2,974 research outputs found
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
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}
Robust Network Routing under Cascading Failures
We propose a dynamical model for cascading failures in single-commodity
network flows. In the proposed model, the network state consists of flows and
activation status of the links. Network dynamics is determined by a, possibly
state-dependent and adversarial, disturbance process that reduces flow capacity
on the links, and routing policies at the nodes that have access to the network
state, but are oblivious to the presence of disturbance. Under the proposed
dynamics, a link becomes irreversibly inactive either due to overload condition
on itself or on all of its immediate downstream links. The coupling between
link activation and flow dynamics implies that links to become inactive
successively are not necessarily adjacent to each other, and hence the pattern
of cascading failure under our model is qualitatively different than standard
cascade models. The magnitude of a disturbance process is defined as the sum of
cumulative capacity reductions across time and links of the network, and the
margin of resilience of the network is defined as the infimum over the
magnitude of all disturbance processes under which the links at the origin node
become inactive. We propose an algorithm to compute an upper bound on the
margin of resilience for the setting where the routing policy only has access
to information about the local state of the network. For the limiting case when
the routing policies update their action as fast as network dynamics, we
identify sufficient conditions on network parameters under which the upper
bound is tight under an appropriate routing policy. Our analysis relies on
making connections between network parameters and monotonicity in network state
evolution under proposed dynamics
Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs
We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation).
Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions
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