82,371 research outputs found
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover
On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions
We study the non-autonomously forced Burgers equation
on the space interval with two sets of the boundary conditions:
the Dirichlet and periodic ones. For both situations we prove that there exists
the unique bounded trajectory of this equation defined for all . Moreover we demonstrate that this trajectory attracts all
trajectories both in pullback and forward sense. We also prove that for the
Dirichlet case this attraction is exponential
BPS counting for knots and combinatorics on words
We discuss relations between quantum BPS invariants defined in terms of a
product decomposition of certain series, and difference equations (quantum
A-polynomials) that annihilate such series. We construct combinatorial models
whose structure is encoded in the form of such difference equations, and whose
generating functions (Hilbert-Poincar\'e series) are solutions to those
equations and reproduce generating series that encode BPS invariants.
Furthermore, BPS invariants in question are expressed in terms of Lyndon words
in an appropriate language, thereby relating counting of BPS states to the
branch of mathematics referred to as combinatorics on words. We illustrate
these results in the framework of colored extremal knot polynomials: among
others we determine dual quantum extremal A-polynomials for various knots,
present associated combinatorial models, find corresponding BPS invariants
(extremal Labastida-Mari\~no-Ooguri-Vafa invariants) and discuss their
integrality.Comment: 41 pages, 1 figure, a supplementary Mathematica file attache
Knots, BPS states, and algebraic curves
We analyze relations between BPS degeneracies related to
Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated
to knots. We introduce a new class of such curves that we call extremal
A-polynomials, discuss their special properties, and determine exact and
asymptotic formulas for the corresponding (extremal) BPS degeneracies. These
formulas lead to nontrivial integrality statements in number theory, as well as
to an improved integrality conjecture stronger than the known M-theory
integrality predictions. Furthermore we determine the BPS degeneracies encoded
in augmentation polynomials and show their consistency with known colored
HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots,
determine them from the knowledge of super-A-polynomials, and verify their
integrality. We illustrate our results with twist knots, torus knots, and
various other knots with up to 10 crossings.Comment: 43 pages, 6 figure
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