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Combinatorial optimization
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<h2 id="applications">Applications</h2> <p>Basic applications of combinatorial optimization include, but are not limited to: </p> <ul><li><a href="/facts/Logistics/0Hr2w55W">Logistics</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li><a href="/facts/Supply_chain_optimization/gSpJdjIs">Supply chain optimization</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li>Developing the best airline network of spokes and destinations</li> <li>Deciding which taxis in a fleet to route to pick up fares</li> <li>Determining the optimal way to deliver packages</li> <li>Allocating jobs to people optimally</li> <li>Designing water distribution networks</li> <li><a href="/facts/Earth_science/rQsXCSbM">Earth science</a> problems (e.g. <a href="/facts/Reservoir/2FewszmR">reservoir</a> flow-rates)<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul> <h2 id="methods">Methods</h2> <p>There is a large amount of literature on <a href="/facts/Polynomial-time_algorithm/77T62gmf">polynomial-time algorithms</a> for certain special classes of discrete optimization. A considerable amount of it is unified by the theory of <a href="/facts/Linear_programming/GduXFQxT">linear programming</a>. Some examples of combinatorial optimization problems that are covered by this framework are <a href="/facts/Shortest_path/g4nsrswr">shortest paths</a> and <a href="/facts/Shortest-path_tree/9GDrhwmY">shortest-path trees</a>, <a href="/facts/Flow_network/k0LCoRwD">flows and circulations</a>, <a href="/facts/Spanning_tree/qQPN9taR">spanning trees</a>, <a href="/facts/Matching_(graph_theory)/s0pUJ7hd">matching</a>, and <a href="/facts/Matroid/5bLcGlRF">matroid</a> problems. </p><p>For <a href="/facts/NP-complete/NFPv56DU">NP-complete</a> discrete optimization problems, current research literature includes the following topics: </p> <ul><li>polynomial-time exactly solvable special cases of the problem at hand (e.g. <a href="/facts/Fixed-parameter_tractable/RoyaJGgJ">fixed-parameter tractable</a> problems)</li> <li>algorithms that perform well on "random" instances (e.g. for the <a href="/facts/Traveling_salesman_problem/2WEKVLnV">traveling salesman problem</a>)</li> <li><a href="/facts/Approximation_algorithm/BWlf7M32">approximation algorithms</a> that run in polynomial time and find a solution that is close to optimal</li> <li><a href="/facts/Parameterized_approximation_algorithm/k38q7Ile">parameterized approximation algorithms</a> that run in <a href="/facts/Fixed-parameter_tractability/RoyaJGgJ">FPT</a> time and find a solution close to the optimum</li> <li>solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior of in NP-complete problems (e.g. real-world TSP instances with tens of thousands of nodes<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>).</li></ul> <p>Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of <a href="/facts/Search_algorithm/eJuw0nyz">search algorithm</a> or <a href="/facts/Metaheuristic/ECF71Rez">metaheuristic</a> can be used to solve them. Widely applicable approaches include <a href="/facts/Branch_and_bound/mhALtav7">branch-and-bound</a> (an exact algorithm which can be stopped at any point in time to serve as heuristic), <a href="/facts/Branch_and_cut/YJ07W6eP">branch-and-cut</a> (uses linear optimisation to generate bounds), <a href="/facts/Dynamic_programming/CLcabtmk">dynamic programming</a> (a recursive solution construction with limited search window) and <a href="/facts/Tabu_search/olw8qTSl">tabu search</a> (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are <a href="/facts/NP-complete/NFPv56DU">NP-complete</a>, such as the traveling salesman (decision) problem,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> this is expected unless <a href="/facts/P%253DNP/xoC7Ruda">P=NP</a>. </p><p>For each combinatorial optimization problem, there is a corresponding <a href="/facts/Decision_problem/cQNWjbdE">decision problem</a> that asks whether there is a feasible solution for some particular measure m 0 {\displaystyle m_{0}} . For example, if there is a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> G {\displaystyle G} which contains vertices u {\displaystyle u} and v {\displaystyle v} , an optimization problem might be "find a path from u {\displaystyle u} to v {\displaystyle v} that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from u {\displaystyle u} to v {\displaystyle v} that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'. </p><p>The field of <a href="/facts/Approximation_algorithm/BWlf7M32">approximation algorithms</a> deals with algorithms to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is then more naturally characterized as an optimization problem.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="np-optimization-problem">NP optimization problem</h2> <p>An <i>NP-optimization problem</i> (NPO) is a combinatorial optimization problem with the following additional conditions.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Note that the below referred <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances. </p> <ul><li>the size of every feasible solution y ∈ f ( x ) {\displaystyle y\in f(x)} is polynomially <a href="/facts/Bounded_set/yCldjtoy">bounded</a> in the size of the given instance x {\displaystyle x} ,</li> <li>the languages { x ∣ x ∈ I } {\displaystyle \{\,x\,\mid \,x\in I\,\}} and { ( x , y ) ∣ y ∈ f ( x ) } {\displaystyle \{\,(x,y)\,\mid \,y\in f(x)\,\}} can be <a href="/facts/Decidable_language/4R0zBWJE">recognized</a> in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a>, and</li> <li> m {\displaystyle m} is <a href="/facts/Polynomial_time/77T62gmf">polynomial-time computable</a>.</li></ul> <p>This implies that the corresponding decision problem is in <a href="/facts/NP_(complexity)/6muUfVMI">NP</a>. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are <a href="/facts/NP-completeness/NFPv56DU">NP-complete</a>. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual <a href="/facts/Turing_reduction/qczcQTbz">Turing</a> and <a href="/facts/Karp_reduction/bBvfFxyM">Karp reductions</a>. An example of such a reduction would be <a href="/facts/L-reduction/k71t0luq">L-reduction</a>. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>NPO is divided into the following subclasses according to their approximability:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <ul><li><i>NPO(I)</i>: Equals <a href="/facts/FPTAS/52Chjx9y">FPTAS</a>. Contains the <a href="/facts/Knapsack_problem/HMIdVJH5">Knapsack problem</a>.</li> <li><i>NPO(II)</i>: Equals <a href="/facts/Polynomial-time_approximation_scheme/52Chjx9y">PTAS</a>. Contains the <a href="/facts/Makespan/w1HEjJaQ">Makespan</a> scheduling problem.</li> <li><i>NPO(III)</i>: The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most <i>c</i> times the optimal cost (for minimization problems) or a cost at least 1 / c {\displaystyle 1/c} of the optimal cost (for maximization problems). In <a href="/facts/Juraj_Hromkovi%25C4%258D/nA3f0xck">Hromkovič</a>'s book[<i>which?</i>], excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains <a href="/facts/MAX-SAT/evkYeaAD">MAX-SAT</a> and metric <a href="/facts/Travelling_salesman_problem/2WEKVLnV">TSP</a>.</li> <li><i>NPO(IV)</i>: The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovič's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the <a href="/facts/Set_cover/MVzyBYs1">set cover</a> problem.</li> <li><i>NPO(V)</i>: The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the <a href="/facts/Travelling_salesman_problem/2WEKVLnV">TSP</a> and <a href="/facts/Clique_problem/cVVBNP08">clique problem</a>.</li></ul> <p>An NPO problem is called <i>polynomially bounded</i> (PB) if, for every instance x {\displaystyle x} and for every solution y ∈ f ( x ) {\displaystyle y\in f(x)} , the measure m ( x , y ) {\displaystyle m(x,y)} is bounded by a polynomial function of the size of x {\displaystyle x} . The class NPOPB is the class of NPO problems that are polynomially-bounded. </p> <h2 id="specific-problems">Specific problems</h2> <p class="note">Further information: Category:Combinatorial optimization</p> <p class="note">This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.</p> <ul><li><a href="/facts/Assignment_problem/dIMpM0aW">Assignment problem</a></li> <li><a href="/facts/Bin_packing_problem/T67Qe934">Bin packing problem</a></li> <li><a href="/facts/Chinese_postman_problem/zh0EhCsU">Chinese postman problem</a></li> <li><a href="/facts/Closure_problem/nBgu8YXo">Closure problem</a></li> <li><a href="/facts/Constraint_satisfaction_problem/oKCeIV2L">Constraint satisfaction problem</a></li> <li><a href="/facts/Cutting_stock_problem/aGw6fSEd">Cutting stock problem</a></li> <li><a href="/facts/Dominating_set/3xX4kSnP">Dominating set</a> problem</li> <li><a href="/facts/Integer_programming/MDaaadoA">Integer programming</a></li> <li><a href="/facts/Job_shop_scheduling/1JjAvD3m">Job shop scheduling</a></li> <li><a href="/facts/Knapsack_problem/HMIdVJH5">Knapsack problem</a></li> <li><a href="/facts/Metric_k-center/Uvt3a9Xw">Metric k-center</a> / vertex k-center problem</li> <li><a href="/facts/Minimum_relevant_variables_in_linear_system/3KK9pxDd">Minimum relevant variables in linear system</a></li> <li><a href="/facts/Minimum_spanning_tree/kgkGmY9s">Minimum spanning tree</a></li> <li><a href="/facts/Nurse_scheduling_problem/X9p6eVep">Nurse scheduling problem</a></li> <li><a href="/facts/Ring_star_problem/QpCdqPSF">Ring star problem</a></li> <li><a href="/facts/Set_cover_problem/MVzyBYs1">Set cover problem</a></li> <li><a href="/facts/Talent_scheduling/eJQ6GbFu">Talent scheduling</a></li> <li><a href="/facts/Traveling_salesman_problem/2WEKVLnV">Traveling salesman problem</a></li> <li><a href="/facts/Vehicle_rescheduling_problem/VGsVa0SF">Vehicle rescheduling problem</a></li> <li><a href="/facts/Vehicle_routing_problem/TS5pWNdR">Vehicle routing problem</a></li> <li><a href="/facts/Weapon_target_assignment_problem/1Q5pDfL6">Weapon target assignment problem</a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Constraint_composite_graph/JYrYbw9t">Constraint composite graph</a> – Node-weighted undirected graph associated with a given combinatorial optimization problem</li></ul> <h2 id="notes">Notes</h2> <ul><li>Beasley, J. E. <a href="http://people.brunel.ac.uk/~mastjjb/jeb/or/ip.html">"Integer programming"</a> (lecture notes).</li></ul> <ul><li><a href="/facts/William_J._Cook/n14EYleg">Cook, William J.</a>; Cunningham, William H.; <a href="/facts/William_R._Pulleyblank/vC30DT70">Pulleyblank, William R.</a>; <a href="/facts/Alexander_Schrijver/eJHJNRGz">Schrijver, Alexander</a> (1997). <i>Combinatorial Optimization</i>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-55894-X.</li></ul> <ul><li>Cook, William (2016). <a href="http://www.tsp.gatech.edu/optimal/index.html">"Optimal TSP Tours"</a>. <a href="/facts/University_of_Waterloo/NANSHN5Y">University of Waterloo</a>. <i>(Information on the largest TSP instances solved to date.)</i></li></ul> <ul><li>Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; <a href="/facts/Marek_Karpinski/XPxNRbPA">Karpinski, Marek</a>; <a href="/facts/Gerhard_J._Woeginger/beUa9kJr">Woeginger, Gerhard</a> (eds.). <a href="https://www.csc.kth.se/%7Eviggo/wwwcompendium/">"A Compendium of NP Optimization Problems"</a>. <i>(This is a continuously updated catalog of approximability results for NP optimization problems.)</i></li></ul> <ul><li>Das, Arnab; <a href="/facts/Bikas_K_Chakrabarti/svWvzLUy">Chakrabarti, Bikas K</a>, eds. (2005). <i>Quantum Annealing and Related Optimization Methods</i>. Lecture Notes in Physics. Vol. 679. Springer. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005qnro.book.....D">2005qnro.book.....D</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-27987-7.</li></ul> <ul><li>Das, Arnab; Chakrabarti, Bikas K (2008). "Colloquium: Quantum annealing and analog quantum computation". <i>Rev. Mod. Phys</i>. 80 (3): 1061. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0801.2193">0801.2193</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008RvMP...80.1061D">2008RvMP...80.1061D</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.563.9990">10.1.1.563.9990</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FRevModPhys.80.1061">10.1103/RevModPhys.80.1061</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14255125">14255125</a>.</li></ul> <ul><li><a href="/facts/Eugene_Lawler/pcQgHbej">Lawler, Eugene</a> (2001). <i>Combinatorial Optimization: Networks and Matroids</i>. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-41453-1.</li></ul> <ul><li><a href="/facts/Jon_Lee_(mathematician)/hnfK1FQN">Lee, Jon</a> (2004). <a href="https://books.google.com/books?id=3pL1B7WVYnAC"><i>A First Course in Combinatorial Optimization</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-01012-8.</li></ul> <ul><li>Papadimitriou, Christos H.; <a href="/facts/Kenneth_Steiglitz/zIFEom0n">Steiglitz, Kenneth</a> (July 1998). <i>Combinatorial Optimization : Algorithms and Complexity</i>. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-40258-4.</li></ul> <ul><li>Schrijver, Alexander (2003). <a href="https://books.google.com/books?id=mqGeSQ6dJycC"><i>Combinatorial Optimization: Polyhedra and Efficiency</i></a>. Algorithms and Combinatorics. Vol. 24. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783540443896.</li></ul> <ul><li>Schrijver, Alexander (2005). <a href="http://homepages.cwi.nl/~lex/files/histco.pdf">"On the history of combinatorial optimization (till 1960)"</a> (PDF). In <a href="/facts/Karen_Aardal/C2Y31j5W">Aardal, K.</a>; Nemhauser, G.L.; Weismantel, R. (eds.). <i>Handbook of Discrete Optimization</i>. Elsevier. pp. 1–68.</li></ul> <ul><li>Schrijver, Alexander (February 1, 2006). <a href="http://homepages.cwi.nl/~lex/files/dict.pdf"><i>A Course in Combinatorial Optimization</i></a> (PDF).</li></ul> <ul><li>Sierksma, Gerard; Ghosh, Diptesh (2010). <i>Networks in Action; Text and Computer Exercises in Network Optimization</i>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-5512-8.</li></ul> <ul><li>Gerard Sierksma; Yori Zwols (2015). <i>Linear and Integer Optimization: Theory and Practice</i>. CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-498-71016-9.</li></ul> <ul><li>Pintea, C-M. (2014). <a href="https://www.springer.com/la/book/9783642401787"><i>Advances in Bio-inspired Computing for Combinatorial Optimization Problem</i></a>. Intelligent Systems Reference Library. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-40178-7.</li></ul> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Combinatorial optimization. <ul><li><a href="https://www.springer.com/mathematics/journal/10878">Journal of Combinatorial Optimization</a></li> <li><a href="http://www.iasi.cnr.it/aussois">The Aussois Combinatorial Optimization Workshop</a></li> <li><a href="http://sourceforge.net/projects/jcop/">Java Combinatorial Optimization Platform</a> (open source code)</li> <li><a href="https://www.mjc2.com/staff-planning-complexity.htm">Why is scheduling people hard?</a></li> <li><a href="https://web.archive.org/web/20170830023312/https://www7.in.tum.de/~kugele/files/jobsis.pdf">Complexity classes for optimization problems / Stefan Kugele</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Schrijver 2003, p. 1. - Schrijver, Alexander (2003). Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics. Vol. 24. Springer. ISBN 9783540443896. <a href="https://books.google.com/books?id=mqGeSQ6dJycC" target="_blank">https://books.google.com/books?id=mqGeSQ6dJycC</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Sbihi, Abdelkader; Eglese, Richard W. (2007). "Combinatorial optimization and Green Logistics" (PDF). 4OR. 5 (2): 99–116. doi:10.1007/s10288-007-0047-3. S2CID 207070217. Archived (PDF) from the original on 2019-12-26. Retrieved 2019-12-26. <a href="https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf" target="_blank">https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Eskandarpour, Majid; Dejax, Pierre; Miemczyk, Joe; Péton, Olivier (2015). "Sustainable supply chain network design: An optimization-oriented review" (PDF). Omega. 54: 11–32. doi:10.1016/j.omega.2015.01.006. Archived (PDF) from the original on 2019-12-26. Retrieved 2019-12-26. <a href="https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf" target="_blank">https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hobé, Alex; Vogler, Daniel; Seybold, Martin P.; Ebigbo, Anozie; Settgast, Randolph R.; Saar, Martin O. (2018). "Estimating fluid flow rates through fracture networks using combinatorial optimization". Advances in Water Resources. 122: 85–97. arXiv:1801.08321. Bibcode:2018AdWR..122...85H. doi:10.1016/j.advwatres.2018.10.002. S2CID 119476042. Archived from the original on 2020-08-21. Retrieved 2020-09-16. <a href="https://www.sciencedirect.com/science/article/abs/pii/S0309170818300666" target="_blank">https://www.sciencedirect.com/science/article/abs/pii/S0309170818300666</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Cook 2016. - Cook, William (2016). "Optimal TSP Tours". University of Waterloo. <a href="http://www.tsp.gatech.edu/optimal/index.html" target="_blank">http://www.tsp.gatech.edu/optimal/index.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Approximation-TSP" (PDF). Archived (PDF) from the original on 2022-03-01. Retrieved 2022-02-17. <a href="https://www.csd.uoc.gr/~hy583/papers/ch11.pdf" target="_blank">https://www.csd.uoc.gr/~hy583/papers/ch11.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Ausiello, Giorgio; et al. (2003), Complexity and Approximation (Corrected ed.), Springer, ISBN 978-3-540-65431-5 <a href="978-3-540-65431-5" target="_blank">978-3-540-65431-5</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hromkovic, Juraj (2002), Algorithmics for Hard Problems, Texts in Theoretical Computer Science (2nd ed.), Springer, ISBN 978-3-540-44134-2 <a href="978-3-540-44134-2" target="_blank">978-3-540-44134-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems, Royal Institute of Technology, Sweden, ISBN 91-7170-082-X <a href="91-7170-082-X" target="_blank">91-7170-082-X</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hromkovic, Juraj (2002), Algorithmics for Hard Problems, Texts in Theoretical Computer Science (2nd ed.), Springer, ISBN 978-3-540-44134-2 <a href="978-3-540-44134-2" target="_blank">978-3-540-44134-2</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>