111 research outputs found
A revised model of fluid transport optimization in Physarum polycephalum
Optimization of fluid transport in the slime mold Physarum polycephalum has
been the subject of several modeling efforts in recent literature. Existing
models assume that the tube adaptation mechanism in P. polycephalum's tubular
network is controlled by the sheer amount of fluid flow through the tubes. We
put forward the hypothesis that the controlling variable may instead be the
flow's pressure gradient along the tube. We carry out the stability analysis of
such a revised mathematical model for a parallel-edge network, proving that the
revised model supports the global flow-optimizing behavior of the slime mold
for a substantially wider class of response functions compared to previous
models. Simulations also suggest that the same conclusion may be valid for
arbitrary network topologies.Comment: To appear in Journal of Mathematical Biolog
A Laplacian Approach to -Norm Minimization
We propose a novel differentiable reformulation of the linearly-constrained
minimization problem, also known as the basis pursuit problem. The
reformulation is inspired by the Laplacian paradigm of network theory and leads
to a new family of gradient-based methods for the solution of
minimization problems. We analyze the iteration complexity of a natural
solution approach to the reformulation, based on a multiplicative weights
update scheme, as well as the iteration complexity of an accelerated gradient
scheme. The results can be seen as bounds on the complexity of iteratively
reweighted least squares (IRLS) type methods of basis pursuit
Pooling or sampling: Collective dynamics for electrical flow estimation
The computation of electrical flows is a crucial primitive for many recently proposed optimization algorithms on weighted networks. While typically implemented as a centralized subroutine, the ability to perform this task in a fully decentralized way is implicit in a number of biological systems. Thus, a natural question is whether this task can provably be accomplished in an efficient way by a network of agents executing a simple protocol. We provide a positive answer, proposing two distributed approaches to electrical flow computation on a weighted network: a deterministic process mimicking Jacobi's iterative method for solving linear systems, and a randomized token diffusion process, based on revisiting a classical random walk process on a graph with an absorbing node. We show that both processes converge to a solution of Kirchhoff's node potential equations, derive bounds on their convergence rates in terms of the weights of the network, and analyze their time and message complexity
On the Convergence Time of a Natural Dynamics for Linear Programming
We consider a system of nonlinear ordinary differential equations for the solution of linear programming (LP) problems that was first proposed in the mathematical biology literature as a model for the foraging behavior of acellular slime mold Physarum polycephalum, and more recently considered as a method to solve LP instances. We study the convergence time of the continuous Physarum dynamics in the context of the linear programming problem, and derive a new time bound to approximate optimality that depends on the relative entropy between projected versions of the optimal point and of the initial point. The bound scales logarithmically with the LP cost coefficients and linearly with the inverse of the relative accuracy, establishing the efficiency of the dynamics for arbitrary LP instances with positive costs
Algorithms for Hierarchical and Semi-Partitioned Parallel Scheduling
We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming formulation of the problem and leverage its fractional relaxation to obtain a polynomial-time 2-approximation algorithm. Extensions that incorporate memory capacity constraints are also discussed
Feasibility Tests for Recurrent Real-Time Tasks in the Sporadic DAG Model
A model has been proposed in [Baruah et al., in Proceedings of the IEEE
Real-Time Systems Symposium 2012] for representing recurrent
precedence-constrained tasks to be executed on multiprocessor platforms, where
each recurrent task is modeled by a directed acyclic graph (DAG), a period, and
a relative deadline. Each vertex of the DAG represents a sequential job, while
the edges of the DAG represent precedence constraints between these jobs. All
the jobs of the DAG are released simultaneously and have to be completed within
some specified relative deadline. The task may release jobs in this manner an
unbounded number of times, with successive releases occurring at least the
specified period apart. The feasibility problem is to determine whether such a
recurrent task can be scheduled to always meet all deadlines on a specified
number of dedicated processors.
The case of a single task has been considered in [Baruah et al., 2012]. The
main contribution of this paper is to consider the case of multiple tasks. We
show that EDF has a speedup bound of 2-1/m, where m is the number of
processors. Moreover, we present polynomial and pseudopolynomial schedulability
tests, of differing effectiveness, for determining whether a set of sporadic
DAG tasks can be scheduled by EDF to meet all deadlines on a specified number
of processors
Two Results on Slime Mold Computations
We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can -approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on , and a number of steps
with quartic dependence on , where is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ().
We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of , and the number of steps depends logarithmically
on and quadratically on
ILP-based approaches to partitioning recurrent workloads upon heterogeneous multiprocessors
The problem of partitioning systems of independent constrained-deadline sporadic tasks upon heterogeneous multiprocessor platforms is considered. Several different integer linear program (ILP) formulations of this problem, offering different tradeoffs between effectiveness (as quantified by speedup bound) and running time efficiency, are presented
On the compatibility of exact schedulability tests for global fixed priority pre-emptive scheduling with Audsley’s optimal priority assignment algorithm
Audsley's optimal priority assignment (OPA) algorithm can be applied to multiprocessor scheduling provided that three conditions hold with respect to the schedulability tests used. In this short paper, we prove that no exact test for global fixed priority pre-emptive scheduling of sporadic tasks can be compatible with Audsley's algorithm, and hence the OPA algorithm cannot be used to obtain an optimal priority assignment for such systems
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