3,910 research outputs found
Greedy Algorithms for Steiner Forest
In the Steiner Forest problem, we are given terminal pairs ,
and need to find the cheapest subgraph which connects each of the terminal
pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson
gave primal-dual constant-factor approximation algorithms for this problem;
until now, the only constant-factor approximations we know are via linear
programming relaxations.
We consider the following greedy algorithm: Given terminal pairs in a metric
space, call a terminal "active" if its distance to its partner is non-zero.
Pick the two closest active terminals (say ), set the distance
between them to zero, and buy a path connecting them. Recompute the metric, and
repeat. Our main result is that this algorithm is a constant-factor
approximation.
We also use this algorithm to give new, simpler constructions of cost-sharing
schemes for Steiner forest. In particular, the first "group-strict" cost-shares
for this problem implies a very simple combinatorial sampling-based algorithm
for stochastic Steiner forest
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
Travelling on Graphs with Small Highway Dimension
We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP)
in graphs of low highway dimension. This graph parameter was introduced by
Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP
and STP naturally occur for various applications in logistics. It was
previously shown [Feldmann et al. ICALP 2015] that these problems admit a
quasi-polynomial time approximation scheme (QPTAS) on graphs of constant
highway dimension. We demonstrate that a significant improvement is possible in
the special case when the highway dimension is 1, for which we present a
fully-polynomial time approximation scheme (FPTAS). We also prove that STP is
weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for
graphs of highway dimension 6, which answers an open problem posed in [Feldmann
et al. ICALP 2015]
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Online Steiner Tree with Deletions
In the online Steiner tree problem, the input is a set of vertices that
appear one-by-one, and we have to maintain a Steiner tree on the current set of
vertices. The cost of the tree is the total length of edges in the tree, and we
want this cost to be close to the cost of the optimal Steiner tree at all
points in time. If we are allowed to only add edges, a tight bound of
on the competitiveness is known. Recently it was shown that if
we can add one new edge and make one edge swap upon every vertex arrival, we
can maintain a constant-competitive tree online.
But what if the set of vertices sees both additions and deletions? Again, we
would like to obtain a low-cost Steiner tree with as few edge changes as
possible. The original paper of Imase and Waxman had also considered this
model, and it gave a greedy algorithm that maintained a constant-competitive
tree online, and made at most edge changes for the first
requests. In this paper give the following two results.
Our first result is an online algorithm that maintains a Steiner tree only
under deletions: we start off with a set of vertices, and at each time one of
the vertices is removed from this set: our Steiner tree no longer has to span
this vertex. We give an algorithm that changes only a constant number of edges
upon each request, and maintains a constant-competitive tree at all times. Our
algorithm uses the primal-dual framework and a global charging argument to
carefully make these constant number of changes.
We then study the natural greedy algorithm proposed by Imase and Waxman that
maintains a constant-competitive Steiner tree in the fully-dynamic model (where
each request either adds or deletes a vertex). Our second result shows that
this algorithm makes only a constant number of changes per request in an
amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc
Approximation Algorithms for Generalized MST and TSP in Grid Clusters
We consider a special case of the generalized minimum spanning tree problem
(GMST) and the generalized travelling salesman problem (GTSP) where we are
given a set of points inside the integer grid (in Euclidean plane) where each
grid cell is . In the MST version of the problem, the goal is to
find a minimum tree that contains exactly one point from each non-empty grid
cell (cluster). Similarly, in the TSP version of the problem, the goal is to
find a minimum weight cycle containing one point from each non-empty grid cell.
We give a and -approximation
algorithm for these two problems in the described setting, respectively.
Our motivation is based on the problem posed in [7] for a constant
approximation algorithm. The authors designed a PTAS for the more special case
of the GMST where non-empty cells are connected end dense enough. However,
their algorithm heavily relies on this connectivity restriction and is
unpractical. Our results develop the topic further
Dial a Ride from k-forest
The k-forest problem is a common generalization of both the k-MST and the
dense--subgraph problems. Formally, given a metric space on vertices
, with demand pairs and a ``target'' ,
the goal is to find a minimum cost subgraph that connects at least demand
pairs. In this paper, we give an -approximation
algorithm for -forest, improving on the previous best ratio of
by Segev & Segev.
We then apply our algorithm for k-forest to obtain approximation algorithms
for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the
following: given an point metric space with objects each with its own
source and destination, and a vehicle capable of carrying at most objects
at any time, find the minimum length tour that uses this vehicle to move each
object from its source to destination. We prove that an -approximation
algorithm for the -forest problem implies an
-approximation algorithm for Dial-a-Ride. Using our
results for -forest, we get an -
approximation algorithm for Dial-a-Ride. The only previous result known for
Dial-a-Ride was an -approximation by Charikar &
Raghavachari; our results give a different proof of a similar approximation
guarantee--in fact, when the vehicle capacity is large, we give a slight
improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200
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