7,772 research outputs found
Minimum multicuts and Steiner forests for Okamura-Seymour graphs
We study the problem of finding minimum multicuts for an undirected planar
graph, where all the terminal vertices are on the boundary of the outer face.
This is known as an Okamura-Seymour instance. We show that for such an
instance, the minimum multicut problem can be reduced to the minimum-cost
Steiner forest problem on a suitably defined dual graph. The minimum-cost
Steiner forest problem has a 2-approximation algorithm. Hence, the minimum
multicut problem has a 2-approximation algorithm for an Okamura-Seymour
instance.Comment: 6 pages, 1 figur
Approximations for the Steiner Multicycle Problem
The Steiner Multicycle problem consists of, given a complete graph, a weight
function on its vertices, and a collection of pairwise disjoint non-unitary
sets called terminal sets, finding a minimum weight collection of
vertex-disjoint cycles in the graph such that, for every terminal set, all of
its vertices are in a same cycle of the collection. This problem generalizes
the Traveling Salesman problem and therefore is hard to approximate in general.
On the practical side, it models a collaborative less-than-truckload problem
with pickup and delivery locations. Using an algorithm for the Survivable
Network Design problem and T -joins, we obtain a 3-approximation for the metric
case, improving on the previous best 4-approximation. Furthermore, we present
an (11/9)-approximation for the particular case of the Steiner Multicycle in
which each edge weight is 1 or 2. This algorithm can be adapted to obtain a
(7/6)-approximation when every terminal set contains at least 4 vertices.
Finally, we devise an O(lg n)-approximation algorithm for the asymmetric
version of the problem
Steiner connectivity problems in hypergraphs
We say that a tree is an -Steiner tree if and a
hypergraph is an -Steiner hypertree if it can be trimmed to an -Steiner
tree. We prove that it is NP-hard to decide, given a hypergraph
and some , whether there is a subhypergraph of
which is an -Steiner hypertree. As corollaries, we give two
negative results for two Steiner orientation problems in hypergraphs. Firstly,
we show that it is NP-hard to decide, given a hypergraph , some and some , whether this
hypergraph has an orientation in which every vertex of is reachable from
. Secondly, we show that it is NP-hard to decide, given a hypergraph
and some , whether this hypergraph
has an orientation in which any two vertices in are mutually reachable from
each other. This answers a longstanding open question of the Egerv\'ary
Research group. On the positive side, we show that the problem of finding a
Steiner hypertree and the first orientation problem can be solved in polynomial
time if the number of terminals is fixed
The Minimum Wiener Connector
The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph and a set
of query vertices, find a subgraph of that connects all
query vertices and has minimum Wiener index.
We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless . Our main contribution is a
constant-factor approximation algorithm running in time
.
A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set a small number of important vertices
(i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International
Conference on Management of Dat
Minimum Convex Partitions and Maximum Empty Polytopes
Let be a set of points in . A Steiner convex partition
is a tiling of with empty convex bodies. For every integer ,
we show that admits a Steiner convex partition with at most tiles. This bound is the best possible for points in general
position in the plane, and it is best possible apart from constant factors in
every fixed dimension . We also give the first constant-factor
approximation algorithm for computing a minimum Steiner convex partition of a
planar point set in general position. Establishing a tight lower bound for the
maximum volume of a tile in a Steiner convex partition of any points in the
unit cube is equivalent to a famous problem of Danzer and Rogers. It is
conjectured that the volume of the largest tile is .
Here we give a -approximation algorithm for computing the
maximum volume of an empty convex body amidst given points in the
-dimensional unit box .Comment: 16 pages, 4 figures; revised write-up with some running times
improve
Minimum feature size preserving decompositions
The minimum feature size of a crossing-free straight line drawing is the
minimum distance between a vertex and a non-incident edge. This quantity
measures the resolution needed to display a figure or the tool size needed to
mill the figure. The spread is the ratio of the diameter to the minimum feature
size. While many algorithms (particularly in meshing) depend on the spread of
the input, none explicitly consider finding a mesh whose spread is similar to
the input. When a polygon is partitioned into smaller regions, such as
triangles or quadrangles, the degradation is the ratio of original to final
spread (the final spread is always greater).
Here we present an algorithm to quadrangulate a simple n-gon, while achieving
constant degradation. Note that although all faces have a quadrangular shape,
the number of edges bounding each face may be larger. This method uses Theta(n)
Steiner points and produces Theta(n) quadrangles. In fact to obtain constant
degradation, Omega(n) Steiner points are required by any algorithm.
We also show that, for some polygons, a constant factor cannot be achieved by
any triangulation, even with an unbounded number of Steiner points. The
specific lower bounds depend on whether Steiner vertices are used or not.Comment: 12 pages, 4 figure
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