation algorithms for scheduling unrelated parallel machines”" title="10.1007/BF01585745">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://ir.cwi.nl/pub/18055" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">R. Aharoni, P. Erdös and N. Linial, ”Dual integer linear programs and the relationship between their optima,”Proceedings of the 17th Annual ACM Symposium on Theory of Computing (1985) 476–483.</td> </tr><tr> <td>[2]</td> <td class="space">J.J. Bartholdi III, ”A guaranteed-accuracy round-off algorithm for cyclic scheduling and set covering,”Operations Research 29 (1981) 501–510. · <a href="/0466.90037" class="nowrap">Zbl 0466.90037</a> · <a href="https://doi.org/10.1287/opre.29.3.501" class="nowrap">doi:10.1287/opre.29.3.501</a></td> </tr><tr> <td>[3]</td> <td class="space">J.J. Bartholdi III, J.B. Orlin and H.D. Ratliff, ”Cyclic scheduling via integer programs with circular ones,”Operations Research 28 (1980) 1074–1085. · <a href="/0451.90075" class="nowrap">Zbl 0451.90075</a> · <a href="https://doi.org/10.1287/opre.28.5.1074" class="nowrap">doi:10.1287/opre.28.5.1074</a></td> </tr><tr> <td>[4]</td> <td class="space">S. Baum and L.E. Trotter Jr., ”Integer rounding for polymatroid and branching optimization problems,”SIAM Journal on Algebraic and Discrete Methods 2 (1981) 416–425. · <a href="/0518.90058" class="nowrap">Zbl 0518.90058</a> · <a href="https://doi.org/10.1137/0602044" class="nowrap">doi:10.1137/0602044</a></td> </tr><tr> <td>[5]</td> <td class="space">V. Chvátal, ”A greedy heuristic for the set-covering problem,”Mathematics of Operations Research 4 (1979) 233–235. · <a href="/0443.90066" class="nowrap">Zbl 0443.90066</a> · <a href="https://doi.org/10.1287/moor.4.3.233" class="nowrap">doi:10.1287/moor.4.3.233</a></td> </tr><tr> <td>[6]</td> <td class="space">G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).</td> </tr><tr> <td>[7]</td> <td class="space">E. Davis and J.M. Jaffe, ”Algorithms for scheduling tasks on unrelated processors,”Journal of the Association for Computing Machinery 28 (1981) 721–736. · <a href="/0472.68020" class="nowrap">Zbl 0472.68020</a></td> </tr><tr> <td>[8]</td> <td class="space">M.R. Garey and D.S. Johnson, ”Complexity results for multiprocessor scheduling under resource constraints,”SIAM Journal on Computing 4 (1975) 397–411. · <a href="/0365.90076" class="nowrap">Zbl 0365.90076</a> · <a href="https://doi.org/10.1137/0204035" class="nowrap">doi:10.1137/0204035</a></td> </tr><tr> <td>[9]</td> <td class="space">M.R. Garey and D.S. Johnson, ”Strong NP-completeness results: motivation, examples and implications,”Journal of the Association for Computing Machinery 25 (1978) 499–508. · <a href="/0379.68035" class="nowrap">Zbl 0379.68035</a></td> </tr><tr> <td>[10]</td> <td class="space">M.R. Garey and D.S. Johnson,Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, San Francisco, CA, 1979). · <a href="/0411.68039" class="nowrap">Zbl 0411.68039</a></td> </tr><tr> <td>[11]</td> <td class="space">T. Gonzalez, O.H. Ibarra and S. Sahni, ”Bounds for LPT schedules on uniform processors,”SIAM Journal on Computing 6 (1977) 155–166. · <a href="/0347.68043" class="nowrap">Zbl 0347.68043</a> · <a href="https://doi.org/10.1137/0206013" class="nowrap">doi:10.1137/0206013</a></td> </tr><tr> <td>[12]</td> <td class="space">R.L. Graham, ”Bounds for certain multiprocessing anomalies,”Bell System Technological Journal 45 (1966) 1563–1581. · <a href="/0168.40703" class="nowrap">Zbl 0168.40703</a></td> </tr><tr> <td>[13]</td> <td class="space">R.L. Graham, ”Bounds on multiprocessing timing anomalies,”SIAM Journal on Applied Mathematics 17 (1969) 416–429. · <a href="/0188.23101" class="nowrap">Zbl 0188.23101</a> · <a href="https://doi.org/10.1137/0117039" class="nowrap">doi:10.1137/0117039</a></td> </tr><tr> <td>[14]</td> <td class="space">R.L. Graham, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, ”Optimization and approximation in deterministic sequencing and scheduling: a survey,”Annals of Discrete Mathematics 5 (1979) 287–326. · <a href="/0411.90044" class="nowrap">Zbl 0411.90044</a> · <a href="https://doi.org/10.1016/S0167-5060(08)70356-X" class="nowrap">doi:10.1016/S0167-5060(08)70356-X</a></td> </tr><tr> <td>[15]</td> <td class="space">M. Grötschel, L. Lovász and A. Schrijver,Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988). · <a href="/0634.05001" class="nowrap">Zbl 0634.05001</a></td> </tr><tr> <td>[16]</td> <td class="space">D.S. Hochbaum and D.B. Shmoys, ”Using dual approximation algorithms for scheduling problems: practical and theoretical results,”Journal of the Association for Computing Machinery 34 (1987) 144–162.</td> </tr><tr> <td>[17]</td> <td class="space">D.S. Hochbaum and D.B. Shmoys, ”A polynomial approximation scheme for machine scheduling on uniform processors: using the dual approximation approach,”SIAM Journal on Computing 17 (1988) 539–551. · <a href="/0647.68040" class="nowrap">Zbl 0647.68040</a> · <a href="https://doi.org/10.1137/0217033" class="nowrap">doi:10.1137/0217033</a></td> </tr><tr> <td>[18]</td> <td class="space">E. Horowitz and S. Sahni, ”Exact and approximate algorithms for scheduling nonidentical processors,”Journal of the Association for Computing Machinery 23 (1976) 317–327. · <a href="/0329.68041" class="nowrap">Zbl 0329.68041</a></td> </tr><tr> <td>[19]</td> <td class="space">L.G. Khachian, ”A polynomial time algorithm in linear programming, ”Soviet Mathematics Doklady 20 (1979) 191–194.</td> </tr><tr> <td>[20]</td> <td class="space">L. Lovász, ”On the ratio of optimal and fractional covers,”Discrete Applied Mathematics 13 (1983) 383–390. · <a href="/0323.05127" class="nowrap">Zbl 0323.05127</a></td> </tr><tr> <td>[21]</td> <td class="space">O.M.-C. Marcotte,Topics in Combinatorial Packing and Covering, Ph.D. Thesis, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1983).</td> </tr><tr> <td>[22]</td> <td class="space">K. Numata, ”Approximate and exact algorithms for scheduling independent tasks on unrelated processors,”Journal of the Operations Research Society of Japan 31 (1988) 61–81. · <a href="/0645.90038" class="nowrap">Zbl 0645.90038</a></td> </tr><tr> <td>[23]</td> <td class="space">C.N. Potts, ”Analysis of a linear programming heuristic for scheduling unrelated parallel machines,”Discrete Applied Mathematics 10 (1985) 155–164. · <a href="/0571.90035" class="nowrap">Zbl 0571.90035</a> · <a href="https://doi.org/10.1016/0166-218X(85)90009-5" class="nowrap">doi:10.1016/0166-218X(85)90009-5</a></td> </tr><tr> <td>[24]</td> <td class="space">P. Raghavan, ”Probabilistic construction of deterministic algorithms: approximating packing integer programs,”Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science (1986) 10–18.</td> </tr><tr> <td>[25]</td> <td class="space">P. Raghavan and C.D. Thompson, ”Provably good routing in graphs: regular arrays,”Proceedings of the 17th Annual ACM Symposium on Theory of Computing (1985) 79–87.</td> </tr><tr> <td>[26]</td> <td class="space">S. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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