576 research outputs found
Network Design with Coverage Costs
We study network design with a cost structure motivated by redundancy in data
traffic. We are given a graph, g groups of terminals, and a universe of data
packets. Each group of terminals desires a subset of the packets from its
respective source. The cost of routing traffic on any edge in the network is
proportional to the total size of the distinct packets that the edge carries.
Our goal is to find a minimum cost routing. We focus on two settings. In the
first, the collection of packet sets desired by source-sink pairs is laminar.
For this setting, we present a primal-dual based 2-approximation, improving
upon a logarithmic approximation due to Barman and Chawla (2012). In the second
setting, packet sets can have non-trivial intersection. We focus on the case
where each packet is desired by either a single terminal group or by all of the
groups, and the graph is unweighted. For this setting we present an O(log
g)-approximation.
Our approximation for the second setting is based on a novel spanner-type
construction in unweighted graphs that, given a collection of g vertex subsets,
finds a subgraph of cost only a constant factor more than the minimum spanning
tree of the graph, such that every subset in the collection has a Steiner tree
in the subgraph of cost at most O(log g) that of its minimum Steiner tree in
the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result
Dynamic vs Oblivious Routing in Network Design
Consider the robust network design problem of finding a minimum cost network
with enough capacity to route all traffic demand matrices in a given polytope.
We investigate the impact of different routing models in this robust setting:
in particular, we compare \emph{oblivious} routing, where the routing between
each terminal pair must be fixed in advance, to \emph{dynamic} routing, where
routings may depend arbitrarily on the current demand. Our main result is a
construction that shows that the optimal cost of such a network based on
oblivious routing (fractional or integral) may be a factor of
\BigOmega(\log{n}) more than the cost required when using dynamic routing.
This is true even in the important special case of the asymmetric hose model.
This answers a question in \cite{chekurisurvey07}, and is tight up to constant
factors. Our proof technique builds on a connection between expander graphs and
robust design for single-sink traffic patterns \cite{ChekuriHardness07}
Online Directed Spanners and Steiner Forests
We present online algorithms for directed spanners and Steiner forests. These
problems fall under the unifying framework of online covering linear
programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009),
based on primal-dual techniques. Our results include the following:
For the pairwise spanner problem, in which the pairs of vertices to be
spanned arrive online, we present an efficient randomized
-competitive algorithm for graphs with general lengths,
where is the number of vertices. With uniform lengths, we give an efficient
randomized -competitive algorithm, and an
efficient deterministic -competitive algorithm,
where is the number of terminal pairs. These are the first online
algorithms for directed spanners. In the offline setting, the current best
approximation ratio with uniform lengths is ,
due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020).
For the directed Steiner forest problem with uniform costs, in which the
pairs of vertices to be connected arrive online, we present an efficient
randomized -competitive algorithm. The
state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene,
Krishnaswamy, and Panigrahi (SICOMP 2018) and is -competitive. In the offline version, the current best approximation
ratio with uniform costs is , due to Abboud
and Bodwin (SODA 2018).
A small modification of the online covering framework by Buchbinder and Naor
implies a polynomial-time primal-dual approach with separation oracles, which a
priori might perform exponentially many calls. We convert the online spanner
problem and the online Steiner forest problem into online covering problems and
round in a problem-specific fashion
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