5,937 research outputs found
Combinatorial simplex algorithms can solve mean payoff games
A combinatorial simplex algorithm is an instance of the simplex method in
which the pivoting depends on combinatorial data only. We show that any
algorithm of this kind admits a tropical analogue which can be used to solve
mean payoff games. Moreover, any combinatorial simplex algorithm with a
strongly polynomial complexity (the existence of such an algorithm is open)
would provide in this way a strongly polynomial algorithm solving mean payoff
games. Mean payoff games are known to be in NP and co-NP; whether they can be
solved in polynomial time is an open problem. Our algorithm relies on a
tropical implementation of the simplex method over a real closed field of Hahn
series. One of the key ingredients is a new scheme for symbolic perturbation
which allows us to lift an arbitrary mean payoff game instance into a
non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction
expanded, 18 pages, 3 figure
Crossing Patterns in Nonplanar Road Networks
We define the crossing graph of a given embedded graph (such as a road
network) to be a graph with a vertex for each edge of the embedding, with two
crossing graph vertices adjacent when the corresponding two edges of the
embedding cross each other. In this paper, we study the sparsity properties of
crossing graphs of real-world road networks. We show that, in large road
networks (the Urban Road Network Dataset), the crossing graphs have connected
components that are primarily trees, and that the remaining non-tree components
are typically sparse (technically, that they have bounded degeneracy). We prove
theoretically that when an embedded graph has a sparse crossing graph, it has
other desirable properties that lead to fast algorithms for shortest paths and
other algorithms important in geographic information systems. Notably, these
graphs have polynomial expansion, meaning that they and all their subgraphs
have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems(ACM
SIGSPATIAL 2017
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Many problems in Physics are described by dynamical systems that are
conformally symplectic (e.g., mechanical systems with a friction proportional
to the velocity, variational problems with a small discount or thermostated
systems). Conformally symplectic systems are characterized by the property that
they transform a symplectic form into a multiple of itself. The limit of small
dissipation, which is the object of the present study, is particularly
interesting.
We provide all details for maps, but we present also the modifications needed
to obtain a direct proof for the case of differential equations. We consider a
family of conformally symplectic maps defined on a
-dimensional symplectic manifold with exact symplectic form
; we assume that satisfies
. We assume that the family
depends on a -dimensional parameter (called drift) and also on a small
scalar parameter . Furthermore, we assume that the conformal factor
depends on , in such a way that for we have
(the symplectic case).
We study the domains of analyticity in near of
perturbative expansions (Lindstedt series) of the parameterization of the
quasi--periodic orbits of frequency (assumed to be Diophantine) and of
the parameter . Notice that this is a singular perturbation, since any
friction (no matter how small) reduces the set of quasi-periodic solutions in
the system. We prove that the Lindstedt series are analytic in a domain in the
complex plane, which is obtained by taking from a ball centered at
zero a sequence of smaller balls with center along smooth lines going through
the origin. The radii of the excluded balls decrease faster than any power of
the distance of the center to the origin
Efficiently Controllable Graphs
We investigate graphs that can be disconnected into small components by
removing a vanishingly small fraction of their vertices. We show that when a
quantum network is described by such a graph, the network is efficiently
controllable, in the sense that universal quantum computation can be performed
using a control sequence polynomial in the size of the network while
controlling a vanishingly small fraction of subsystems. We show that networks
corresponding to finite-dimensional lattices are efficently controllable, and
explore generalizations to percolation clusters and random graphs. We show that
the classical computational complexity of estimating the ground state of
Hamiltonians described by controllable graphs is polynomial in the number of
subsystems/qubits
Computing simplicial representatives of homotopy group elements
A central problem of algebraic topology is to understand the homotopy groups
of a topological space . For the computational version of the
problem, it is well known that there is no algorithm to decide whether the
fundamental group of a given finite simplicial complex is
trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex that is simply connected (i.e., with
trivial), compute the higher homotopy group for any given .
%The first such algorithm was given by Brown, and more recently, \v{C}adek et
al.
However, these algorithms come with a caveat: They compute the isomorphism
type of , as an \emph{abstract} finitely generated abelian
group given by generators and relations, but they work with very implicit
representations of the elements of . Converting elements of this
abstract group into explicit geometric maps from the -dimensional sphere
to has been one of the main unsolved problems in the emerging field
of computational homotopy theory.
Here we present an algorithm that, given a~simply connected space ,
computes and represents its elements as simplicial maps from a
suitable triangulation of the -sphere to . For fixed , the
algorithm runs in time exponential in , the number of simplices of
. Moreover, we prove that this is optimal: For every fixed , we
construct a family of simply connected spaces such that for any simplicial
map representing a generator of , the size of the triangulation of
on which the map is defined, is exponential in
Applications of incidence bounds in point covering problems
In the Line Cover problem a set of n points is given and the task is to cover
the points using either the minimum number of lines or at most k lines. In
Curve Cover, a generalization of Line Cover, the task is to cover the points
using curves with d degrees of freedom. Another generalization is the
Hyperplane Cover problem where points in d-dimensional space are to be covered
by hyperplanes. All these problems have kernels of polynomial size, where the
parameter is the minimum number of lines, curves, or hyperplanes needed. First
we give a non-parameterized algorithm for both problems in O*(2^n) (where the
O*(.) notation hides polynomial factors of n) time and polynomial space,
beating a previous exponential-space result. Combining this with incidence
bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover
algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant.
Our result improves the previous best times O*((k/1.35)^k) for Line Cover
(where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds
for covering points by parabolas or conics. We also present an algorithm for
Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving
on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201
Probably Approximately Correct Nash Equilibrium Learning
We consider a multi-agent noncooperative game with agents' objective
functions being affected by uncertainty. Following a data driven paradigm, we
represent uncertainty by means of scenarios and seek a robust Nash equilibrium
solution. We treat the Nash equilibrium computation problem within the realm of
probably approximately correct (PAC) learning. Building upon recent
developments in scenario-based optimization, we accompany the computed Nash
equilibrium with a priori and a posteriori probabilistic robustness
certificates, providing confidence that the computed equilibrium remains
unaffected (in probabilistic terms) when a new uncertainty realization is
encountered. For a wide class of games, we also show that the computation of
the so called compression set - a key concept in scenario-based optimization -
can be directly obtained as a byproduct of the proposed solution methodology.
Finally, we illustrate how to overcome differentiability issues, arising due to
the introduction of scenarios, and compute a Nash equilibrium solution in a
decentralized manner. We demonstrate the efficacy of the proposed approach on
an electric vehicle charging control problem.Comment: Preprint submitted to IEEE Transactions on Automatic Contro
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
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