Document Zbl 1116.90089

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Oper. Res.</a> <a href="/?q=in%3A114259" title="Articles in this Issue">32, No. 9, 2271-2284 (2005)</a>. </div> <div class="abstract">Summary: The knapsack problem is believed to be one of the “easier” $NP$-hard problems. Not only can it be solved in pseudo-polynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature. The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems. These problems are constructed either by using standard benchmark instances with larger coefficients, or by introducing new classes of instances for which most upper bounds perform badly. The first group of problems challenge the dynamic programming algorithms while the other group of problems are focused towards branch-and-bound algorithms. Numerous computational experiments with all recent state-of-the-art codes are used to show that (KP) is still difficult to solve for a wide number of problems. One could say that the previous benchmark tests were limited to a few highly structured instances, which do not show the full characteristics of knapsack problems.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A2160310">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A2160310">Cited in <strong>74</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C27" title="MSC2020">90C27</a> </td> <td class="space"> Combinatorial optimization </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C39" title="MSC2020">90C39</a> </td> <td class="space"> Dynamic programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C57" title="MSC2020">90C57</a> </td> <td class="space"> Polyhedral combinatorics, branch-and-bound, branch-and-cut </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AKnapsack+problem">Knapsack problem</a>; <a href="/?q=ut%3ADynamic+programming">Dynamic programming</a>; <a href="/?q=ut%3ABranch-and-bound">Branch-and-bound</a>; <a href="/?q=ut%3ATest+instances">Test instances</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4723">Knapsack</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 1116.90089" data-ciurl="/ci/02160310" data-biburl="/bibtex/02160310.bib" data-amsurl="/amsrefs/02160310.bib" data-xmlurl="/xml/02160310.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02160310.pdf" title="Zbl 1116.90089 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/10.1016/j.cor.2004.03.002" aria-label="DOI for “Where are the hard knapsack problems?”" title="10.1016/j.cor.2004.03.002">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Martello, S.; Pisinger, D.; Toth, P., New trends in exact algorithms for the 0-1 knapsack problem, European Journal of Operational Research, 123, 325-332 (2000) · <a href="/0961.90090" class="nowrap">Zbl 0961.90090</a></td> </tr><tr> <td>[2]</td> <td class="space">Kellerer, H.; Pferschy, U.; Pisinger, D., Knapsack problems (2004), Springer: Springer Berlin · <a href="/1103.90003" class="nowrap">Zbl 1103.90003</a></td> </tr><tr> <td>[3]</td> <td class="space">Meyer auf der Heide, F., A polynomial linear search algorithm for the $n$-dimensional knapsack problem, Journal of the ACM, 31, 668-676 (1984) · <a href="/0631.68037" class="nowrap">Zbl 0631.68037</a></td> </tr><tr> <td>[4]</td> <td class="space">Fournier H, Koiran P. Are lower bounds easier over the reals? In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC), 1998. p. 507-13.; Fournier H, Koiran P. Are lower bounds easier over the reals? In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC), 1998. p. 507-13. · <a href="/1028.68051" class="nowrap">Zbl 1028.68051</a></td> </tr><tr> <td>[5]</td> <td class="space">Balas, E.; Zemel, E., An algorithm for large zero-one knapsack problems, Operations Research, 28, 1130-1154 (1980) · <a href="/0449.90064" class="nowrap">Zbl 0449.90064</a></td> </tr><tr> <td>[6]</td> <td class="space">Martello, S.; Toth, P., A new algorithm for the 0-1 knapsack problem, Management Science, 34, 633-644 (1988) · <a href="/0645.90054" class="nowrap">Zbl 0645.90054</a></td> </tr><tr> <td>[7]</td> <td class="space">Pisinger, D., An expanding-core algorithm for the exact 0-1 knapsack problem, European Journal of Operational Research, 87, 175-187 (1995) · <a href="/0914.90199" class="nowrap">Zbl 0914.90199</a></td> </tr><tr> <td>[8]</td> <td class="space">Martello, S.; Toth, P., Upper bounds and algorithms for hard 0-1 knapsack problems, Operations Research, 45, 768-778 (1997) · <a href="/0902.90125" class="nowrap">Zbl 0902.90125</a></td> </tr><tr> <td>[9]</td> <td class="space">Bellman, R. E., Dynamic programming (1957), Princeton University Press: Princeton University Press Princeton, NJ · <a href="/0077.13605" class="nowrap">Zbl 0077.13605</a></td> </tr><tr> <td>[10]</td> <td class="space">Pisinger, D., Dynamic programming on the word RAM, Algorithmica, 35, 128-145 (2003) · <a href="/1033.90145" class="nowrap">Zbl 1033.90145</a></td> </tr><tr> <td>[11]</td> <td class="space">Pisinger, D., Linear time algorithms for knapsack problems with bounded weights, Journal of Algorithms, 33, 1-14 (1999) · <a href="/0951.90047" class="nowrap">Zbl 0951.90047</a></td> </tr><tr> <td>[12]</td> <td class="space">Pisinger, D., A minimal algorithm for the 0-1 knapsack problem, Operations Research, 45, 758-767 (1997) · <a href="/0902.90126" class="nowrap">Zbl 0902.90126</a></td> </tr><tr> <td>[13]</td> <td class="space">Martello, S.; Pisinger, D.; Toth, P., Dynamic programming and strong bounds for the 0-1 knapsack problem, Management Science, 45, 414-424 (1999) · <a href="/1231.90338" class="nowrap">Zbl 1231.90338</a></td> </tr><tr> <td>[14]</td> <td class="space">Martello, S.; Toth, P., Knapsack problems: algorithms and computer implementations (1990), Wiley: Wiley New York · <a href="/0708.68002" class="nowrap">Zbl 0708.68002</a></td> </tr><tr> <td>[15]</td> <td class="space">Pisinger, D., Core problems in knapsack algorithms, Operations Research, 47, 570-575 (1999) · <a href="/0979.90091" class="nowrap">Zbl 0979.90091</a></td> </tr><tr> <td>[16]</td> <td class="space">Weismantel, R., On the 0/1 knapsack polytope, Mathematical Programming, 77, 49-68 (1997) · <a href="/0891.90130" class="nowrap">Zbl 0891.90130</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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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