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Complete linear descriptions of small asymmetric traveling salesman polytopes

R. Euler, H. Le Verge
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Using a refined version of Chernikova's algorithm we determined a complete and irredundant linear description of small asymmetric traveling salesman polytopes.  ...  We present such a description for the monotone polytope on 5 nodes consisting of 7615 facet-defining inequalities, and we present 319 015 facet-defining inequalities which, together with 11 equations,  ...  By splitting up the cone induced by a given vertex and its adjacent vertices a complete description of the symmetric traveling salesman polytope on 8 nodes has been obtained in [6] hereby detecting 3  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(95)98954-y">doi:10.1016/0166-218x(95)98954-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/daj3iu574raozbuzm54xqvitdy">fatcat:daj3iu574raozbuzm54xqvitdy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190318105310/https://core.ac.uk/download/pdf/82748360.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/66/3d/663d6b1f33fd23e65194ce45492b20a6f085dbc7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(95)98954-y"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 1390 of Mathematical Reviews Vol. 58, Issue 2 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
problem, structure of the STSP polytope, lifting theo- rems, facets of the STSP polytope), 3: the asymmetric travelling salesman problem (ATSP) (assignment problem, optimum branch- ing problem, structure  ...  of the ATSP polytope, lifting theorems, facets of the ATSP polytope), 4: an example: solution of a 120- STSP.  ... 
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Page 1414 of Mathematical Reviews Vol. , Issue 99b [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
, for a facet of the monotone completion to define a facet of the original polytope itself.”  ...  For the important cases of the traveling salesman (TS) poly- tope in both its symmetric and asymmetric variants, and of the linear ordering polytope, we give sufficient conditions trivially easy to verify  ... 
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On the Matrix-Cut Rank of Polyhedra

William Cook, Sanjeeb Dash
<span title="">2001</span> <i title="Institute for Operations Research and the Management Sciences (INFORMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cizxujlrtbbxnixq7eue6moupy" style="color: black;">Mathematics of Operations Research</a> </i> &nbsp;
Thi.s second example is used to show ihai the standard linear programming relaxation of a A-city traveling salesman problem requires at least [k/^l iterations of the combined operaior; Lhis bound is best  ...  COOK AND SANJEEB DASH Lov^sz and Schrijver (1991) described a semidcfinile operator for generating strong valid inequalities for the 0-! vectors in a prescribed polyhedron.  ...  \ation of tbe traveling salesman problem requires at least [A78J iterations of the combined operator, where it is the number of cities.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/moor.26.1.19.10593">doi:10.1287/moor.26.1.19.10593</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/trm25uj6hva6vbolnu5ikch6pi">fatcat:trm25uj6hva6vbolnu5ikch6pi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20100202030702/http://www2.isye.gatech.edu:80/~wcook/papers/matrix.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ad/7c/ad7c12a7a5cc63cb46469d5ca16645b936991b33.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/moor.26.1.19.10593"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

The Symmetric Traveling Salesman Polytope: New Facets from the Graphical Relaxation

Denis Naddef, Giovanni Rinaldi
<span title="">2007</span> <i title="Institute for Operations Research and the Management Sciences (INFORMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cizxujlrtbbxnixq7eue6moupy" style="color: black;">Mathematics of Operations Research</a> </i> &nbsp;
The path, the wheelbarrow, and the bicycle inequalities have been shown by Cornuéjols, Fonlupt, and Naddef to be facetdefining for the graphical relaxation of STSP n , the polytope of the symmetric traveling  ...  In conclusion, we characterize a large family of facet-defining inequalities for STSP n that include, as special cases, most of the inequalities currently known to have this property as the comb, the clique  ...  This work was supported in part by the Marie Curie Research Training Network "Algorithmic Discrete Optimization" (ADONET), Grant MRTN-CT-2003-504438, of the European Union Research Commission, which we  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/moor.1060.0244">doi:10.1287/moor.1060.0244</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dmyqo4rqqrfm5n627cttiauldi">fatcat:dmyqo4rqqrfm5n627cttiauldi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808141059/http://pagesperso.g-scop.grenoble-inp.fr/~naddefd/papers/MathOR.32.2007.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f0/cc/f0cc7559ae4360bbd88d65c387375847a16a797a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1287/moor.1060.0244"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 4812 of Mathematical Reviews Vol. , Issue 2003f [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This procedure can also be used to derive new facets for the triangle-free subgraph polytopes.  ...  Summary: “Symmetric and antisymmetric distance matrices in the single agent traveling salesman problem are not the only distance matrices to generate elementary landscapes for ‘swap’ and *2-opt’ neighborhoods  ... 
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Selected topics on assignment problems

Rainer E. Burkard
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
We survey recent developments in the ÿelds of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment  ...  He considers the role of odd cycles for a class of facets of this polytope.  ...  It turns out that the traveling salesman polytope and the linear ordering polytope are projections of QAP n , and that QAP n is a face of the Boolean quadric polytope, see [91] . x C ij = 1 if (i; j)  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(01)00343-2">doi:10.1016/s0166-218x(01)00343-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xmwjqmt53zheldnspyjbali54y">fatcat:xmwjqmt53zheldnspyjbali54y</a> </span>
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On the Power of Symmetric Linear Programs [article]

Albert Atserias and Anuj Dawar and Joanna Ochremiak
<span title="2019-01-23">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral  ...  In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs  ...  The second author was partially supported by a Fellowship of the Alan Turing Institute under the EPSRC grant EP/N510129/1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1901.07825v1">arXiv:1901.07825v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/homhzmxau5ad3ecb37qnolljim">fatcat:homhzmxau5ad3ecb37qnolljim</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191017233236/https://arxiv.org/pdf/1901.07825v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/73/0a/730a9b06c3cf83de036a08ed9f9953fd5032fe08.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1901.07825v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Quadratic Assignment Problem [chapter]

Rainer E. Burkard, Eranda Çela, Panos M. Pardalos, Leonidas S. Pitsoulis
<span title="">1998</span> <i title="Springer US"> Handbook of Combinatorial Optimization </i> &nbsp;
, e.g. the traveling salesman problem, the graph partitioning problem, etc.  ...  This paper aims at describing the state of the art on quadratic assignment problems (QAPs).  ...  problems like the traveling salesman polytope or the cut polytope (see [102] ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-1-4613-0303-9_27">doi:10.1007/978-1-4613-0303-9_27</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mm3iphjj4bhv5htk2rtljohu2i">fatcat:mm3iphjj4bhv5htk2rtljohu2i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20081031194056/http://www.opt.math.tu-graz.ac.at/~cela/papers/qap_bericht.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/41/b4/41b41fb86021cf2f3b18743423864f7321ce88c9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-1-4613-0303-9_27"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The hypergraph assignment problem

Ralf Borndörfer, Olga Heismann
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/s6sw62lf6zcxrghfhvjqhycu5y" style="color: black;">Discrete Optimization</a> </i> &nbsp;
As test instances we used several LOP and TSP polytopes as well as polytopes describing the target visitation problem , a combination of the linear ordering problem and the traveling salesman problem,  ...  For given arc length, the well-known asymmetric traveling salesman problem (ATSP) consists of finding a Hamiltonian cycle in D n of minimum total length.  ...  Facets of the HAP Polytope for G 2,3 We list here one facet from each of the 30 facet classes of P(HAP) for G 2,3 . For an explanation, see Section 5.5.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disopt.2014.11.002">doi:10.1016/j.disopt.2014.11.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rxt6ot7rifdi7g5z62uvgcq4uq">fatcat:rxt6ot7rifdi7g5z62uvgcq4uq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190430074847/https://depositonce.tu-berlin.de/bitstream/11303/4380/1/heismann_olga.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/77/e5/77e511f1b50164fcc68c0354651f6f2ace7d71a4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disopt.2014.11.002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Subdirect decompositions of transformation graphs

M. Yoeli, C. M. Ablow
<span title="">1966</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rvwmmtrjoza55my35c5kvrivg4" style="color: black;">Annual Symposium on Switching and Automata Theory</a> </i> &nbsp;
Combinatorial Theory, Vol. 1, pp. 385-393, 1966. rris, "A Composition Theorem for the Enumeration of Certain Subsets of the Symmetric Semigroup," in Graph Theory and Its Applications, edited by B.  ...  "Families of Sets with Faithful Graph Representations," IBM Research Note, Vol. NC-184, 1962. " A Solvable Case of the Traveling Salesman Problem," Proc. Nut. Acad.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/swat.1966.25">doi:10.1109/swat.1966.25</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/focs/YoeliA66.html">dblp:conf/focs/YoeliA66</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ezldr3cxzzhuzcsvpd6mgcti6y">fatcat:ezldr3cxzzhuzcsvpd6mgcti6y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190304085057/http://pdfs.semanticscholar.org/d8e9/e2306d0fd56cf2c29b09cc4316726a3e1d35.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d8/e9/d8e9e2306d0fd56cf2c29b09cc4316726a3e1d35.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/swat.1966.25"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

The traveling salesman problem: An overview of exact and approximate algorithms

Gilbert Laporte
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2fqqu5s6wbgk5gcucpt3wqm3gy" style="color: black;">European Journal of Operational Research</a> </i> &nbsp;
In this paper, some of the main known algorithms for the traveling salesman problem are surveyed.  ...  The paper is organized as follows: 1) definition; 2) applications; 3) complexity analysis; 4) exact algorithms; 5) heuristic algorithms; 6) conclusion.  ...  Finally, thanks are due to Paolo Toth for his valuable comments on a preliminary version of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0377-2217(92)90138-y">doi:10.1016/0377-2217(92)90138-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3brxff5cfjerhg3t4cixrqgu3u">fatcat:3brxff5cfjerhg3t4cixrqgu3u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810105749/http://web.ist.utl.pt/~ist11038/CD_Casquilho/TSP1992EJOR_Laporte.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5e/b8/5eb81ff00095182f05ddc65665a038ce9fbc8cc4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0377-2217(92)90138-y"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

New semidefinite programming relaxations for the Linear Ordering and the Traveling Salesman Problem

Philipp Hungerländer
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
In this paper we strengthen the relaxation proposed by Newman [43] and conduct a polyhedral study of the corresponding polytope.  ...  Furthermore we relate the relaxation to other linear and semidefinite relaxations for the (LOP) and for the Traveling Salesman Problem and elaborate on its connection to the Max Cut problem.  ...  the relaxation strength of various linear and semidefinite relaxations on facets of the traveling salesman polytope for n = 8.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.07.013">doi:10.1016/j.dam.2016.07.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uq34rjzk7bfedmqkn4j2un7hca">fatcat:uq34rjzk7bfedmqkn4j2un7hca</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809131821/http://www.optimization-online.org/DB_FILE/2015/01/4743.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9d/d9/9dd902a3b43fe4549f0791b66d02aea0dd52b932.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.07.013"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

A semidefinite optimization approach to the Target Visitation Problem

P. Hungerländer
<span title="2014-11-11">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5dkx5l3pkjclfd46nyh4eftsrq" style="color: black;">Optimization Letters</a> </i> &nbsp;
We propose an exact algorithm for the Target Visitation Problem (TVP). The (TVP) is a composition of the Linear Ordering Problem and the Traveling Salesman Problem.  ...  It has several military and non-military applications, where two important, often competing factors are the overall distance traveled (e.g. by an unmanned aerial vehicle) and the visiting sequence of the  ...  It is a composition of the Linear Ordering Problem (LOP) and the Traveling Salesman Problem (TSP).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11590-014-0824-9">doi:10.1007/s11590-014-0824-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xntr7nhozjehpiymmmmbk7kx54">fatcat:xntr7nhozjehpiymmmmbk7kx54</a> </span>
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On lexicographic approximations of integer programs [article]

Michael Eldredge, Akshay Gupte
<span title="2017-10-28">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Since the construction of our bounds depends on the computation of lex optima, we derive explicit formulae for lex optima of some special polytopes, such as polytopes that are monotone with respect to  ...  The former result yields a structural characterization for the optimum of 0 1 programs, with connections to matroid optimization, and a heuristic for general integer programs.  ...  We prove the first statement for the cutset polytope X tsp of a traveling salesman problem defined on the complete graph K n .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1610.06470v3">arXiv:1610.06470v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jwqazyg3vnh3bhbfwjaj7tzngu">fatcat:jwqazyg3vnh3bhbfwjaj7tzngu</a> </span>
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