Document Zbl 1499.68364

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ICALP 2018, Prague, Czech Republic, July 9–13, 2018. Proceedings. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 107, Article 48, 13 p. (2018). </div> <div class="abstract">Summary: Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than $k$ different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph $G$ whose vertices are colored by color sets, two designated vertices $s,t\in V(G)$, and $k\in\mathbb{N}$, is there an $s$-$t$ path in $G$ that uses at most $k$ colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.<br class="zbmathjax-paragraph">We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by $k$. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless $\mathrm{W[2]}= \mathrm{FPT}$, the problem cannot be approximated in FPT time to within a factor that is a function of $k$. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem.<br class="zbmathjax-paragraph">We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex $v$, there exists a set of paths whose cardinality is upper bounded by a function of $k$, that “represents” the valid $s$-$t$ paths containing subsets of colors from $v$. We employ these structural results to design an FPT algorithm for the problem parameterized by both $k$ and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both $k$ and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1392.68012">Zbl 1392.68012</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7375975">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%3A68U05" title="MSC2020">68U05</a> </td> <td class="space"> Computer graphics; computational geometry (digital and algorithmic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q17" title="MSC2020">68Q17</a> </td> <td class="space"> Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q27" title="MSC2020">68Q27</a> </td> <td class="space"> Parameterized complexity, tractability and kernelization </td> </tr><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%3A68W05" title="MSC2020">68W05</a> </td> <td class="space"> Nonnumerical algorithms </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aparameterized+complexity">parameterized complexity</a>; <a href="/?q=ut%3Amotion+planning">motion planning</a>; <a href="/?q=ut%3Abarrier+coverage">barrier coverage</a>; <a href="/?q=ut%3Abarrier+resilience">barrier resilience</a>; <a href="/?q=ut%3Acolored+path">colored path</a>; <a href="/?q=ut%3Aminimum+constraint+removal">minimum constraint removal</a>; <a href="/?q=ut%3Aplanar+graph">planar graph</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 1499.68364" data-ciurl="/ci/07375975" data-biburl="/bibtex/07375975.bib" data-amsurl="/amsrefs/07375975.bib" data-xmlurl="/xml/07375975.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07375975.pdf" title="Zbl 1499.68364 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/10.4230/LIPIcs.ICALP.2018.48" aria-label="DOI for “How to navigate through obstacles?”" title="10.4230/LIPIcs.ICALP.2018.48">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1712.04043"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">H. Alt, S. Cabello, P. Giannopoulos, and C. Knauer. Minimum cell connection in line segment arrangements. {\it International Journal of Computational Geometry and Applications}, 27(3):159-176, 2017. · <a href="/1423.68532" class="nowrap">Zbl 1423.68532</a></td> </tr><tr> <td>[2]</td> <td class="space">S. Bereg and D. Kirkpatrick. Approximating barrier resilience in wireless sensor networks. In {\it Proceedings of ALGOSENSORS}, pages 29-40, 2009.</td> </tr><tr> <td>[3]</td> <td class="space">R. Carr, S. Doddi, G. Konjevod, and M. Marathe. On the red-blue set cover problem. In {\it Proceedings of SODA}, pages 345-353, 2000. · <a href="/0952.90026" class="nowrap">Zbl 0952.90026</a></td> </tr><tr> <td>[4]</td> <td class="space">D. Chan and D. Kirkpatrick.Multi-path algorithms for minimum-colour path problems with applications to approximating barrier resilience. {\it Theoretical Computer Science}, 553:74-90, 2014. · <a href="/1417.68283" class="nowrap">Zbl 1417.68283</a></td> </tr><tr> <td>[5]</td> <td class="space">R. Diestel. {\it Graph Theory, 4th Edition}. Springer, 2012.</td> </tr><tr> <td>[6]</td> <td class="space">R. Downey and M. Fellows. {\it Fundamentals of Parameterized Complexity}. Texts in Computer Science. Springer, Berlin, Heidelberg, 2013. · <a href="/1358.68006" class="nowrap">Zbl 1358.68006</a></td> </tr><tr> <td>[7]</td> <td class="space">E. Eiben, J. Gemmell, I. Kanj, and A. Youngdahl. Improved results for minimum constraint removal. In {\it Proceedings of AAAI}. AAAI Press, 2018.</td> </tr><tr> <td>[8]</td> <td class="space">E. Eiben and I. Kanj. How to navigate through obstacles? {\it CoRR}, abs/1712.04043v1, 2017. arXiv:1712.04043.</td> </tr><tr> <td>[9]</td> <td class="space">L. Erickson and S. LaValle. A simple, but NP-hard, motion planning problem. In {\it Proceed-} {\it ings of AAAI}. AAAI Press, 2013.</td> </tr><tr> <td>[10]</td> <td class="space">A. Gorbenko and V. Popov. The discrete minimum constraint removal motion planning problem. In {\it Proceedings of the American Institute of Physics}, volume 1648. AIP Press, 2015.</td> </tr><tr> <td>[11]</td> <td class="space">K. Hauser. The minimum constraint removal problem with three robotics applications. {\it International Journal of Robotics Research}, 33(1):5-17, 2014.</td> </tr><tr> <td>[12]</td> <td class="space">M. Korman, M. Löffler, R. Silveira, and D. Strash. On the complexity of barrier resilience for fat regions and bounded ply. {\it Computational Geometry}, 72:34-51, 2018. · <a href="/1443.68204" class="nowrap">Zbl 1443.68204</a></td> </tr><tr> <td>[13]</td> <td class="space">S. Kumar, T. Lai, and A. Arora. Barrier coverage with wireless sensors. In {\it Proceedings of} {\it MOBICOM}, pages 284-298. ACM, 2005.</td> </tr><tr> <td>[14]</td> <td class="space">N. Robertson and P. Seymour. Graph minors. III. planar tree-width. {\it Journal of Combina-} {\it torial Theory, Series B}, 36(1):49-64, 1984. · <a href="/0548.05025" class="nowrap">Zbl 0548.05025</a></td> </tr><tr> <td>[15]</td> <td class="space">K. Tseng and D. Kirkpatrick. On barrier resilience of sensor networks. In {\it Proceedings of} {\it ALGOSENSORS}, pages 130-144, 2012.</td> </tr><tr> <td>[16]</td> <td class="space">S. Yang.{\it Some Path Planning Algorithms in Computational Geometry and Air Traf-} {\it fic Management}.PhD thesis, University of New Yort at Stony Brook. Available at: https://dspace.sunyconnect.suny.edu/handle/1951/59927, 2012.</td> </tr><tr> <td>[17]</td> <td class="space">S. Yuan, S. Varma, and J. Jue.Minimum-color path problems for reliability in mesh networks. In {\it Proceedings of INFOCOM}, pages 2658-2669, 2005.</td> </tr><tr> <td>[18]</td> <td class="space">:13</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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