561 research outputs found
Chapter 10: Algebraic Algorithms
Our Chapter in the upcoming Volume I: Computer Science and Software
Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales
and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic
and numerical, for matrix computations and root-finding for polynomials and
systems of polynomials equations. We cover part of these large subjects and
include basic bibliography for further study. To meet space limitation we cite
books, surveys, and comprehensive articles with pointers to further references,
rather than including all the original technical papers.Comment: 41.1 page
Are two given maps homotopic? An algorithmic viewpoint
This paper presents two algorithms. In their simplest form, the first
algorithm decides the existence of a pointed homotopy between given simplicial
maps f, g from X to Y and the second computes the group of
pointed homotopy classes of maps from a suspension; in both cases, the target Y
is assumed simply connected and the algorithms run in polynomial time when the
dimension of X is fixed. More generally, these algorithms work relative to a
subspace A of X, fibrewise over a simply connected B and also equivariantly
when all spaces are equipped with a free action of a fixed finite group G
Decidability of the extension problem for maps into odd-dimensional spheres
In a recent paper, it was shown that the problem of existence of a continuous
map extending a given map defined on a subspace is undecidable, even for an even-dimensional sphere. In the
present paper, we prove that the same problem for an odd-dimensional sphere
is decidable. More generally, the same holds for any -connected target space
whose homotopy groups are finite for .Comment: 6 page
Isomorphism problem for finitely generated fully residually free groups
We prove that the isomorphism problem for finitely generated fully residually
free groups is decidable. We also show that each finitely generated fully
residually free group G has a decomposition that is invariant under
automorphisms of G, and obtain a structure theorem for the group of outer
automorphisms Out(G).Comment: 26 page
Rational homotopy type and computability
Given a simplicial pair , a simplicial complex , and a map , does have an extension to ? We show that for a fixed , this
question is algorithmically decidable for all , , and if and only if
has the rational homotopy type of an H-space. As a corollary, many
questions related to bundle structures over a finite complex are decidable.Comment: 18 pages, comments welcom
Public key cryptography based on some extensions of group
Bogopolski, Martino and Ventura in [BMV10] introduced a general criteria to
construct groups extensions with unsolvable conjugacy problem using short exact
sequences. We prove that such extensions have always solvable word problem.
This makes the proposed construction a systematic way to obtain finitely
presented groups with solvable word problem and unsolvable conjugacy problem.
It is believed that such groups are important in cryptography. For this, and as
an example, we provide an explicit construction of an extension of Thompson
group F and we propose it as a base for a public key cryptography protocol
Convex Integer Maximization via Graver Bases
We present a new algebraic algorithmic scheme to solve {\em convex integer
maximization} problems of the following form, where is a convex function on
and are linear forms on , This method works for arbitrary input data
. Moreover, for fixed and several important classes of
programs in {\em variable dimension}, we prove that our algorithm runs in {\em
polynomial time}. As a consequence, we obtain polynomial time algorithms for
various types of multi-way transportation problems, packing problems, and
partitioning problems in variable dimension
Group Isomorphism with Fixed Subnormal Chains
In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly
listed -groups of order could be tested in time, roughly a square root of the classical bound. The term is
entirely due to an cost of testing for isomorphisms that match fixed
composition series in the two groups. We focus here on the
fixed-composition-series subproblem and exhibit a polynomial-time algorithm
that is valid for general groups. A subsequent paper will construct canonical
forms within the same time bound
An Algorithmic Blend of LPs and Ring Equations for Promise CSPs
Promise CSPs are a relaxation of constraint satisfaction problems where the
goal is to find an assignment satisfying a relaxed version of the constraints.
Several well-known problems can be cast as promise CSPs including approximate
graph coloring, discrepancy minimization, and interesting variants of
satisfiability. Similar to CSPs, the tractability of promise CSPs can be tied
to the structure of operations on the solution space called polymorphisms,
though in the promise world these operations are much less constrained. Under
the thesis that non-trivial polymorphisms govern tractability, promise CSPs
therefore provide a fertile ground for the discovery of novel algorithms.
In previous work, we classified Boolean promise CSPs when the constraint
predicates are symmetric. In this work, we vastly generalize these algorithmic
results. Specifically, we show that promise CSPs that admit a family of
"regional-periodic" polymorphisms are in P, assuming that determining which
region a point is in can be computed in polynomial time. Such polymorphisms are
quite general and are obtained by gluing together several functions that are
periodic in the Hamming weights in different blocks of the input.
Our algorithm is based on a novel combination of linear programming and
solving linear systems over rings. We also abstract a framework based on
reducing a promise CSP to a CSP over an infinite domain, solving it there, and
then rounding the solution to an assignment for the promise CSP instance. The
rounding step is intimately tied to the family of polymorphisms and clarifies
the connection between polymorphisms and algorithms in this context. As a key
ingredient, we introduce the technique of finding a solution to a linear
program with integer coefficients that lies in a different ring (such as
) to bypass ad-hoc adjustments for lying on a rounding
boundary.Comment: 41 pages, 2 figure
Conditions for Solvability in Chemical Reaction Networks at Quasi-Steady-State
The quasi-steady-state assumption (QSSA) is an approximation that is widely
used in chemistry and chemical engineering to simplify reaction mechanisms. The
key step in the method requires a solution by radicals of a system of
multivariate polynomials. However, Pantea, Gupta, Rawlings, and Craciun showed
that there exist mechanisms for which the associated polynomials are not
solvable by radicals, and hence reduction by QSSA is not possible. In practice,
however, reduction by QSSA always succeeds. To provide some explanation for
this phenomenon, we prove that solvability is guaranteed for a class of common
chemical reaction networks. In the course of establishing this result, we
examine the question of when it is possible to ensure that there are finitely
many (quasi) steady states. We also apply our results to several examples, in
particular demonstrating the minimality of the nonsolvable example presented by
Pantea, Gupta, Rawlings, and Craciun.Comment: 25 pages, 3 figure
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