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==References== {{refbegin|30em}} *{{cite conference |last1= Berman |first1= Piotr |last2= Karpinski |first2= Marek | authorlink2 = Marek Karpinski |last3= Zelikovsky |first3= Alexander |year= 2009 |contribution= 1.25-approximation algorithm for Steiner tree problem with distances 1 and 2 |series= Lecture Notes in Computer Science |title=Algorithms and Data Structures: 11th International Symposium, WADS 2009, Banff, Canada, August 21–23, 2009, Proceedings |volume= 5664 |pages= 86–97 |doi=10.1007/978-3-642-03367-4_8 |arxiv= 0810.1851 |isbn= 978-3-642-03366-7 }} *{{cite journal |last1= Bern |first1= Marshall W. |last2= Graham |first2= Ronald L. | author2-link = Ronald Graham |year= 1989 |title= The shortest-network problem |journal= Scientific American |volume= 260 |pages= 84–89 |doi=10.1038/scientificamerican0189-84 |issue=1 |bibcode= 1989SciAm.260a..84B }} *{{cite conference |last1=Björklund | first1=Andreas |last2=Husfeldt | first2=Thore |last3=Kaski | first3=Petteri |last4=Koivisto | first4=Mikko |year=2007 |contribution=Fourier Meets Möbius: Fast Subset Convolution |title=[[Symposium on Theory of Computing|Proceedings of the 39th ACM Symposium on Theory of Computing]] |pages=67–74 |doi=10.1145/1250790.1250801 |arxiv=cs/0611101| isbn=978-1-59593-631-8 }} *{{cite conference|last1=Byrka|first1=J.|last2=Grandoni|first2=F.|last3=Rothvoß|first3=T.|last4=Sanita|first4=L.|year=2010|contribution=An improved LP-based approximation for Steiner tree|doi=10.1145/1806689.1806769|title = [[Symposium on Theory of Computing|Proceedings of the 42nd ACM Symposium on Theory of Computing]]|pages= 583–592 |isbn=978-1-4503-0050-6 |citeseerx=10.1.1.177.3565}} *{{cite journal|last1=Chlebík|first1=Miroslav|last2=Chlebíková|first2=Janka|author2-link=Janka Chlebíková|year=2008|title=The Steiner tree problem on graphs: Inapproximability results|doi=10.1016/j.tcs.2008.06.046|journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]]|volume=406|issue=3|pages=207–214|doi-access=free}} *{{cite conference | last1 = Chung | first1 = F. R. K. | author1-link = Fan Chung | last2 = Graham | first2 = R. L. | author2-link = Ronald Graham | contribution = A new bound for Euclidean Steiner minimal trees | doi = 10.1111/j.1749-6632.1985.tb14564.x | mr = 809217 | pages = 328–346 | publisher = New York Academy of Science | location = New York | series = Annals of the New York Academy of Science | title = Discrete geometry and convexity (New York, 1982) | volume = 440 | year = 1985 | bibcode = 1985NYASA.440..328C}} *{{cite book |first=Dietmar|last=Cieslik |title=Steiner Minimal Trees |year=1998 |isbn=0-7923-4983-0 |pages=319|publisher=Springer }} *{{Cite web | last1=Crescenzi | first1=Pierluigi | last2=Kann | first2=Viggo | last3=Halldórsson | first3=Magnús | last4=Karpinski | first4=Marek | authorlink4=Marek Karpinski | last5=Woeginger | first5=Gerhard | author5-link = Gerhard J. Woeginger | work=A Compendium of NP Optimization Problems | title=Minimum geometric Steiner tree | url=https://www.csc.kth.se/~viggo/wwwcompendium/node79.html | year=2000 }} *{{cite journal |last1=Cygan | first1=Marek |last2=Dell | first2=Holger |last3=Lokshtanov | first3=Daniel |last4=Marx | first4=Dániel |last5=Nederlof | first5=Jesper |last6=Okamoto | first6=Yoshio |last7=Paturi | first7=Ramamohan |last8=Saurabh | first8=Saket |last9=Wahlström | first9=Magnus |title=On Problems as Hard as CNF-SAT |journal=ACM Transactions on Algorithms | year=2016 |volume=12 |number=3 |pages=41:1–41:24 |doi=10.1145/2925416| s2cid=7320634 | url=http://eprints.sztaki.hu/9047/ |arxiv=1112.2275 }} *{{cite journal |last1=Dom | first1=Michael |last2=Lokshtanov | first2=Daniel |last3=Saurabh | first3=Saket |title=Kernelization Lower Bounds Through Colors and IDs |journal=ACM Transactions on Algorithms |year=2014 |volume=11 |number=2 |pages=13:1–13:20 |doi=10.1145/2650261| s2cid=13570734 }} *{{cite journal |last1=Dreyfus | first1= S.E. |last2=Wagner | first2 = R.A. |year=1971 |title=The Steiner problem in graphs |journal=Networks |volume=1 |issue=3 |pages=195–207 |doi=10.1002/net.3230010302}} *{{cite conference |last1=Fomin | first1=Fedor V. |last2=Kaski | first2=Petteri |last3=Lokshtanov | first3=Daniel |last4=Panolan | first4=Fahad |last5=Saurabh | first5=Saket |year=2015 |title=[[International Colloquium on Automata, Languages and Programming|Automata, Languages, and Programming – 42nd International Colloquium, ICALP 2015, Proceedings, Part I]] | series=Lecture Notes in Computer Science | volume=9134 |contribution=Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree |pages=494–505 |doi=10.1007/978-3-662-47672-7_40|hdl=1956/23311 | isbn=978-3-662-47671-0 |hdl-access=free }} *{{cite journal |last1=Fuchs | first1=Benjamin |last2=Kern | first2=Walter |last3=Mölle | first3=Daniel |last4=Richter | first4=Stefan |last5=Rossmanith | first5=Peter |last6=Wang | first6=Xinhui |title=Dynamic Programming for Minimum Steiner Trees |journal=Theory of Computing Systems |year=2007 |volume=41 |issue=3 |pages=493–500 |doi=10.1007/s00224-007-1324-4| s2cid=7478978 | url=http://e-archive.informatik.uni-koeln.de/492/2/zaik2005-492.pdf }} *{{Cite book | first=Joseph L. | last=Ganley | url=https://xlinux.nist.gov/dads/HTML/steinerratio.html | contribution=Steiner ratio | title=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | year=2004 | access-date=2012-05-24 }} * {{Garey-Johnson}}, {{Page numbers|208–209}}, problems ND12 and ND13. *{{cite journal | last = Hwang | first = F. 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S.|last2=Richards|first3=P.|last3=Winter |title=The Steiner Tree Problem |publisher=[[Elsevier]] |place=[[North-Holland Publishing Company|North-Holland]] |year=1992 |isbn=0-444-89098-X |series=Annals of Discrete Mathematics |volume=53}} * {{cite book |last1= Ivanov |first1= Alexander |last2= Tuzhilin |first2= Alexey |title=Minimal Networks: The Steiner Problem and Its Generalizations |url= https://archive.org/details/minimalnetworkss0000ivan |url-access= registration |publisher=[[CRC Press]] |place=N.W., Boca Raton, Florida |year=1994 |isbn=978-0-8493-8642-8}} * {{cite book |last1= Ivanov |first1= Alexander |last2= Tuzhilin |first2= Alexey |title=Branching solutions to one-dimensional variational problems |publisher=[[World Scientific]] |place=Singapore-New Jersey-London-Hong Kong |year=2000 |isbn=978-981-02-4060-8}} * {{cite book |last1= Ivanov |first1= Alexander |last2= Tuzhilin |first2= Alexey |title=Extreme Networks Theory |place=Moscow-Izhevsk |publisher=Institute of Computer Investigations |year=2003 |language=Russian |isbn=5-93972-292-X}} *{{cite journal |last1= Ivanov |first1= Alexander |last2= Tuzhilin |first2= Alexey |title=The Steiner ratio Gilbert–Pollak conjecture is still open: Clarification statement|journal=Algorithmica|year=2012|volume=62|issue=1–2|pages=630–632|doi=10.1007/s00453-011-9508-3|s2cid= 7486839 }} *{{cite journal |last1= Ivanov |first1= Alexander |last2= Tuzhilin |first2= Alexey |year= 2015 |title= Branched coverings and Steiner ratio |journal= International Transactions in Operational Research |volume= 23 |issue= 5 |pages= 875–882 |doi=10.1111/itor.12182 |arxiv=1412.5433|s2cid= 3386263 }} *{{cite journal|last1=Juhl|first1=D.|last2=Warme|first2=D.M.|last3=Winter |first3=P.|last4=Zachariasen|first4=M.|title=The GeoSteiner Software Package for computing Steiner trees in the plane: an updated computational study|journal= Mathematical Programming Computation|date=January 2018 |volume=10 |issue=4 |pages=487–532|doi=10.1007/s12532-018-0135-8|s2cid=255616114 |url=https://curis.ku.dk/portal/da/publications/the-geosteiner-software-package-for-computing-steiner-trees-in-the-plane(33ef119c-99c9-4511-9455-296ff28ee2eb).html }} *{{cite journal|last1=Rehfeldt|first1=D.|last2=Koch|first2=T.|title=Implications, conflicts, and reductions for Steiner trees|journal= Mathematical Programming|date=February 2023 |volume=197|issue=2|pages=903–966|doi=10.1007/s10107-021-01757-5|s2cid=231842568 |doi-access=free}} * {{cite conference | first1=Marek | last1=Karpinski | first2=Alexander | last2=Zelikovsky | contribution=Approximating dense cases of covering problems | pages=169–178 | year=1998 | contribution-url=https://books.google.com/books?id=IMmuF0RZk1MC&pg=PA169 | title=Proceedings of the DIMACS Workshop on Network Design: Connectivity and Facilities Location | publisher=American Mathematical Society | series=DIMACS Series in Discrete Mathematics and Theoretical Computer Science | volume=40}} *{{cite book | last1=Korte | first1=Bernhard | author1-link = Bernhard Korte | last2=Vygen | first2=Jens | title=Combinatorial Optimization: Theory and Algorithms | publisher=[[Springer Science+Business Media|Springer]] | year=2006 | edition=3rd | isbn=3-540-25684-9 | chapter=Section 20.1 }} *{{cite journal |last1=Kou |first1=L. |last2=Markowsky |first2=G. |last3=Berman |first3=L. |title=A fast algorithm for Steiner trees |journal=Acta Informatica |date=1 June 1981 |volume=15 |issue=2 |pages=141–145 |doi=10.1007/BF00288961|s2cid=21057232 }} *{{cite journal |last1=Levin | first1=A. Yu. |year=1971 |title=Algorithm for the shortest connection of a group of graph vertices |journal=Soviet Mathematics Doklady |volume=12 |pages=1477–1481}} *{{cite conference |last1 = Lokshtanov | first1=Daniel |last2 = Nederlof | first2=Jesper |contribution = Saving space by algebraization |title=[[Symposium on Theory of Computing|Proceedings of the 42nd ACM Symposium on Theory of Computing]] |year=2010 |pages=321–330 |doi=10.1145/1806689.1806735| isbn=978-1-4503-0050-6 }} *{{cite journal|last1=Paolini|first1=E. |last2=Stepanov|first2=E.|title=Existence and regularity results for the Steiner problem|journal=Calc. Var. Partial Diff. Equations|year=2012|volume=46 |issue=3–4 |pages=837–860 |doi=10.1007/s00526-012-0505-4|hdl=2158/600141 |s2cid=55793499 |url=https://flore.unifi.it/bitstream/2158/600141/1/final-for-CalcVar.pdf|hdl-access=free}} *{{cite conference | last1 = Robins | first1 = Gabriel | last2 = Zelikovsky | first2 = Alexander | contribution = Improved Steiner Tree Approximation in Graphs | contribution-url = http://dl.acm.org/citation.cfm?id=338219.338638 | isbn = 0-89871-453-2 | location = Philadelphia, PA, USA | pages = [https://archive.org/details/proceedingsofele0000acms/page/770 770–779] | publisher = Society for Industrial and Applied Mathematics | title = Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00) | year = 2000 | url = https://archive.org/details/proceedingsofele0000acms/page/770 }} *{{cite book|title=Algorithms for VLSI Physical Design Automation|first=Naveed A.|last=Sherwani | publisher = Kluwer Academic Publishers | year = 1993 | isbn = 9781475722192 }} *{{cite book |editor1-first=Ding-Zhu|editor1-last=Du|editor2-first=Frank|editor2-last=Hwang |title=Computing in Euclidean geometry |publisher=World Scientific Publishing Co |location=River Edge, NJ |year=1995 |isbn=981-02-1876-1 |series=Lecture Notes Series on Computing |volume=4 | edition=2nd|contribution=Computational geometry and topological network design|first1=J. M.|last1=Smith|first2=P.|last2=Winter|pages=351–451}} *{{cite journal |last1=Takahashi |first1=Hiromitsu |last2=Matsuyama |first2=Akira |title=An approximate solution for the Steiner problem in graphs |journal=Math. Japonica |date=1980 |volume=24 |issue=6 |pages=573–577}} *{{cite book | last = Vazirani | first = Vijay V. | authorlink = Vijay Vazirani | title = Approximation Algorithms | publisher = Springer | year = 2003 | place = Berlin | isbn = 3-540-65367-8 }} *{{cite book | last1=Wu | first1=Bang Ye | last2=Chao | first2=Kun-Mao | title=Spanning Trees and Optimization Problems | publisher=Chapman & Hall/CRC | year=2004 | isbn=1-58488-436-3 | chapter=Chapter 7 }} *{{cite journal |last1=Wu |first1=Y. F. |last2=Widmayer |first2=P. |last3=Wong |first3=C. K. |title=A faster approximation algorithm for the Steiner problem in graphs |journal=Acta Informatica |date=May 1986 |volume=23 |issue=2 |pages=223–229 |doi=10.1007/bf00289500 |s2cid=7772232 }} {{refend}}
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