1,952 research outputs found
A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs
The Capacitated Vehicle Routing problem is to find a minimum-cost set of
tours that collectively cover clients in a graph, such that each tour starts
and ends at a specified depot and is subject to a capacity bound on the number
of clients it can serve. In this paper, we present a polynomial-time
approximation scheme (PTAS) for instances in which the input graph is planar
and the capacity is bounded. Previously, only a quasipolynomial-time
approximation scheme was known for these instances. To obtain this result, we
show how to embed planar graphs into bounded-treewidth graphs while preserving,
in expectation, the client-to-client distances up to a small additive error
proportional to client distances to the depot
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
On Visibility Representations of Non-planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an
assignment of axis-aligned rectangles to vertices such that for every edge
there exists a horizontal or vertical line of sight between the rectangles
assigned to its endpoints. Testing whether a graph has an RVR is known to be
NP-hard. In this paper, we study the problem of finding an RVR under the
assumption that an embedding in the plane of the input graph is fixed and we
are looking for an RVR that reflects this embedding. We show that in this case
the problem can be solved in polynomial time for general embedded graphs and in
linear time for 1-plane graphs (i.e., embedded graphs having at most one
crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of
1-plane graphs. These forbidden configurations can be tested for in linear
time, and so in linear time we can test whether a 1-plane graph has an RVR and
either compute such a representation or report a negative witness. Finally, we
discuss some extensions of our study to the case when the embedding is not
fixed but the RVR can have at most one crossing per edge
Measured descent: A new embedding method for finite metrics
We devise a new embedding technique, which we call measured descent, based on
decomposing a metric space locally, at varying speeds, according to the density
of some probability measure. This provides a refined and unified framework for
the two primary methods of constructing Frechet embeddings for finite metrics,
due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space
(X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where
alpha_X is a geometric estimate on the decomposability of X. As an immediate
corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding,
where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this
result recovers Bourgain's theorem, but when the metric X is, in a sense,
``low-dimensional,'' improved bounds are achieved.
Our embeddings are volume-respecting for subsets of arbitrary size. One
consequence is the existence of (k, O(log n)) volume-respecting embeddings for
all 1 \leq k \leq n, which is the best possible, and answers positively a
question posed by U. Feige. Our techniques are also used to answer positively a
question of Y. Rabinovich, showing that any weighted n-point planar graph
embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the
dimension is optimal, and improves upon the previously known bound of O((log
n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric &
Functional Analysis. This version fixes a subtle error in Section 2.
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