5,033 research outputs found
On Randomized Algorithms for Matching in the Online Preemptive Model
We investigate the power of randomized algorithms for the maximum cardinality
matching (MCM) and the maximum weight matching (MWM) problems in the online
preemptive model. In this model, the edges of a graph are revealed one by one
and the algorithm is required to always maintain a valid matching. On seeing an
edge, the algorithm has to either accept or reject the edge. If accepted, then
the adjacent edges are discarded. The complexity of the problem is settled for
deterministic algorithms.
Almost nothing is known for randomized algorithms. A lower bound of
is known for MCM with a trivial upper bound of . An upper bound of
is known for MWM. We initiate a systematic study of the same in this paper with
an aim to isolate and understand the difficulty. We begin with a primal-dual
analysis of the deterministic algorithm due to McGregor. All deterministic
lower bounds are on instances which are trees at every step. For this class of
(unweighted) graphs we present a randomized algorithm which is
-competitive. The analysis is a considerable extension of the
(simple) primal-dual analysis for the deterministic case. The key new technique
is that the distribution of primal charge to dual variables depends on the
"neighborhood" and needs to be done after having seen the entire input. The
assignment is asymmetric: in that edges may assign different charges to the two
end-points. Also the proof depends on a non-trivial structural statement on the
performance of the algorithm on the input tree.
The other main result of this paper is an extension of the deterministic
lower bound of Varadaraja to a natural class of randomized algorithms which
decide whether to accept a new edge or not using independent random choices
Complex-linear invariants of biochemical networks
The nonlinearities found in molecular networks usually prevent mathematical
analysis of network behaviour, which has largely been studied by numerical
simulation. This can lead to difficult problems of parameter determination.
However, molecular networks give rise, through mass-action kinetics, to
polynomial dynamical systems, whose steady states are zeros of a set of
polynomial equations. These equations may be analysed by algebraic methods, in
which parameters are treated as symbolic expressions whose numerical values do
not have to be known in advance. For instance, an "invariant" of a network is a
polynomial expression on selected state variables that vanishes in any steady
state. Invariants have been found that encode key network properties and that
discriminate between different network structures. Although invariants may be
calculated by computational algebraic methods, such as Gr\"obner bases, these
become computationally infeasible for biologically realistic networks. Here, we
exploit Chemical Reaction Network Theory (CRNT) to develop an efficient
procedure for calculating invariants that are linear combinations of
"complexes", or the monomials coming from mass action. We show how this
procedure can be used in proving earlier results of Horn and Jackson and of
Shinar and Feinberg for networks of deficiency at most one. We then apply our
method to enzyme bifunctionality, including the bacterial EnvZ/OmpR osmolarity
regulator and the mammalian
6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase glycolytic regulator,
whose networks have deficiencies up to four. We show that bifunctionality leads
to different forms of concentration control that are robust to changes in
initial conditions or total amounts. Finally, we outline a systematic procedure
for using complex-linear invariants to analyse molecular networks of any
deficiency.Comment: 36 pages, 6 figure
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
More Than 1700 Years of Word Equations
Geometry and Diophantine equations have been ever-present in mathematics.
Diophantus of Alexandria was born in the 3rd century (as far as we know), but a
systematic mathematical study of word equations began only in the 20th century.
So, the title of the present article does not seem to be justified at all.
However, a linear Diophantine equation can be viewed as a special case of a
system of word equations over a unary alphabet, and, more importantly, a word
equation can be viewed as a special case of a Diophantine equation. Hence, the
problem WordEquations: "Is a given word equation solvable?" is intimately
related to Hilbert's 10th problem on the solvability of Diophantine equations.
This became clear to the Russian school of mathematics at the latest in the mid
1960s, after which a systematic study of that relation began.
Here, we review some recent developments which led to an amazingly simple
decision procedure for WordEquations, and to the description of the set of all
solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings
of CAI 2015, Stuttgart, Germany, September 1 - 4, 201
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
Statistical Mechanics of Steiner trees
The Minimum Weight Steiner Tree (MST) is an important combinatorial
optimization problem over networks that has applications in a wide range of
fields. Here we discuss a general technique to translate the imposed global
connectivity constrain into many local ones that can be analyzed with cavity
equation techniques. This approach leads to a new optimization algorithm for
MST and allows to analyze the statistical mechanics properties of MST on random
graphs of various types
Collective Identity and Voice at the Australian Citizens' Parliament
This paper examines the role of collective identity and collective voice in political life. We argue that persons have an underlying predisposition to use collective dimensions, such as common identities and a public voice, in thinking and expressing themselves politically. This collective orientation, however, can be either fostered or weakened by citizensâ political experiences. Although the collective level is an important dimension in contemporary politics, conventional democratic practices do not foster it. Deliberative democracy is suggested as an environment that might allow more ground for citizens to express themselves not only in individual but also in collective terms. We examine this theoretical perspective through a case study of the Australian Citizensâ Parliament, in which transcripts are analyzed to determine the extent to which collective identities and common voice surfaced in actual discourse. We analyze the dynamics involved in the advent of collective dimensions in the deliberative process and highlight the factorsâdeliberation, nature of the discussion, and exceptional opportunityâthat potentially facilitated the rise of group identities and common voice. In spite of the strong individualistic character of the Australian cultural identity, we nonetheless found evidence of collective identity and voice at the Citizensâ Parliament, expressed in terms of national, state, and community levels. In the conclusion, we discuss the implications of those findings for future research and practice of public deliberation
The solution of a mixed boundary value problem in the theory of diffraction by a semi-infinite plane
A solution is obtained for the problem of the diffraction of a plane wave sound source by a semi-infinite half plane. One surface of the half plane has a soft (pressure release) boundary condition, and the other surface a rigid boundary condition. Two unusual features arise in this boundary value problem. The first is the edge field singularity. It is found to be more singular than that associated with either a completely rigid or a completely soft semi-infinite half plane. The second is that the normal Wiener-Hopf method (which is the standard technique to solve half plane problems) has to be modified to give the solution to the present mixed boundary value problem. The mathematical problem which is solved is an approximate model for a rigid noise barrier, one face of which is treated with an absorbing lining. It is shown that the optimum attenuation in the shadow region is obtained when the absorbing lining is on the side of the screen which makes the smallest angle to the source or the receiver from the edge
Geometric gauge potentials and forces in low-dimensional scattering systems
We introduce and analyze several low-dimensional scattering systems that
exhibit geometric phase phenomena. The systems are fully solvable and we
compare exact solutions of them with those obtained in a Born-Oppenheimer
projection approximation. We illustrate how geometric magnetism manifests in
them, and explore the relationship between solutions obtained in the diabatic
and adiabatic pictures. We provide an example, involving a neutral atom dressed
by an external field, in which the system mimics the behavior of a charged
particle that interacts with, and is scattered by, a ferromagnetic material. We
also introduce a similar system that exhibits Aharonov-Bohm scattering. We
propose some practical applications. We provide a theoretical approach that
underscores universality in the appearance of geometric gauge forces. We do not
insist on degeneracies in the adiabatic Hamiltonian, and we posit that the
emergence of geometric gauge forces is a consequence of symmetry breaking in
the latter.Comment: (Final version, published in Phy. Rev. A. 86, 042704 (2012
Dynamic Range Majority Data Structures
Given a set of coloured points on the real line, we study the problem of
answering range -majority (or "heavy hitter") queries on . More
specifically, for a query range , we want to return each colour that is
assigned to more than an -fraction of the points contained in . We
present a new data structure for answering range -majority queries on a
dynamic set of points, where . Our data structure uses O(n)
space, supports queries in time, and updates in amortized time. If the coordinates of the points are integers,
then the query time can be improved to . For constant values of , this improved query
time matches an existing lower bound, for any data structure with
polylogarithmic update time. We also generalize our data structure to handle
sets of points in d-dimensions, for , as well as dynamic arrays, in
which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201
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