{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article12156","name":"Connecting the Dots (with Minimum Crossings)","abstract":"We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L'subseteq L such that the graph G=(P,L') is isomorphic to a graph in F and L' has at most k crossings. By G=(P,L'), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L'. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build\/draw\/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links\/roads (such as highways) may be cheaper to build and faster to traverse, and signals\/moving objects would collide\/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces.\r\nAs a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP.","keywords":["crossing minimization","parameterized complexity","FPT algorithm","polynomial kernel","W[1]-hardness"],"author":[{"@type":"Person","name":"Agrawal, Akanksha","givenName":"Akanksha","familyName":"Agrawal","email":"mailto:agrawal@post.bgu.ac.il","affiliation":"Ben-Gurion University, Beer-Sheva, Israel","funding":"The work was carried out when the author was employed at Hungarian Academy of Sciences, and was supported by ERC Consolidator Grant SYSTEMATICGRAPH (No. 46 725978)."},{"@type":"Person","name":"Gu\u015bpiel, Grzegorz","givenName":"Grzegorz","familyName":"Gu\u015bpiel","email":"mailto:guspiel@tcs.uj.edu.pl","affiliation":"Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, {Krak\\texorpdfstring\u00f3{o}w, Poland}","funding":"Partially supported by the MNiSW grant DI2013 000443."},{"@type":"Person","name":"Madathil, Jayakrishnan","givenName":"Jayakrishnan","familyName":"Madathil","email":"mailto:jayakrishnanm@imsc.res.in","affiliation":"The Institute of Mathematical Sciences, HBNI, Chennai, India"},{"@type":"Person","name":"Saurabh, Saket","givenName":"Saket","familyName":"Saurabh","email":"mailto:saket@imsc.res.in","affiliation":"The Institute of Mathematical Sciences, HBNI, Chennai, India","funding":"Supported by ERC Consolidator Grant LOPPRE (No. 819416)."},{"@type":"Person","name":"Zehavi, Meirav","givenName":"Meirav","familyName":"Zehavi","email":"mailto:meiravze@bgu.ac.il","affiliation":"Ben-Gurion University, Beer-Sheva, Israel","funding":"Supported by ISF grant no. 1176\/18."}],"position":7,"pageStart":"7:1","pageEnd":"7:17","dateCreated":"2019-06-11","datePublished":"2019-06-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agrawal, Akanksha","givenName":"Akanksha","familyName":"Agrawal","email":"mailto:agrawal@post.bgu.ac.il","affiliation":"Ben-Gurion University, Beer-Sheva, Israel","funding":"The work was carried out when the author was employed at Hungarian Academy of Sciences, and was supported by ERC Consolidator Grant SYSTEMATICGRAPH (No. 46 725978)."},{"@type":"Person","name":"Gu\u015bpiel, Grzegorz","givenName":"Grzegorz","familyName":"Gu\u015bpiel","email":"mailto:guspiel@tcs.uj.edu.pl","affiliation":"Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, {Krak\\texorpdfstring\u00f3{o}w, Poland}","funding":"Partially supported by the MNiSW grant DI2013 000443."},{"@type":"Person","name":"Madathil, Jayakrishnan","givenName":"Jayakrishnan","familyName":"Madathil","email":"mailto:jayakrishnanm@imsc.res.in","affiliation":"The Institute of Mathematical Sciences, HBNI, Chennai, India"},{"@type":"Person","name":"Saurabh, Saket","givenName":"Saket","familyName":"Saurabh","email":"mailto:saket@imsc.res.in","affiliation":"The Institute of Mathematical Sciences, HBNI, Chennai, India","funding":"Supported by ERC Consolidator Grant LOPPRE (No. 819416)."},{"@type":"Person","name":"Zehavi, Meirav","givenName":"Meirav","familyName":"Zehavi","email":"mailto:meiravze@bgu.ac.il","affiliation":"Ben-Gurion University, Beer-Sheva, Israel","funding":"Supported by ISF grant no. 1176\/18."}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2019.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/orderandgeometry2016.tcs.uj.edu.pl\/docs\/OG2016-ProblemBooklet.pdf","isPartOf":{"@type":"PublicationVolume","@id":"#volume6332","volumeNumber":129,"name":"35th International Symposium on Computational Geometry (SoCG 2019)","dateCreated":"2019-06-11","datePublished":"2019-06-11","editor":[{"@type":"Person","name":"Barequet, Gill","givenName":"Gill","familyName":"Barequet","email":"mailto:barequet@cs.technion.ac.il","affiliation":"Technion - Israel Inst. of Technology, Haifa, Israel"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang","email":"mailto:yusu@cse.ohio-state.edu","affiliation":"The Ohio State University, Ohio, USA"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article12156","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6332"}}}