1,933 research outputs found
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
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
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
Traffic-Redundancy Aware Network Design
We consider network design problems for information networks where routers
can replicate data but cannot alter it. This functionality allows the network
to eliminate data-redundancy in traffic, thereby saving on routing costs. We
consider two problems within this framework and design approximation
algorithms.
The first problem we study is the traffic-redundancy aware network design
(RAND) problem. We are given a weighted graph over a single server and many
clients. The server owns a number of different data packets and each client
desires a subset of the packets; the client demand sets form a laminar set
system. Our goal is to connect every client to the source via a single path,
such that the collective cost of the resulting network is minimized. Here the
transportation cost over an edge is its weight times times the number of
distinct packets that it carries.
The second problem is a facility location problem that we call RAFL. Here the
goal is to find an assignment from clients to facilities such that the total
cost of routing packets from the facilities to clients (along unshared paths),
plus the total cost of "producing" one copy of each desired packet at each
facility is minimized.
We present a constant factor approximation for the RAFL and an O(log P)
approximation for RAND, where P is the total number of distinct packets. We
remark that P is always at most the number of different demand sets desired or
the number of clients, and is generally much smaller.Comment: 17 pages. To be published in the proceedings of the Twenty-Third
Annual ACM-SIAM Symposium on Discrete Algorithm
Hypergraphic LP Relaxations for Steiner Trees
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Koenemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following. Structural results: We extend the
technique of uncrossing, usually applied to families of sets, to families of
partitions. As a consequence we show that any basic feasible solution to the
partition LP formulation has sparse support. Although the number of variables
could be exponential, the number of positive variables is at most the number of
terminals. Relations with other relaxations: We show the equivalence of the
partition LP relaxation with other known hypergraphic relaxations. We also show
that these hypergraphic relaxations are equivalent to the well studied
bidirected cut relaxation, if the instance is quasibipartite. Integrality gap
upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap
of these hypergraph relaxations in general graphs. In the special case of
uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~
1.216. By our equivalence theorem, the latter result implies an improved upper
bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Minimum Makespan Multi-vehicle Dial-a-Ride
Dial a ride problems consist of a metric space (denoting travel time between
vertices) and a set of m objects represented as source-destination pairs, where
each object requires to be moved from its source to destination vertex. We
consider the multi-vehicle Dial a ride problem, with each vehicle having
capacity k and its own depot-vertex, where the objective is to minimize the
maximum completion time (makespan) of the vehicles. We study the "preemptive"
version of the problem, where an object may be left at intermediate vertices
and transported by more than one vehicle, while being moved from source to
destination. Our main results are an O(log^3 n)-approximation algorithm for
preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation
for its special case when there is no capacity constraint. We also show that
the approximation ratios improve by a log-factor when the underlying metric is
induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200
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