19,535 research outputs found
Facility Location in Evolving Metrics
Understanding the dynamics of evolving social or infrastructure networks is a
challenge in applied areas such as epidemiology, viral marketing, or urban
planning. During the past decade, data has been collected on such networks but
has yet to be fully analyzed. We propose to use information on the dynamics of
the data to find stable partitions of the network into groups. For that
purpose, we introduce a time-dependent, dynamic version of the facility
location problem, that includes a switching cost when a client's assignment
changes from one facility to another. This might provide a better
representation of an evolving network, emphasizing the abrupt change of
relationships between subjects rather than the continuous evolution of the
underlying network. We show that in realistic examples this model yields indeed
better fitting solutions than optimizing every snapshot independently. We
present an -approximation algorithm and a matching hardness result,
where is the number of clients and the number of time steps. We also
give an other algorithms with approximation ratio for the variant
where one pays at each time step (leasing) for each open facility
The reverse greedy algorithm for the metric k-median problem
The Reverse Greedy algorithm (RGreedy) for the k-median problem works as
follows. It starts by placing facilities on all nodes. At each step, it removes
a facility to minimize the resulting total distance from the customers to the
remaining facilities. It stops when k facilities remain. We prove that, if the
distance function is metric, then the approximation ratio of RGreedy is between
?(log n/ log log n) and O(log n).Comment: to appear in IPL. preliminary version in COCOON '0
Incremental Medians via Online Bidding
In the k-median problem we are given sets of facilities and customers, and
distances between them. For a given set F of facilities, the cost of serving a
customer u is the minimum distance between u and a facility in F. The goal is
to find a set F of k facilities that minimizes the sum, over all customers, of
their service costs.
Following Mettu and Plaxton, we study the incremental medians problem, where
k is not known in advance, and the algorithm produces a nested sequence of
facility sets where the kth set has size k. The algorithm is c-cost-competitive
if the cost of each set is at most c times the cost of the optimum set of size
k. We give improved incremental algorithms for the metric version: an
8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive
randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic
algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized
algorithm.
The algorithm is s-size-competitive if the cost of the kth set is at most the
minimum cost of any set of size k, and has size at most s k. The optimal
size-competitive ratios for this problem are 4 (deterministic) and e
(randomized). We present the first poly-time O(log m)-size-approximation
algorithm for the offline problem and first poly-time O(log m)-size-competitive
algorithm for the incremental problem.
Our proofs reduce incremental medians to the following online bidding
problem: faced with an unknown threshold T, an algorithm submits "bids" until
it submits a bid that is at least the threshold. It pays the sum of all its
bids. We prove that folklore algorithms for online bidding are optimally
competitive.Comment: conference version appeared in LATIN 2006 as "Oblivious Medians via
Online Bidding
General Bounds for Incremental Maximization
We propose a theoretical framework to capture incremental solutions to
cardinality constrained maximization problems. The defining characteristic of
our framework is that the cardinality/support of the solution is bounded by a
value that grows over time, and we allow the solution to be
extended one element at a time. We investigate the best-possible competitive
ratio of such an incremental solution, i.e., the worst ratio over all
between the incremental solution after steps and an optimum solution of
cardinality . We define a large class of problems that contains many
important cardinality constrained maximization problems like maximum matching,
knapsack, and packing/covering problems. We provide a general
-competitive incremental algorithm for this class of problems, and show
that no algorithm can have competitive ratio below in general.
In the second part of the paper, we focus on the inherently incremental
greedy algorithm that increases the objective value as much as possible in each
step. This algorithm is known to be -competitive for submodular objective
functions, but it has unbounded competitive ratio for the class of incremental
problems mentioned above. We define a relaxed submodularity condition for the
objective function, capturing problems like maximum (weighted) (-)matching
and a variant of the maximum flow problem. We show that the greedy algorithm
has competitive ratio (exactly) for the class of problems that satisfy
this relaxed submodularity condition.
Note that our upper bounds on the competitive ratios translate to
approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo
Dynamic Facility Location via Exponential Clocks
The \emph{dynamic facility location problem} is a generalization of the
classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel
to model the dynamics of evolving social/infrastructure networks. The
generalization lies in that the distance metric between clients and facilities
changes over time. This leads to a trade-off between optimizing the classic
objective function and the "stability" of the solution: there is a switching
cost charged every time a client changes the facility to which it is connected.
While the standard linear program (LP) relaxation for the classic problem
naturally extends to this problem, traditional LP-rounding techniques do not,
as they are often sensitive to small changes in the metric resulting in
frequent switches.
We present a new LP-rounding algorithm for facility location problems, which
yields the first constant approximation algorithm for the dynamic facility
location problem. Our algorithm installs competing exponential clocks on the
clients and facilities, and connect every client by the path that repeatedly
follows the smallest clock in the neighborhood. The use of exponential clocks
gives rise to several properties that distinguish our approach from previous
LP-roundings for facility location problems. In particular, we use \emph{no
clustering} and we allow clients to connect through paths of \emph{arbitrary
lengths}. In fact, the clustering-free nature of our algorithm is crucial for
applying our LP-rounding approach to the dynamic problem
Robust Legged Robot State Estimation Using Factor Graph Optimization
Legged robots, specifically quadrupeds, are becoming increasingly attractive
for industrial applications such as inspection. However, to leave the
laboratory and to become useful to an end user requires reliability in harsh
conditions. From the perspective of state estimation, it is essential to be
able to accurately estimate the robot's state despite challenges such as uneven
or slippery terrain, textureless and reflective scenes, as well as dynamic
camera occlusions. We are motivated to reduce the dependency on foot contact
classifications, which fail when slipping, and to reduce position drift during
dynamic motions such as trotting. To this end, we present a factor graph
optimization method for state estimation which tightly fuses and smooths
inertial navigation, leg odometry and visual odometry. The effectiveness of the
approach is demonstrated using the ANYmal quadruped robot navigating in a
realistic outdoor industrial environment. This experiment included trotting,
walking, crossing obstacles and ascending a staircase. The proposed approach
decreased the relative position error by up to 55% and absolute position error
by 76% compared to kinematic-inertial odometry.Comment: 8 pages, 12 figures. Accepted to RA-L + IROS 2019, July 201
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