14 research outputs found
Changing Bases: Multistage Optimization for Matroids and Matchings
This paper is motivated by the fact that many systems need to be maintained
continually while the underlying costs change over time. The challenge is to
continually maintain near-optimal solutions to the underlying optimization
problems, without creating too much churn in the solution itself. We model this
as a multistage combinatorial optimization problem where the input is a
sequence of cost functions (one for each time step); while we can change the
solution from step to step, we incur an additional cost for every such change.
We study the multistage matroid maintenance problem, where we need to maintain
a base of a matroid in each time step under the changing cost functions and
acquisition costs for adding new elements. The online version of this problem
generalizes online paging. E.g., given a graph, we need to maintain a spanning
tree at each step: we pay for the cost of the tree at time
, and also for the number of edges changed at
this step. Our main result is an -approximation, where is
the number of elements/edges and is the rank of the matroid. We also give
an approximation for the offline version of the problem. These
bounds hold when the acquisition costs are non-uniform, in which caseboth these
results are the best possible unless P=NP.
We also study the perfect matching version of the problem, where we must
maintain a perfect matching at each step under changing cost functions and
costs for adding new elements. Surprisingly, the hardness drastically
increases: for any constant , there is no
-approximation to the multistage matching maintenance
problem, even in the offline case
Competitive Online Peak-Demand Minimization Using Energy Storage
We study the problem of online peak-demand minimization under energy storage
constraints. It is motivated by an increasingly popular scenario where
large-load customers utilize energy storage to reduce the peak procurement from
the grid, which accounts for up to of their electric bills. The problem
is uniquely challenging due to (i) the coupling of online decisions across time
imposed by the inventory constraints and (ii) the noncumulative nature of the
peak procurement. In this paper, we develop an optimal online algorithm for the
problem, attaining the best possible competitive ratio (CR) among all
deterministic and randomized algorithms. We show that the optimal CR can be
computed in polynomial time, by solving a linear number of linear-fractional
problems. More importantly, we generalize our approach to develop an
\emph{anytime-optimal} online algorithm that achieves the best possible CR at
any epoch, given the inputs and online decisions so far. The algorithm retains
the optimal worst-case performance and achieves adaptive average-case
performance. Simulation results based on real-world traces show that, under
typical settings, our algorithms improve peak reduction by over as
compared to baseline alternatives