825 research outputs found
Checking Dynamic Consistency of Conditional Hyper Temporal Networks via Mean Payoff Games (Hardness and (pseudo) Singly-Exponential Time Algorithm)
In this work we introduce the \emph{Conditional Hyper Temporal Network
(CHyTN)} model, which is a natural extension and generalization of both the
\CSTN and the \HTN model. Our contribution goes as follows. We show that
deciding whether a given \CSTN or CHyTN is dynamically consistent is
\coNP-hard. Then, we offer a proof that deciding whether a given CHyTN is
dynamically consistent is \PSPACE-hard, provided that the input instances are
allowed to include both multi-head and multi-tail hyperarcs. In light of this,
we continue our study by focusing on CHyTNs that allow only multi-head or only
multi-tail hyperarcs, and we offer the first deterministic (pseudo)
singly-exponential time algorithm for the problem of checking the
dynamic-consistency of such CHyTNs, also producing a dynamic execution strategy
whenever the input CHyTN is dynamically consistent. Since \CSTN{s} are a
special case of CHyTNs, this provides as a byproduct the first
sound-and-complete (pseudo) singly-exponential time algorithm for checking
dynamic-consistency in CSTNs. The proposed algorithm is based on a novel
connection between CSTN{s}/CHyTN{s} and Mean Payoff Games. The presentation of
the connection between \CSTN{s}/CHyTNs and \MPG{s} is mediated by the \HTN
model. In order to analyze the algorithm, we introduce a refined notion of
dynamic-consistency, named -dynamic-consistency, and present a sharp
lower bounding analysis on the critical value of the reaction time
where a \CSTN/CHyTN transits from being, to not being,
dynamically consistent. The proof technique introduced in this analysis of
is applicable more generally when dealing with linear
difference constraints which include strict inequalities.Comment: arXiv admin note: text overlap with arXiv:1505.0082
On the frontiers of polynomial computations in tropical geometry
We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
-hardness and #-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio
Recent advances in real geometric reasoning
In the 1930s Tarski showed that real quantifier elimination was possible, and
in 1975 Collins gave a remotely practicable method, albeit with
doubly-exponential complexity, which was later shown to be inherent. We discuss
some of the recent major advances in Collins method: such as an alternative
approach based on passing via the complexes, and advances which come closer to
"solving the question asked" rather than "solving all problems to do with these
polynomials"
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
We present a polynomial time algorithm to approximately scale tensors of any
format to arbitrary prescribed marginals (whenever possible). This unifies and
generalizes a sequence of past works on matrix, operator and tensor scaling.
Our algorithm provides an efficient weak membership oracle for the associated
moment polytopes, an important family of implicitly-defined convex polytopes
with exponentially many facets and a wide range of applications. These include
the entanglement polytopes from quantum information theory (in particular, we
obtain an efficient solution to the notorious one-body quantum marginal
problem) and the Kronecker polytopes from representation theory (which capture
the asymptotic support of Kronecker coefficients). Our algorithm can be applied
to succinct descriptions of the input tensor whenever the marginals can be
efficiently computed, as in the important case of matrix product states or
tensor-train decompositions, widely used in computational physics and numerical
mathematics.
We strengthen and generalize the alternating minimization approach of
previous papers by introducing the theory of highest weight vectors from
representation theory into the numerical optimization framework. We show that
highest weight vectors are natural potential functions for scaling algorithms
and prove new bounds on their evaluations to obtain polynomial-time
convergence. Our techniques are general and we believe that they will be
instrumental to obtain efficient algorithms for moment polytopes beyond the
ones consider here, and more broadly, for other optimization problems
possessing natural symmetries
Constraint Satisfaction Problems over Numeric Domains
We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra
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