87 research outputs found
3:1 Nesting Rules in Redistricting
In legislative redistricting, most states draw their House and Senate maps
separately. Ohio and Wisconsin require that their Senate districts be made with
a 3:1 nesting rule, i.e., out of triplets of adjacent House districts. We seek
to study the impact of this requirement on redistricting, specifically on the
number of seats won by a particular political party. We compare two ensembles
generated using Markov Chain Monte Carlo methods; one which uses the ReCom
chain to generate Senate maps without a nesting requirement, and the other
which uses a chain that generates Senate maps with a 3:1 nesting requirement.
We find that requiring a 3:1 nesting rule has minimal impact on the
distribution of seats won. Moreover, we study the impact the chosen House map
has on the distribution of nested Senate maps, and find that an extreme seat
bias at the House level does not significantly impact the distribution of seats
won at the Senate level.Comment: 15 pages, 9 figures. For associated code, see
https://github.com/cdonnay/nesting_OH_WI. Submitted to Statistics and Public
Polic
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Connected Coordinated Motion Planning with Bounded Stretch
We consider the problem of connected coordinated motion planning for a large
collective of simple, identical robots: From a given start grid configuration
of robots, we need to reach a desired target configuration via a sequence of
parallel, collision-free robot motions, such that the set of robots induces a
connected grid graph at all integer times. The objective is to minimize the
makespan of the motion schedule, i.e., to reach the new configuration in a
minimum amount of time. We show that this problem is NP-complete, even for
deciding whether a makespan of 2 can be achieved, while it is possible to check
in polynomial time whether a makespan of 1 can be achieved. On the algorithmic
side, we establish simultaneous constant-factor approximation for two
fundamental parameters, by achieving constant stretch for constant scale.
Scaled shapes (which arise by increasing all dimensions of a given object by
the same multiplicative factor) have been considered in previous seminal work
on self-assembly, often with unbounded or logarithmic scale factors; we provide
methods for a generalized scale factor, bounded by a constant. Moreover, our
algorithm achieves a constant stretch factor: If mapping the start
configuration to the target configuration requires a maximum Manhattan distance
of , then the total duration of our overall schedule is ,
which is optimal up to constant factors.Comment: 28 pages, 18 figures, full version of an extended abstract that
appeared in the proceedings of the 32nd International Symposium on Algorithms
and Computation (ISAAC 2021); revised version (more details added, and typing
errors corrected
Efficiently Reconfiguring a Connected Swarm of Labeled Robots
When considering motion planning for a swarm of labeled robots, we need
to rearrange a given start configuration into a desired target configuration
via a sequence of parallel, continuous, collision-free robot motions. The
objective is to reach the new configuration in a minimum amount of time; an
important constraint is to keep the swarm connected at all times. Problems of
this type have been considered before, with recent notable results achieving
constant stretch for not necessarily connected reconfiguration: If mapping the
start configuration to the target configuration requires a maximum Manhattan
distance of , the total duration of an overall schedule can be bounded to
, which is optimal up to constant factors. However, constant
stretch could only be achieved if disconnected reconfiguration is allowed, or
for scaled configurations (which arise by increasing all dimensions of a given
object by the same multiplicative factor) of unlabeled robots.
We resolve these major open problems by (1) establishing a lower bound of
for connected, labeled reconfiguration and, most
importantly, by (2) proving that for scaled arrangements, constant stretch for
connected reconfiguration can be achieved. In addition, we show that (3) it is
NP-hard to decide whether a makespan of 2 can be achieved, while it is possible
to check in polynomial time whether a makespan of 1 can be achieved.Comment: 26 pages, 17 figures, full version of an extended abstract accepted
for publication in the proceedings of the 33rd International Symposium on
Algorithms and Computation (ISAAC 2022
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials
Combinatorial problems arising in puzzles, origami, and (meta)material design
have rare sets of solutions, which define complex and sharply delineated
boundaries in configuration space. These boundaries are difficult to capture
with conventional statistical and numerical methods. Here we show that
convolutional neural networks can learn to recognize these boundaries for
combinatorial mechanical metamaterials, down to finest detail, despite using
heavily undersampled training sets, and can successfully generalize. This
suggests that the network infers the underlying combinatorial rules from the
sparse training set, opening up new possibilities for complex design of
(meta)materials
Connected Coordinated Motion Planning with Bounded Stretch
We consider the problem of coordinated motion planning for a swarm of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, continuous, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-hard, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.
On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is ?(d), which is optimal up to constant factors
Analyse de quelques algorithmes probabilistes à délais aléatoires
Dans la première partie de cette étude, nous proposons et analysons des algorithmes probabilistes d’élection uniforme dans des graphes de types arbres, les k-arbres et les polyominoïdes. Ces algorithmes utilisent des durées de vie aléatoires associées aux sommets découverts (sommets feuilles ou simpliciaux). Ces durées sont des variables aléatoires indépendantes et sont localement engendrées au fur et à mesure que les sommets sont découverts. Dans la seconde partie, nous analysons un algorithme probabiliste de synchronisation pour le problème de rendez-vous avec agendas dynamiques. L’objectif est de trouver un couplage maximal dans un graphe donné. Ensuite, nous proposons et étudions un modèle de diffusion à délai aléatoire pour la transmission d’un message dans un réseau. Finalement, dans la dernière partie, nous exposons les outils utilisés pour implémenter la simulation des algorithmes distribués.In the first part of this study, we propose and analyze a probabilistic algorithms of uniform election in graphs of structures of the trees type, k-trees and polyominoids. These algorithms use random delay associated to discovered vertices (leaf vertices or simplicial vertices). These delays are independent random variables and are locally generated as and when the vertices are discovered. In the second part, we analyze a probabilistic algorithm of synchronization for the problem of rendezvous with dynamic agendas. The goal is to find a maximal matching in a given graph. Then, we propose and study a model of diffusion with random delay for the transmission of a message in a network. Finally, in the last part, we expose the tools used to implement the simulation of the distributed algorithms
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