23 research outputs found
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
We consider the problem of partitioning the set of vertices of a given unit
disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and
various constant factor approximations are known, with the current best ratio
of 3. Our main result is a {\em weakly robust} polynomial time approximation
scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a
clique partition or (ii) gives a certificate that the graph is not a UDG; for
the case (i) that it computes a clique partition, we show that it is guaranteed
to be within (1+\eps) ratio of the optimum if the input is UDG; however if
the input is not a UDG it either computes a clique partition as in case (i)
with no guarantee on the quality of the clique partition or detects that it is
not a UDG. Noting that recognition of UDG's is NP-hard even if we are given
edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be
transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS.
We consider a weighted version of the clique partition problem on vertex
weighted UDGs that generalizes the problem. We note some key distinctions with
the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet,
surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted
case where the graph is expressed, say, as an adjacency matrix. This improves
on the best known 8-approximation for the {\em unweighted} case for UDGs
expressed in standard form.Comment: 21 pages, 9 figure
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
A separator theorem for string graphs and its applications
A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with in edges can be separated into two parts of roughly equal size by the removal of O(m(3/4)root log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph K-t,K-t has at most c(t)n edges, where c(t) is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any epsilon > 0, there is an integer g(epsilon) such that every string graph with n vertices and girth at least g(epsilon) has at most (1 + epsilon)n edges. Furthermore, the number of such labelled graphs is at most (1 + epsilon)(n) T(n), where T(n) = n(n-2) is the number of labelled trees on n vertices
Weak epsilon-nets for points on a hypersphere
We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NP-hard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing the results to optimum solutions
Weak #epsilon#-nets for points on a hypersphere
Available from TIB Hannover: RR 1912(95-1-029) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman