412 research outputs found
Counting Solutions of a Polynomial System Locally and Exactly
We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system , with
, and a polydisc
, our method aims to certify the existence
of solutions (counted with multiplicity) within the polydisc.
In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
is sufficiently small and well-isolating for a -fold
solution of the system.
Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of as well
as the distances between and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in of the original system to the problem of solving a
truncated system of degree . In particular, if the multiplicity of
is small compared to the total degrees of the polynomials ,
our method considerably improves upon known complete and certified methods.
For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method
Low Diameter Graph Decompositions by Approximate Distance Computation
In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation.
The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded
Efficiently Computing Real Roots of Sparse Polynomials
We propose an efficient algorithm to compute the real roots of a sparse
polynomial having non-zero real-valued coefficients. It
is assumed that arbitrarily good approximations of the non-zero coefficients
are given by means of a coefficient oracle. For a given positive integer ,
our algorithm returns disjoint disks
, with , centered at the
real axis and of radius less than together with positive integers
such that each disk contains exactly
roots of counted with multiplicity. In addition, it is ensured
that each real root of is contained in one of the disks. If has only
simple real roots, our algorithm can also be used to isolate all real roots.
The bit complexity of our algorithm is polynomial in and , and
near-linear in and , where and constitute
lower and upper bounds on the absolute values of the non-zero coefficients of
, and is the degree of . For root isolation, the bit complexity is
polynomial in and , and near-linear in and
, where denotes the separation of the real roots
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
Two Results on Slime Mold Computations
We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can -approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on , and a number of steps
with quartic dependence on , where is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ().
We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of , and the number of steps depends logarithmically
on and quadratically on
Improved Distributed Algorithms for Exact Shortest Paths
Computing shortest paths is one of the central problems in the theory of
distributed computing. For the last few years, substantial progress has been
made on the approximate single source shortest paths problem, culminating in an
algorithm of Becker et al. [DISC'17] which deterministically computes
-approximate shortest paths in time, where
is the hop-diameter of the graph. Up to logarithmic factors, this time
complexity is optimal, matching the lower bound of Elkin [STOC'04].
The question of exact shortest paths however saw no algorithmic progress for
decades, until the recent breakthrough of Elkin [STOC'17], which established a
sublinear-time algorithm for exact single source shortest paths on undirected
graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for
exact all pairs shortest paths problem on directed graphs.
In this paper, we present a new single-source shortest path algorithm with
complexity . For polylogarithmic , this improves
on Elkin's bound and gets closer to the
lower bound of Elkin [STOC'04]. For larger values of
, we present an improved variant of our algorithm which achieves complexity
, and
thus compares favorably with Elkin's bound of in essentially the entire range of parameters. This
algorithm provides also a qualitative improvement, because it works for the
more challenging case of directed graphs (i.e., graphs where the two directions
of an edge can have different weights), constituting the first sublinear-time
algorithm for directed graphs. Our algorithm also extends to the case of exact
-source shortest paths...Comment: 26 page
On Flows, Paths, Roots, and Zeros
This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt
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