15 research outputs found
An Overview of Transience Bounds in Max-Plus Algebra
We survey and discuss upper bounds on the length of the transient phase of
max-plus linear systems and sequences of max-plus matrix powers. In particular,
we explain how to extend a result by Nachtigall to yield a new approach for
proving such bounds and we state an asymptotic tightness result by using an
example given by Hartmann and Arguelles.Comment: 13 pages, 2 figure
Faster Parametric Shortest Path and Minimum Balance Algorithms
The parametric shortest path problem is to find the shortest paths in graph
where the edge costs are of the form w_ij+lambda where each w_ij is constant
and lambda is a parameter that varies. The problem is to find shortest path
trees for every possible value of lambda.
The minimum-balance problem is to find a ``weighting'' of the vertices so
that adjusting the edge costs by the vertex weights yields a graph in which,
for every cut, the minimum weight of any edge crossing the cut in one direction
equals the minimum weight of any edge crossing the cut in the other direction.
The paper presents fast algorithms for both problems. The algorithms run in
O(nm+n^2 log n) time. The paper also describes empirical studies of the
algorithms on random graphs, suggesting that the expected time for finding a
minimum-mean cycle (an important special case of both problems) is O(n log(n) +
m)
Analysis of a Classical Matrix Preconditioning Algorithm
We study a classical iterative algorithm for balancing matrices in the
norm via a scaling transformation. This algorithm, which goes back
to Osborne and Parlett \& Reinsch in the 1960s, is implemented as a standard
preconditioner in many numerical linear algebra packages. Surprisingly, despite
its widespread use over several decades, no bounds were known on its rate of
convergence. In this paper we prove that, for any irreducible (real
or complex) input matrix~, a natural variant of the algorithm converges in
elementary balancing operations, where
measures the initial imbalance of~ and is the target imbalance
of the output matrix. (The imbalance of~ is , where
are the maximum entries in magnitude in the
th row and column respectively.) This bound is tight up to the
factor. A balancing operation scales the th row and column so that their
maximum entries are equal, and requires arithmetic operations on
average, where is the number of non-zero elements in~. Thus the running
time of the iterative algorithm is . This is the first time
bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. We
also prove a conjecture of Chen that characterizes those matrices for which the
limit of the balancing process is independent of the order in which balancing
operations are performed.Comment: The previous version (1) (see also STOC'15) handled UB ("unique
balance") input matrices. In this version (2) we extend the work to handle
all input matrice
On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers
We study the transients of matrices in max-plus algebra. Our approach is
based on the weak CSR expansion. Using this expansion, the transient can be
expressed by , where is the weak CSR threshold and
is the time after which the purely pseudoperiodic CSR terms start to dominate
in the expansion. Various bounds have been derived for and ,
naturally leading to the question which matrices, if any, attain these bounds.
In the present paper we characterize the matrices attaining two particular
bounds on , which are generalizations of the bounds of Wielandt and
Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to
a characterization of tightness for the same bounds on the transients of
critical rows and columns. The characterizations themselves are generalizations
of those for the non-weighted case.Comment: 42 pages, 9 figure
Max-balanced flows in oriented matroids
Let M=(E,O) be an oriented matroid on the ground set E. A real-valued vector x defined on E is a max-balanced flow for M if for every signed cocircuit YâOâ„, we have maxeΔY+Xe=maxeΔYâXe. We extend the admissibility and decomposition theorems of Hamacher from regular to general oriented matroids in the case of max-balanced flows, which gives necessary and sufficient conditions for the existence of a max-balanced flow x satisfying lâ©œĂâ©œu. We further investigate the semilattice of such flows under the usual coordinate partial order, and obtain structural results for the minimal elements. We also give necessary and sufficient conditions for the existence of such a flow when we are allowed to reverse the signs on a subset FâE. The proofs of all of our results are constructive, and yield polynomial algorithms in case M is coordinatized by a rational matrix A. In this same setting, we describe a polynomial algorithm that for a given vector w defined on E, either finds a potential p such that wâČ=w+pA is max-balanced, or a certificate that M has no max-balanced flow
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
Weak CSR expansions and transience bounds in max-plus algebra
This paper aims to unify and extend existing techniques for deriving upper
bounds on the transient of max-plus matrix powers. To this aim, we introduce
the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of
the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR
expansion to hold, which does not depend on the values of the entries of the
matrix but only on its pattern, and (ii) a bound for the CS^tR term to
dominate. To improve and analyze (i), we consider various cycle replacement
techniques and show that some of the known bounds for indices and exponents of
digraphs apply here. We also show how to make use of various parameters of
digraphs. To improve and analyze (ii), we introduce three different kinds of
weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle
Threshold). As a result, we obtain a collection of bounds, in general
incomparable to one another, but better than the bounds found in the
literature.Comment: 32 page