259 research outputs found
Non-preemptive Scheduling in a Smart Grid Model and its Implications on Machine Minimization
We study a scheduling problem arising in demand response management in smart
grid. Consumers send in power requests with a flexible feasible time interval
during which their requests can be served. The grid controller, upon receiving
power requests, schedules each request within the specified interval. The
electricity cost is measured by a convex function of the load in each timeslot.
The objective is to schedule all requests with the minimum total electricity
cost. Previous work has studied cases where jobs have unit power requirement
and unit duration. We extend the study to arbitrary power requirement and
duration, which has been shown to be NP-hard. We give the first online
algorithm for the general problem, and prove that the problem is fixed
parameter tractable. We also show that the online algorithm is asymptotically
optimal when the objective is to minimize the peak load. In addition, we
observe that the classical non-preemptive machine minimization problem is a
special case of the smart grid problem with min-peak objective, and show that
we can solve the non-preemptive machine minimization problem asymptotically
optimally
An Improved Online Algorithm for the Traveling Repairperson Problem on a Line
In the online variant of the traveling repairperson problem (TRP), requests arrive in time at points of a metric space X and must be eventually visited by a server. The server starts at a designated point of X and travels at most at unit speed. Each request has a given weight and once the server visits its position, the request is considered serviced; we call such time completion time of the request. The goal is to minimize the weighted sum of completion times of all requests.
In this paper, we give a 5.429-competitive deterministic algorithm for line metrics improving over 5.829-competitive solution by Krumke et al. (TCS 2003). Our result is obtained by modifying the schedule by serving requests that are close to the origin first. To compute the competitive ratio of our approach, we use a charging scheme, and later evaluate its properties using a factor-revealing linear program which upper-bounds the competitive ratio
Phase diagram of a frustrated asymmetric ferromagnetic spin ladder
We perform a systematic investigation on the ground state of an asymmetric
two-leg spin ladder (where exchange couplings of the legs are unequal) with
ferromagnetic (FM) nearest-neighbor interaction and diagonal anti-ferromagnetic
frustration using the Density Matrix Renormalization Group (DMRG) method. When
the ladder is strongly asymmetric with moderate frustration, a magnetic canted
state is observed between a FM state and a singlet dimerized state. The phase
boundaries are dependent on the asymmetric strength . On the other
hand, when the asymmetric strength is intermediate, a so-called spin-stripe
state (spins align parallel on same legs, but antiparallel on rungs) is
discovered, and the system experiences a first-order phase transition from the
FM state to the spin-stripe state upon increasing frustration. We present
numerical evidence to interpret the phase diagram in terms of frustration and
the asymmetric strength.Comment: 14 pages, 8 figure
Traveling Repairperson, Unrelated Machines, and Other Stories About Average Completion Times
We present a unified framework for minimizing average completion time for many seemingly disparate online scheduling problems, such as the traveling repairperson problems (TRP), dial-a-ride problems (DARP), and scheduling on unrelated machines.
We construct a simple algorithm that handles all these scheduling problems, by computing and later executing auxiliary schedules, each optimizing a certain function on already seen prefix of the input. The optimized function resembles a prize-collecting variant of the original scheduling problem. By a careful analysis of the interplay between these auxiliary schedules, and later employing the resulting inequalities in a factor-revealing linear program, we obtain improved bounds on the competitive ratio for all these scheduling problems.
In particular, our techniques yield a 4-competitive deterministic algorithm for all previously studied variants of online TRP and DARP, and a 3-competitive one for the scheduling on unrelated machines (also with precedence constraints). This improves over currently best ratios for these problems that are 5.14 and 4, respectively. We also show how to use randomization to further reduce the competitive ratios to 1+2/ln 3 < 2.821 and 1+1/ln 2 < 2.443, respectively. The randomized bounds also substantially improve the current state of the art. Our upper bound for DARP contradicts the lower bound of 3 given by Fink et al. (Inf. Process. Lett. 2009); we pinpoint a flaw in their proof
Instance complexity of Boolean functions
In the area of query complexity of Boolean functions, the most widely studied
cost measure of an algorithm is the worst-case number of queries made by it on
an input. Motivated by the most natural cost measure studied in online
algorithms, the competitive ratio, we consider a different cost measure for
query algorithms for Boolean functions that captures the ratio of the cost of
the algorithm and the cost of an optimal algorithm that knows the input in
advance. The cost of an algorithm is its largest cost over all inputs.
Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean
functions, and dubbed it instance complexity. Grossman et al. showed, among
other results, that monotone Boolean functions with instance complexity 1 are
precisely those that depend on one or two variables.
We complement the above-mentioned result of Grossman et al. by completely
characterizing the instance complexity of symmetric Boolean functions. As a
corollary we conclude that the only symmetric Boolean functions with instance
complexity 1 are the Parity function and its complement. We also study the
instance complexity of some graph properties like Connectivity and k-clique
containment.
In all the Boolean functions we study above, and those studied by Grossman et
al., the instance complexity turns out to be the ratio of query complexity to
minimum certificate complexity. It is a natural question to ask if this is the
correct bound for all Boolean functions. We show a negative answer in a very
strong sense, by analyzing the instance complexity of the Greater-Than and
Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the
input size for both of these functions, while we exhibit algorithms for which
the instance complexity is a constant
Optimal Nonpreemptive Scheduling in a Smart Grid Model
We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. Previous work has studied cases where jobs have unit power requirement and unit duration. We extend the study to arbitrary power requirement and duration, which has been shown to be NP-hard. We give the first online algorithm for the general problem, and prove that the worst case competitive ratio is asymptotically optimal. We also prove that the problem is fixed parameter tractable. Due to space limit, the missing proofs are presented in the full paper
- …