12,693 research outputs found
La obra de Leslie Valiant
Este año Leslie VALIANT cumple 65 años y nosotros queremos celebrar este importante aniversario con este trabajo en el que se analiza su obra. Centramos nuestra atención en aquellos de sus trabajos en los que una clara influencia de Volker STRASSEN puede ser detectada. Es patente la influencia de Strassen en la obra de Valiant, pero esto no quiere decir que el trabajo de Valiant, complejo y multifacético, sea un simple corolario a la obra del primero.
Para citar este artículo: J. Andrés Montoya, The work of Leslie Valiant: alle die Strassen führen nach Strassen, Rev. Integr. Temas Mat. 32 (2014), no. 2, 153-168
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
The Minrank of Random Graphs
The minrank of a graph is the minimum rank of a matrix that can be
obtained from the adjacency matrix of by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
for all regimes of . In particular, for any constant ,
we show that with high probability,
where is chosen from . This bound gives a near quadratic
improvement over the previous best lower bound of (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits
Efficient, Safe, and Probably Approximately Complete Learning of Action Models
In this paper we explore the theoretical boundaries of planning in a setting
where no model of the agent's actions is given. Instead of an action model, a
set of successfully executed plans are given and the task is to generate a plan
that is safe, i.e., guaranteed to achieve the goal without failing. To this
end, we show how to learn a conservative model of the world in which actions
are guaranteed to be applicable. This conservative model is then given to an
off-the-shelf classical planner, resulting in a plan that is guaranteed to
achieve the goal. However, this reduction from a model-free planning to a
model-based planning is not complete: in some cases a plan will not be found
even when such exists. We analyze the relation between the number of observed
plans and the likelihood that our conservative approach will indeed fail to
solve a solvable problem. Our analysis show that the number of trajectories
needed scales gracefully
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
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