Applications of matrix methods to the theory of lower bounds in computational complexity

COMBINATORICA ◽  
1990 ◽  
Vol 10 (1) ◽  
pp. 81-93 ◽  
Author(s):  
A. A. Razborov
Keyword(s):  
Computational Complexity ◽  
Lower Bounds ◽  
Matrix Methods

scholarly journals Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

2017 ◽  
Vol 5 (1) ◽  
pp. 202-224 ◽  
Author(s):  
Dimitrios Christou ◽  
Marilena Mitrouli ◽  
Dimitrios Triantafyllou
Keyword(s):  
Computational Complexity ◽  
Qr Decomposition ◽  
Greatest Common Divisor ◽  
Computational Simulations ◽  
Matrix Representations ◽  
Basis Matrix ◽  
Matrix Methods ◽  
Structured Matrix ◽  
Bezout Matrix ◽  
Column Pivoting

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

scholarly journals The Complexity of Finite Memory Programs with Recursion

1976 ◽  
Vol 5 (68) ◽  
Author(s):  
Neil D. Jones ◽  
Steven S. Muchnick
Keyword(s):  
Computational Complexity ◽  
Programming Language ◽  
Lower Bounds ◽  
Equivalence Problem ◽  
Theoretical Interest ◽  
Exponential Time ◽  
Sophisticated Model ◽  
Finite Memory ◽  
Pushdown Automata ◽  
Exponential Space

<p>In an earlier paper (JACM, 1976) we studied the computational complexity of a number of questions of both programming and theoretical interest (e.g. halting, looping, equivalence) concerning the behaviour of programs written in an extremely simple programming language. These finite memory programs or fmps model the behaviour of FORTRAN-like programs with a finite memory whose size can be determined by examination of the program itself.</p><p>The present paper is a continuation in which we extend the analysis to include ALGOL-like programs (called fmp^(rec) s) with the finite memory augmented by an implicit pushdown stack used to support recursion.</p><p>Our major results are the following. First, we show that at least deterministic exponential time is required to determine whether a program in the basic fmpr~C model accepts a nonempty set. Then we show that a model with a limited version of call-by-name requires exponential space to determine acceptance of a nonempty set, and that a more sophisticated model with rewritable conditional formal parametershas an undecidable halting problem. The same lower bounds apply to the equivalence problem, which in contrast to the situation for the basic fmp model is not known to be decidable (since it is not known whether equivalence of deterministic pushdown automata is decidable).</p>

scholarly journals Simplified group activity selection with group size constraints

2021 ◽  
Author(s):  
Andreas Darmann ◽  
Janosch Döcker ◽  
Britta Dorn ◽  
Sebastian Schneckenburger
Keyword(s):  
Computational Complexity ◽  
Simple Model ◽  
Lower Bounds ◽  
Group Activity ◽  
Core Stability ◽  
Upper And Lower Bounds ◽  
Size Constraints ◽  
Broad Variety ◽  
Special Case ◽  
Additional Constraints

AbstractSeveral real-world situations can be represented in terms of agents that have preferences over activities in which they may participate. Often, the agents can take part in at most one activity (for instance, since these take place simultaneously), and there are additional constraints on the number of agents that can participate in an activity. In such a setting, we consider the task of assigning agents to activities in a reasonable way. We introduce the simplified group activity selection problem providing a general yet simple model for a broad variety of settings, and start investigating its special case where upper and lower bounds of the groups have to be taken into account. We apply different solution concepts such as envy-freeness and core stability to our setting and provide a computational complexity study for the problem of finding such solutions.

Lower bounds in algebraic computational complexity

1985 ◽  
Vol 29 (4) ◽  
pp. 1388-1425 ◽  
Author(s):  
D. Yu. Grigor'ev
Keyword(s):  
Computational Complexity ◽  
Lower Bounds

scholarly journals Complexity of Some Problems Concerning Lindenmayer Systems. (Revised version)

1979 ◽  
Vol 7 (85) ◽  
Author(s):  
Neil D. Jones ◽  
Sven Skyum
Keyword(s):  
Computational Complexity ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Turing Machines ◽  
Lindenmayer Systems ◽  
L Systems

<p>We determine the computational complexity of membership, emptiness and infiniteness for several types of L systems. The L systems we consider are EDOL, EOL, EDTOL, and ETOL, with and without empty productions. For each problem and each type of system we establish both upper and lower bounds on the time or memory required for solution by Turing machines.</p><p>Revised version (first version 1978 under the title <em>Complexity of Some Problems Concerning: Lindenmayer Systems</em>)</p>

scholarly journals Decomposition-Guided Reductions for Argumentation and Treewidth

2021 ◽  
Author(s):  
Johannes Fichte ◽  
Markus Hecher ◽  
Yasir Mahmood ◽  
Arne Meier
Keyword(s):  
Computational Complexity ◽  
Lower Bounds ◽  
Complexity Analysis ◽  
Upper Bounds ◽  
Abstract Argumentation

Argumentation is a widely applied framework for modeling and evaluating arguments and its reasoning with various applications. Popular frameworks are abstract argumentation (Dung’s framework) or logic-based argumentation (Besnard-Hunter’s framework). Their computational complexity has been studied quite in-depth. Incorporating treewidth into the complexity analysis is particularly interesting, as solvers oftentimes employ SAT-based solvers, which can solve instances of low treewidth fast. In this paper, we address whether one can design reductions from argumentation problems to SAT-problems while linearly preserving the treewidth, which results in decomposition-guided (DG) reductions. It turns out that the linear treewidth overhead caused by our DG reductions, cannot be significantly improved under reasonable assumptions. Finally, we consider logic-based argumentation and establish new upper bounds using DG reductions and lower bounds.

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