101 research outputs found
Polynomial formulations as a barrier for reduction-based hardness proofs
The Strong Exponential Time Hypothesis (SETH) asserts that for every
there exists such that -SAT requires time
. The field of fine-grained complexity has leveraged SETH to
prove quite tight conditional lower bounds for dozens of problems in various
domains and complexity classes, including Edit Distance, Graph Diameter,
Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been
repeatedly asked in the literature whether SETH-hardness results can be proven
for other fundamental problems such as Hamiltonian Path, Independent Set,
Chromatic Number, MAX--SAT, and Set Cover.
In this paper, we show that fine-grained reductions implying even
-hardness of these problems from SETH for any , would
imply new circuit lower bounds: super-linear lower bounds for Boolean
series-parallel circuits or polynomial lower bounds for arithmetic circuits
(each of which is a four-decade open question).
We also extend this barrier result to the class of parameterized problems.
Namely, for every we conditionally rule out fine-grained reductions
implying SETH-based lower bounds of for a number of problems
parameterized by the solution size .
Our main technical tool is a new concept called polynomial formulations. In
particular, we show that many problems can be represented by relatively
succinct low-degree polynomials, and that any problem with such a
representation cannot be proven SETH-hard (without proving new circuit lower
bounds)
Polynomial-Time Solvers for the Discrete -Optimal Transport Problems
In this note, we propose polynomial-time algorithms solving the Monge and
Kantorovich formulations of the -optimal transport problem in the
discrete and finite setting. It is the first time, to the best of our
knowledge, that efficient numerical methods for these problems have been
proposed
A Quantum Interior Point Method for LPs and SDPs
We present a quantum interior point method with worst case running time
for
SDPs and for LPs, where the output of our algorithm is a pair of matrices
that are -optimal -approximate SDP solutions. The factor
is at most for SDPs and for LP's, and is
an upper bound on the condition number of the intermediate solution matrices.
For the case where the intermediate matrices for the interior point method are
well conditioned, our method provides a polynomial speedup over the best known
classical SDP solvers and interior point based LP solvers, which have a worst
case running time of and respectively. Our results
build upon recently developed techniques for quantum linear algebra and pave
the way for the development of quantum algorithms for a variety of applications
in optimization and machine learning.Comment: 32 page
An Algorithmic Theory of Integer Programming
We study the general integer programming problem where the number of
variables is a variable part of the input. We consider two natural
parameters of the constraint matrix : its numeric measure and its
sparsity measure . We show that integer programming can be solved in time
, where is some computable function of the
parameters and , and is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
and , and is solvable in polynomial time for every fixed and .
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time , independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix , which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for -fold, tree-fold, and -stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith
A (3/2+ɛ) approximation algorithm for scheduling malleable and non-malleable parallel tasks
In this paper we study a scheduling problem with malleable and non-malleable parallel tasks on processors. A non-malleable parallel task is one that runs in parallel on a specific given number of processors. The goal is to find a non-preemptive schedule on the processors which minimizes the makespan, or the latest task completion time. The previous best result is the list scheduling algorithm with an absolute approximation ratio of . On the other hand, there does not exist an approximation algorithm for scheduling non-malleable parallel tasks with ratio smaller than , unless . In this paper we show that a schedule with length can be computed for the scheduling problem in time . Furthermore we present an approximation algorithm for scheduling malleable parallel tasks. Finally, we show how to extend our algorithms to the variant with additional release dates
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