8,263 research outputs found
Smoothed Efficient Algorithms and Reductions for Network Coordination Games
Worst-case hardness results for most equilibrium computation problems have
raised the need for beyond-worst-case analysis. To this end, we study the
smoothed complexity of finding pure Nash equilibria in Network Coordination
Games, a PLS-complete problem in the worst case. This is a potential game where
the sequential-better-response algorithm is known to converge to a pure NE,
albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial)
smoothed complexity when the underlying game graph is a complete (resp.
arbitrary) graph, and every player has constantly many strategies. We note that
the complete graph case is reminiscent of perturbing all parameters, a common
assumption in most known smoothed analysis results.
Second, we define a notion of smoothness-preserving reduction among search
problems, and obtain reductions from -strategy network coordination games to
local-max-cut, and from -strategy games (with arbitrary ) to
local-max-cut up to two flips. The former together with the recent result of
[BCC18] gives an alternate -time smoothed algorithm for the
-strategy case. This notion of reduction allows for the extension of
smoothed efficient algorithms from one problem to another.
For the first set of results, we develop techniques to bound the probability
that an (adversarial) better-response sequence makes slow improvements on the
potential. Our approach combines and generalizes the local-max-cut approaches
of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful
definition of the matrix which captures the increase in potential, a tighter
union bound on adversarial sequences, and balancing it with good enough rank
bounds. We believe that the approach and notions developed herein could be of
interest in addressing the smoothed complexity of other potential and/or
congestion games
Bounds on the number of connected components for tropical prevarieties
For a tropical prevariety in Rn given by a system of k tropical polynomials in n variables with degrees at most d, we prove that its number of connected components is less than k+7n−
Volume-preserving normal forms of Hopf-zero singularity
A practical method is described for computing the unique generator of the
algebra of first integrals associated with a large class of Hopf-zero
singularity. The set of all volume-preserving classical normal forms of this
singularity is introduced via a Lie algebra description. This is a maximal
vector space of classical normal forms with first integral; this is whence our
approach works. Systems with a non-zero condition on their quadratic parts are
considered. The algebra of all first integrals for any such system has a unique
(modulo scalar multiplication) generator. The infinite level volume-preserving
parametric normal forms of any non-degenerate perturbation within the Lie
algebra of any such system is computed, where it can have rich dynamics. The
associated unique generator of the algebra of first integrals are derived. The
symmetry group of the infinite level normal forms are also discussed. Some
necessary formulas are derived and applied to appropriately modified
R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the
applicability of our theoretical results. An approach (introduced by Iooss and
Lombardi) is applied to find an optimal truncation for the first level normal
forms of these examples with exponentially small remainders. The numerically
suggested radius of convergence (for the first integral) associated with a
hypernormalization step is discussed for the truncated first level normal forms
of the examples. This is achieved by an efficient implementation of the results
using Maple
Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
Invariant tori play a fundamental role in the dynamics of symplectic and
volume-preserving maps. Codimension-one tori are particularly important as they
form barriers to transport. Such tori foliate the phase space of integrable,
volume-preserving maps with one action and angles. For the area-preserving
case, Greene's residue criterion is often used to predict the destruction of
tori from the properties of nearby periodic orbits. Even though KAM theory
applies to the three-dimensional case, the robustness of tori in such systems
is still poorly understood. We study a three-dimensional, reversible,
volume-preserving analogue of Chirikov's standard map with one action and two
angles. We investigate the preservation and destruction of tori under
perturbation by computing the "residue" of nearby periodic orbits. We find tori
with Diophantine rotation vectors in the "spiral mean" cubic algebraic field.
The residue is used to generate the critical function of the map and find a
candidate for the most robust torus.Comment: laTeX, 40 pages, 26 figure
Multiplicities of Noetherian deformations
The \emph{Noetherian class} is a wide class of functions defined in terms of
polynomial partial differential equations. It includes functions appearing
naturally in various branches of mathematics (exponential, elliptic, modular,
etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets
defined using Noetherian equations admits effective estimates analogous to the
effective \emph{global} bounds of algebraic geometry.
We make a major step in the development of the theory of Noetherian functions
by providing an effective upper bound for the local number of isolated
solutions of a Noetherian system of equations depending on a parameter
, which remains valid even when the system degenerates at
. An estimate of this sort has played the key role in the
development of the theory of Pfaffian functions, and is expected to lead to
similar results in the Noetherian setting. We illustrate this by deducing from
our main result an effective form of the Lojasiewicz inequality for Noetherian
functions.Comment: v2: reworked last section, accepted to GAF
Morse theory on spaces of braids and Lagrangian dynamics
In the first half of the paper we construct a Morse-type theory on certain
spaces of braid diagrams. We define a topological invariant of closed positive
braids which is correlated with the existence of invariant sets of parabolic
flows defined on discretized braid spaces. Parabolic flows, a type of
one-dimensional lattice dynamics, evolve singular braid diagrams in such a way
as to decrease their topological complexity; algebraic lengths decrease
monotonically. This topological invariant is derived from a Morse-Conley
homotopy index and provides a gloablization of `lap number' techniques used in
scalar parabolic PDEs.
In the second half of the paper we apply this technology to second order
Lagrangians via a discrete formulation of the variational problem. This
culminates in a very general forcing theorem for the existence of infinitely
many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification
of two proofs and one definition; 55 pages, 20 figure
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