27,762 research outputs found
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
Geo-Social Group Queries with Minimum Acquaintance Constraint
The prosperity of location-based social networking services enables
geo-social group queries for group-based activity planning and marketing. This
paper proposes a new family of geo-social group queries with minimum
acquaintance constraint (GSGQs), which are more appealing than existing
geo-social group queries in terms of producing a cohesive group that guarantees
the worst-case acquaintance level. GSGQs, also specified with various spatial
constraints, are more complex than conventional spatial queries; particularly,
those with a strict NN spatial constraint are proved to be NP-hard. For
efficient processing of general GSGQ queries on large location-based social
networks, we devise two social-aware index structures, namely SaR-tree and
SaR*-tree. The latter features a novel clustering technique that considers both
spatial and social factors. Based on SaR-tree and SaR*-tree, efficient
algorithms are developed to process various GSGQs. Extensive experiments on
real-world Gowalla and Dianping datasets show that our proposed methods
substantially outperform the baseline algorithms based on R-tree.Comment: This is the preprint version that is accepted by the Very Large Data
Bases Journa
Optimal Bounds for the -cut Problem
In the -cut problem, we want to find the smallest set of edges whose
deletion breaks a given (multi)graph into connected components. Algorithms
of Karger & Stein and Thorup showed how to find such a minimum -cut in time
approximately . The best lower bounds come from conjectures about
the solvability of the -clique problem, and show that solving -cut is
likely to require time . Recent results of Gupta, Lee & Li have
given special-purpose algorithms that solve the problem in time , and ones that have better performance for special classes of graphs
(e.g., for small integer weights).
In this work, we resolve the problem for general graphs, by showing that the
Contraction Algorithm of Karger outputs any fixed -cut of weight with probability , where
denotes the minimum -cut size. This also gives an extremal bound of
on the number of minimum -cuts and an algorithm to compute a
minimum -cut in similar runtime. Both are tight up to lower-order factors,
with the algorithmic lower bound assuming hardness of max-weight -clique.
The first main ingredient in our result is a fine-grained analysis of how the
graph shrinks -- and how the average degree evolves -- in the Karger process.
The second ingredient is an extremal bound on the number of cuts of size less
than , using the Sunflower lemma.Comment: Final version of arXiv:1911.09165 with new and more general proof
Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
In this work we consider triangulations of point sets in the Euclidean plane,
i.e., maximal straight-line crossing-free graphs on a finite set of points.
Given a triangulation of a point set, an edge flip is the operation of removing
one edge and adding another one, such that the resulting graph is again a
triangulation. Flips are a major way of locally transforming triangular meshes.
We show that, given a point set in the Euclidean plane and two
triangulations and of , it is an APX-hard problem to minimize
the number of edge flips to transform to .Comment: A previous version only showed NP-completeness of the corresponding
decision problem. The current version is the one of the accepted manuscrip
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