295 research outputs found
Rotor-routing reachability is easy, chip-firing reachability is hard
Chip-firing and rotor-routing are two well-studied examples of Abelian
networks. We study the complexity of their respective reachability problems. We
show that the rotor-routing reachability problem is decidable in polynomial
time, and we give a simple characterization of when a chip-and-rotor
configuration is reachable from another one. For chip-firing, it has been known
that the reachability problem is in P if we have a class of graphs whose period
length is polynomial (for example, Eulerian digraphs). Here we show that in the
general case, chip-firing reachability is hard in the sense that if the
chip-firing reachability problem were in P for general digraphs, then the
polynomial hierarchy would collapse to NP
Abelian networks IV. Dynamics of nonhalting networks
An abelian network is a collection of communicating automata whose state
transitions and message passing each satisfy a local commutativity condition.
This paper is a continuation of the abelian networks series of Bond and Levine
(2016), for which we extend the theory of abelian networks that halt on all
inputs to networks that can run forever. A nonhalting abelian network can be
realized as a discrete dynamical system in many different ways, depending on
the update order. We show that certain features of the dynamics, such as
minimal period length, have intrinsic definitions that do not require
specifying an update order.
We give an intrinsic definition of the \emph{torsion group} of a finite
irreducible (halting or nonhalting) abelian network, and show that it coincides
with the critical group of Bond and Levine (2016) if the network is halting. We
show that the torsion group acts freely on the set of invertible recurrent
components of the trajectory digraph, and identify when this action is
transitive.
This perspective leads to new results even in the classical case of sinkless
rotor networks (deterministic analogues of random walks). In Holroyd et. al
(2008) it was shown that the recurrent configurations of a sinkless rotor
network with just one chip are precisely the unicycles (spanning subgraphs with
a unique oriented cycle, with the chip on the cycle). We generalize this result
to abelian mobile agent networks with any number of chips. We give formulas for
generating series such as where is the number of recurrent chip-and-rotor configurations with
chips; is the diagonal matrix of outdegrees, and is the adjacency
matrix. A consequence is that the sequence completely
determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure
Advances in Evolutionary Algorithms
With the recent trends towards massive data sets and significant computational power, combined with evolutionary algorithmic advances evolutionary computation is becoming much more relevant to practice. Aim of the book is to present recent improvements, innovative ideas and concepts in a part of a huge EA field
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