Document Zbl 1248.68376

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Fourth international computer science symposium in Russia, CSR 2009, Novosibirsk, Russia, August 18–23, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-03350-6/pbk). Lecture Notes in Computer Science 5675, 92-104 (2009). </div> <div class="abstract">Summary: A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. We study the approximability properties of the Weighted Maximum $H$-Colourable Subgraph problem (Max $H$-Col). The instances of this problem are edge-weighted graphs $G$ and the objective is to find a subgraph of $G$ that has maximal total edge weight, under the condition that the subgraph has a homomorphism to $H$; note that for $H = K _{k }$ this problem is equivalent to Max $k$-Cut. To this end, we introduce a metric structure on the space of graphs which allows us to extend previously known approximability results to larger classes of graphs. Specifically, the approximation algorithms for Max Cut by Goemans and Williamson and Max $k$-Cut by Frieze and Jerrum can be used to yield non-trivial approximation results for Max $H$-Col. For a variety of graphs, we show near-optimality results under the unique games conjecture. We also use our method for comparing the performance of Frieze & Jerrum’s algorithm with Håstad’s approximation algorithm for general Max 2-Csp. This comparison is, in most cases, favourable to Frieze & Jerrum.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1169.68003">Zbl 1169.68003</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5596326">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C15" title="MSC2020">05C15</a> </td> <td class="space"> Coloring of graphs and hypergraphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q25" title="MSC2020">68Q25</a> </td> <td class="space"> Analysis of algorithms and problem complexity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimisation">optimisation</a>; <a href="/?q=ut%3Aapproximability">approximability</a>; <a href="/?q=ut%3Agraph+homomorphism">graph homomorphism</a>; <a href="/?q=ut%3Agraph+%5C%28H%5C%29-colouring">graph $H$-colouring</a>; <a href="/?q=ut%3Acomputational+complexity">computational complexity</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 1248.68376" data-ciurl="/ci/05596326" data-biburl="/bibtex/05596326.bib" data-amsurl="/amsrefs/05596326.bib" data-xmlurl="/xml/05596326.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05596326.pdf" title="Zbl 1248.68376 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/10.1007/978-3-642-03351-3_11" aria-label="DOI for “Approximability distance in the space of $H$-colourability problems”" title="10.1007/978-3-642-03351-3_11">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">Alon, N.: Bipartite subgraph. Combinatorica 16, 301–311 (1996) · <a href="/0860.05043" class="nowrap">Zbl 0860.05043</a> · <a href="https://doi.org/10.1007/BF01261315" class="nowrap">doi:10.1007/BF01261315</a></td> </tr><tr> <td>[2]</td> <td class="space">Alon, N., Krivelevich, M.: The concentration of the chromatic number of random graphs. Combinatorica 17, 303–313 (1997) · <a href="/0910.05058" class="nowrap">Zbl 0910.05058</a> · <a href="https://doi.org/10.1007/BF01215914" class="nowrap">doi:10.1007/BF01215914</a></td> </tr><tr> <td>[3]</td> <td class="space">Berman, A., Zhang, X.-D.: Bipartite density of cubic graphs. Discrete Mathematics 260, 27–35 (2003) · <a href="/1012.05110" class="nowrap">Zbl 1012.05110</a> · <a href="https://doi.org/10.1016/S0012-365X(02)00490-9" class="nowrap">doi:10.1016/S0012-365X(02)00490-9</a></td> </tr><tr> <td>[4]</td> <td class="space">Bollobás, B.: The chromatic number of random graphs. Combinatorica 8(1), 49–55 (1988) · <a href="/0666.05033" class="nowrap">Zbl 0666.05033</a> · <a href="https://doi.org/10.1007/BF02122551" class="nowrap">doi:10.1007/BF02122551</a></td> </tr><tr> <td>[5]</td> <td class="space">Bollobás, B., Erdös, P.: Cliques in random graphs. Mathematical Proceedings of the Cambridge Philosophical Society 80(419), 419–427 (1976) · <a href="/0344.05155" class="nowrap">Zbl 0344.05155</a> · <a href="https://doi.org/10.1017/S0305004100053056" class="nowrap">doi:10.1017/S0305004100053056</a></td> </tr><tr> <td>[6]</td> <td class="space">Bondy, J., Locke, S.: Largest bipartite subgraphs in triangle-free graphs with maximum degree three. Journal of Graph Theory 10, 477–504 (1986) · <a href="/0609.05046" class="nowrap">Zbl 0609.05046</a> · <a href="https://doi.org/10.1002/jgt.3190100407" class="nowrap">doi:10.1002/jgt.3190100407</a></td> </tr><tr> <td>[7]</td> <td class="space">Borodin, O., Kim, S.-J., Kostochka, A., West, D.: Homomorphisms from sparse graphs with large girth. Journal of Combinatorial Theory, ser. B 90, 147–159 (2004) · <a href="/1033.05037" class="nowrap">Zbl 1033.05037</a> · <a href="https://doi.org/10.1016/S0095-8956(03)00081-9" class="nowrap">doi:10.1016/S0095-8956(03)00081-9</a></td> </tr><tr> <td>[8]</td> <td class="space">Coja-Oghlan, A., Moore, C., Sanwalani, V.: MAX k-CUT and approximating the chromatic number of random graphs. Random Structures and Algorithms 28, 289–322 (2005) · <a href="/1094.68118" class="nowrap">Zbl 1094.68118</a> · <a href="https://doi.org/10.1002/rsa.20096" class="nowrap">doi:10.1002/rsa.20096</a></td> </tr><tr> <td>[9]</td> <td class="space">de Klerk, E., Pasechnik, D., Warners, J.: Approximate graph colouring and MAX-k-CUT algorithms based on the {$\theta$} function. Journal of Combinatorial Optimization 8, 267–294 (2004) · <a href="/1084.68142" class="nowrap">Zbl 1084.68142</a> · <a href="https://doi.org/10.1023/B:JOCO.0000038911.67280.3f" class="nowrap">doi:10.1023/B:JOCO.0000038911.67280.3f</a></td> </tr><tr> <td>[10]</td> <td class="space">Dutton, R., Brigham, R.: Edges in graphs with large girth. Graphs and Combinatorics 7(4), 315–321 (1991) · <a href="/0755.05048" class="nowrap">Zbl 0755.05048</a> · <a href="https://doi.org/10.1007/BF01787638" class="nowrap">doi:10.1007/BF01787638</a></td> </tr><tr> <td>[11]</td> <td class="space">Dvořák, Z., Škrekovski, R., Valla, T.: Planar graphs of odd-girth at least 9 are homomorphic to the Petersen graph. To appear in SIAM Journal on Discrete Mathematics · <a href="/1167.05313" class="nowrap">Zbl 1167.05313</a></td> </tr><tr> <td>[12]</td> <td class="space">Engström, R.: Approximability distances between circular complete graphs. Master’s thesis, Linköping University. LITH-IDA-EX-A–08/062–SE (2008)</td> </tr><tr> <td>[13]</td> <td class="space">Erdös, P.: Graph theory and probability. Canadian Journal of Mathematics 11, 34–38 (1959) · <a href="/0084.39602" class="nowrap">Zbl 0084.39602</a> · <a href="https://doi.org/10.4153/CJM-1959-003-9" class="nowrap">doi:10.4153/CJM-1959-003-9</a></td> </tr><tr> <td>[14]</td> <td class="space">Erdös, P., Rényi, A.: On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 5, 17–61 (1960) · <a href="/0103.16301" class="nowrap">Zbl 0103.16301</a></td> </tr><tr> <td>[15]</td> <td class="space">Frieze, A., Jerrum, M.: Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18(1), 67–81 (1997) · <a href="/0873.68078" class="nowrap">Zbl 0873.68078</a> · <a href="https://doi.org/10.1007/BF02523688" class="nowrap">doi:10.1007/BF02523688</a></td> </tr><tr> <td>[16]</td> <td class="space">Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995) · <a href="/0885.68088" class="nowrap">Zbl 0885.68088</a> · <a href="https://doi.org/10.1145/227683.227684" class="nowrap">doi:10.1145/227683.227684</a></td> </tr><tr> <td>[17]</td> <td class="space">Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2004) · <a href="https://doi.org/10.1093/acprof:oso/9780198528173.001.0001" class="nowrap">doi:10.1093/acprof:oso/9780198528173.001.0001</a></td> </tr><tr> <td>[18]</td> <td class="space">Hophkins, G., Staton, W.: Extremal bipartite subgraphs of cubic triangle-free graphs. Journal of Graph Theory 6, 115–121 (1982) · <a href="/0486.05036" class="nowrap">Zbl 0486.05036</a> · <a href="https://doi.org/10.1002/jgt.3190060205" class="nowrap">doi:10.1002/jgt.3190060205</a></td> </tr><tr> <td>[19]</td> <td class="space">Håstad, J.: Every 2-CSP allows nontrivial approximation. In: Proceedings of the 37th Annual ACM Symposium on the Theory of Computing (STOC 2005), pp. 740–746 (2005) · <a href="/1192.90135" class="nowrap">Zbl 1192.90135</a> · <a href="https://doi.org/10.1145/1060590.1060700" class="nowrap">doi:10.1145/1060590.1060700</a></td> </tr><tr> <td>[20]</td> <td class="space">Jonsson, P., Krokhin, A., Kuivinen, F.: Ruling out polynomial-time approximation schemes for hard constraint satisfaction problems. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 182–193. Springer, Heidelberg (2007), http://www.dur.ac.uk/andrei.krokhin/papers/hardgap.pdf · <a href="/1188.68153" class="nowrap">Zbl 1188.68153</a> · <a href="https://doi.org/10.1007/978-3-540-74510-5_20" class="nowrap">doi:10.1007/978-3-540-74510-5_20</a></td> </tr><tr> <td>[21]</td> <td class="space">Kann, V., Khanna, S., Lagergren, J., Panconesi, A.: On the hardness of approximating MAX k-CUT and its dual. Chicago Journal of Theoretical Computer Science 1997(2) (1997) · <a href="/0924.68013" class="nowrap">Zbl 0924.68013</a></td> </tr><tr> <td>[22]</td> <td class="space">Kaporis, A., Kirousis, L., Stavropoulos, E.: Approximating almost all instances of Max-cut within a ratio above the Håstad threshold. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 432–443. Springer, Heidelberg (2006) · <a href="/1131.05315" class="nowrap">Zbl 1131.05315</a> · <a href="https://doi.org/10.1007/11841036_40" class="nowrap">doi:10.1007/11841036_40</a></td> </tr><tr> <td>[23]</td> <td class="space">Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th Annual ACM Symposium on the Theory of Computing (STOC 2002), pp. 767–775 (2002) · <a href="/1192.68367" class="nowrap">Zbl 1192.68367</a></td> </tr><tr> <td>[24]</td> <td class="space">Khot, S., Kindler, G., Mossel, E., O’Donnel, R.: Optimal inapproximability results for MAX-CUT and other two-variable CSPs? SIAM Journal of Computing 37(1), 319–357 (2007) · <a href="/1135.68019" class="nowrap">Zbl 1135.68019</a> · <a href="https://doi.org/10.1137/S0097539705447372" class="nowrap">doi:10.1137/S0097539705447372</a></td> </tr><tr> <td>[25]</td> <td class="space">Lai, H.-J., Liu, B.: Graph homomorphism into an odd cycle. Bulletin of the Institute of Combinatorics and its Applications 28, 19–24 (2000) · <a href="/0943.05050" class="nowrap">Zbl 0943.05050</a></td> </tr><tr> <td>[26]</td> <td class="space">Langberg, M., Rabani, Y., Swamy, C.: Approximation algorithms for graph homomorphism problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 176–187. Springer, Heidelberg (2006) · <a href="/1155.68581" class="nowrap">Zbl 1155.68581</a> · <a href="https://doi.org/10.1007/11830924_18" class="nowrap">doi:10.1007/11830924_18</a></td> </tr><tr> <td>[27]</td> <td class="space">Locke, S.: A note on bipartite subgraphs of triangle-free regular graphs. Journal of Graph Theory 14, 181–185 (1990) · <a href="/0731.05025" class="nowrap">Zbl 0731.05025</a> · <a href="https://doi.org/10.1002/jgt.3190140206" class="nowrap">doi:10.1002/jgt.3190140206</a></td> </tr><tr> <td>[28]</td> <td class="space">Łuczak, T.: The chromatic number of random graphs. Combinatorica 11(1), 45–54 (1991) · <a href="/0771.05090" class="nowrap">Zbl 0771.05090</a> · <a href="https://doi.org/10.1007/BF01375472" class="nowrap">doi:10.1007/BF01375472</a></td> </tr><tr> <td>[29]</td> <td class="space">Matula, D.: The employee party problem. Notices of the American Mathematical Society 19, A – 382 (1972)</td> </tr><tr> <td>[30]</td> <td class="space">Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th annual ACM symposium on the Theory of Computing (STOC 2008), pp. 245–254 (2008) · <a href="/1231.68142" class="nowrap">Zbl 1231.68142</a> · <a href="https://doi.org/10.1145/1374376.1374414" class="nowrap">doi:10.1145/1374376.1374414</a></td> </tr><tr> <td>[31]</td> <td class="space">Raghavendra, P., Steurer, D.: How to round any CSP (2009) (manuscript) · <a href="/1292.90231" class="nowrap">Zbl 1292.90231</a></td> </tr><tr> <td>[32]</td> <td class="space">Šámal, R.: Fractional covering by cuts. In: Proceedings of the 7th International Colloquium on Graph Theory (ICGT 2005), pp. 455–459 (2005) · <a href="/1200.05165" class="nowrap">Zbl 1200.05165</a> · <a href="https://doi.org/10.1016/j.endm.2005.06.099" class="nowrap">doi:10.1016/j.endm.2005.06.099</a></td> </tr><tr> <td>[33]</td> <td class="space">Šámal, R.: On XY mappings. PhD thesis, Charles University in Prague (2006)</td> </tr><tr> <td>[34]</td> <td class="space">Trevisan, L.: Max Cut and the smallest eigenvalue. CoRR, abs/0806.1978 (2008)</td> </tr><tr> <td>[35]</td> <td class="space">Turán, P.: On an extremal problem in graph theory. Matematicko Fizicki Lapok 48, 436–452 (1941)</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. 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