1,095 research outputs found
Intersection representation of digraphs in trees with few leaves
The leafage of a digraph is the minimum number of leaves in a host tree in
which it has a subtree intersection representation. We discuss bounds on the
leafage in terms of other parameters (including Ferrers dimension), obtaining a
string of sharp inequalities.Comment: 12 pages, 3 included figure
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
-product and -threshold graphs
This paper is the continuation of the research of the author and his
colleagues of the {\it canonical} decomposition of graphs. The idea of the
canonical decomposition is to define the binary operation on the set of graphs
and to represent the graph under study as a product of prime elements with
respect to this operation. We consider the graph together with the arbitrary
partition of its vertex set into subsets (-partitioned graph). On the
set of -partitioned graphs distinguished up to isomorphism we consider the
binary algebraic operation (-product of graphs), determined by the
digraph . It is proved, that every operation defines the unique
factorization as a product of prime factors. We define -threshold graphs as
graphs, which could be represented as the product of one-vertex
factors, and the threshold-width of the graph as the minimum size of
such, that is -threshold. -threshold graphs generalize the classes of
threshold graphs and difference graphs and extend their properties. We show,
that the threshold-width is defined for all graphs, and give the
characterization of graphs with fixed threshold-width. We study in detail the
graphs with threshold-widths 1 and 2
Decentralized Observability with Limited Communication between Sensors
In this paper, we study the problem of jointly retrieving the state of a
dynamical system, as well as the state of the sensors deployed to estimate it.
We assume that the sensors possess a simple computational unit that is capable
of performing simple operations, such as retaining the current state and model
of the system in its memory.
We assume the system to be observable (given all the measurements of the
sensors), and we ask whether each sub-collection of sensors can retrieve the
state of the underlying physical system, as well as the state of the remaining
sensors. To this end, we consider communication between neighboring sensors,
whose adjacency is captured by a communication graph. We then propose a linear
update strategy that encodes the sensor measurements as states in an augmented
state space, with which we provide the solution to the problem of retrieving
the system and sensor states.
The present paper contains three main contributions. First, we provide
necessary and sufficient conditions to ensure observability of the system and
sensor states from any sensor. Second, we address the problem of adding
communication between sensors when the necessary and sufficient conditions are
not satisfied, and devise a strategy to this end. Third, we extend the former
case to include different costs of communication between sensors. Finally, the
concepts defined and the method proposed are used to assess the state of an
example of approximate structural brain dynamics through linearized
measurements.Comment: 15 pages, 5 figures, extended version of paper accepted at IEEE
Conference on Decision and Control 201
On the Complexity of the Constrained Input Selection Problem for Structural Linear Systems
This paper studies the problem of, given the structure of a linear-time
invariant system and a set of possible inputs, finding the smallest subset of
input vectors that ensures system's structural controllability. We refer to
this problem as the minimum constrained input selection (minCIS) problem, since
the selection has to be performed on an initial given set of possible inputs.
We prove that the minCIS problem is NP-hard, which addresses a recent open
question of whether there exist polynomial algorithms (in the size of the
system plant matrices) that solve the minCIS problem. To this end, we show that
the associated decision problem, to be referred to as the CIS, of determining
whether a subset (of a given collection of inputs) with a prescribed
cardinality exists that ensures structural controllability, is NP-complete.
Further, we explore in detail practically important subclasses of the minCIS
obtained by introducing more specific assumptions either on the system dynamics
or the input set instances for which systematic solution methods are provided
by constructing explicit reductions to well known computational problems. The
analytical findings are illustrated through examples in multi-agent
leader-follower type control problems
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