27 research outputs found
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Monadic second-order model-checking on decomposable matroids
A notion of branch-width, which generalizes the one known for graphs, can be
defined for matroids. We first give a proof of the polynomial time
model-checking of monadic second-order formulas on representable matroids of
bounded branch-width, by reduction to monadic second-order formulas on trees.
This proof is much simpler than the one previously known. We also provide a
link between our logical approach and a grammar that allows to build matroids
of bounded branch-width. Finally, we introduce a new class of non-necessarily
representable matroids, described by a grammar and on which monadic
second-order formulas can be checked in linear time.Comment: 32 pages, journal paper. Revision: the last part has been removed and
the writing improve
Enumeration of the Monomials of a Polynomial and Related Complexity Classes
We study the problem of generating monomials of a polynomial in the context
of enumeration complexity. In this setting, the complexity measure is the delay
between two solutions and the total time. We present two new algorithms for
restricted classes of polynomials, which have a good delay and the same global
running time as the classical ones. Moreover they are simple to describe, use
little evaluation points and one of them is parallelizable. We introduce three
new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic
counterparts of the most common classes for enumeration problems, hoping that
randomization will be a tool as strong for enumeration as it is for decision.
Our interpolation algorithms proves that a lot of interesting problems are in
these classes like the enumeration of the spanning hypertrees of a 3-uniform
hypergraph.
Finally we give a method to interpolate a degree 2 polynomials with an
acceptable (incremental) delay. We also prove that finding a specified monomial
in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no
algorithm with a delay as good (polynomial) as the one we achieve for
multilinear polynomials
Scheduling periodic messages on a shared link
Cloud-RAN is a recent architecture for mobile networks where the processing
units are located in distant data centers while, until now, they were attached
to antennas. The main challenge, to fulfill protocol constraints, is to
guarantee low latency for the periodic messages sent from each antenna to its
processing unit and back. The problem we address is to find a periodic sending
scheme of these messages \emph{without contention nor buffering}, when all
messages are of the same size and the period is fixed.
We study the periodic message assignment problem modeling this situation on a
common topology, where contention arises from a single link shared by all
antennas. The problem is reminiscent of coupled-task scheduling, but the
periodicity introduces a new twist. We study how the problem behaves with
regard to the \emph{load} of the shared link. The main contributions are
polynomial-time algorithms which \emph{always} find a solution for an arbitrary
size of messages and load at most or for messages of size one and load at
most , the golden ratio conjugate. We also prove that a randomized
greedy algorithm finds a solution on almost all instances with high
probability, explaining why most greedy algorithms work so well in practice.Comment: 23 pages, 18 figure
Enumeration Complexity of Logical Query Problems with Second-order Variables
We consider query problems defined by first order formulas of the form F(x,T) with free first order and second order variables and study the data complexity of enumerating results of such queries. By considering the number of alternations in the quantifier prefixes of formulas, we show that such query problems either admit a constant delay or a polynomial delay enumeration algorithm or are hard to enumerate.
We also exhibit syntactically defined fragments inside the hard cases that still admit good enumeration algorithms and discuss the case of some restricted classes of database structures as inputs
The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent
Polynomial identity testing and arithmetic circuit lower bounds are two
central questions in algebraic complexity theory. It is an intriguing fact that
these questions are actually related. One of the authors of the present paper
has recently proposed a "real {\tau}-conjecture" which is inspired by this
connection. The real {\tau}-conjecture states that the number of real roots of
a sum of products of sparse univariate polynomials should be polynomially
bounded. It implies a superpolynomial lower bound on the size of arithmetic
circuits computing the permanent polynomial. In this paper we show that the
real {\tau}-conjecture holds true for a restricted class of sums of products of
sparse polynomials. This result yields lower bounds for a restricted class of
depth-4 circuits: we show that polynomial size circuits from this class cannot
compute the permanent, and we also give a deterministic polynomial identity
testing algorithm for the same class of circuits.Comment: 16 page
Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland\u27s Rule
The best algorithm so far for solving Simple Stochastic Games is Ludwig\u27s randomized algorithm [Ludwig, 1995] which works in expected 2^{O(sqrt{n})} time. We first give a simpler iterative variant of this algorithm, using Bland\u27s rule from the simplex algorithm, which uses exponentially less random bits than Ludwig\u27s version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn [Gimbert and Horn, 2008] whose worst case complexity is O(k!), where k is the number of random nodes. Our algorithm has an expected running time of 2^{O(k)}, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs