scholarly journals Exponential Time Complexity of the Permanent and the Tutte Polynomial

2014 ◽  
Vol 10 (4) ◽  
pp. 1-32 ◽  
Author(s):  
Holger Dell ◽  
Thore Husfeldt ◽  
Dániel Marx ◽  
Nina Taslaman ◽  
Martin Wahlén
Keyword(s):  
Time Complexity ◽  
Tutte Polynomial ◽  
Exponential Time

REDUCING SIMPLE GRAMMARS: EXPONENTIAL AGAINST HIGHLY-POLYNOMIAL TIME IN PRACTICE

2007 ◽  
Vol 18 (04) ◽  
pp. 715-725
Author(s):  
CÉDRIC BASTIEN ◽  
JUREK CZYZOWICZ ◽  
WOJCIECH FRACZAK ◽  
WOJCIECH RYTTER
Keyword(s):  
Polynomial Time ◽  
Experimental Analysis ◽  
Time Complexity ◽  
Packet Classification ◽  
State Machines ◽  
Worst Case ◽  
Exponential Time ◽  
Push Down

Simple grammar reduction is an important component in the implementation of Concatenation State Machines (a hardware version of stateless push-down automata designed for wire-speed network packet classification). We present a comparison and experimental analysis of the best-known algorithms for grammar reduction. There are two approaches to this problem: one processing compressed strings without decompression and another one which processes strings explicitly. It turns out that the second approach is more efficient in the considered practical scenario despite having worst-case exponential time complexity (while the first one is polynomial). The study has been conducted in the context of network packet classification, where simple grammars are used for representing the classification policies.

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

scholarly journals Explicit Pieri Inclusions

10.37236/9216 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Markus Hunziker ◽  
John A. Miller ◽  
Mark Sepanski
Keyword(s):  
Time Complexity ◽  
General Linear Group ◽  
Free Resolutions ◽  
Exponential Time ◽  
Exterior Power ◽  
Bounded Number ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Multiplicity Free ◽  
Free Decomposition

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents  are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløystad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.

scholarly journals Complexity of Ising Polynomials

2012 ◽  
Vol 21 (5) ◽  
pp. 743-772 ◽  
Author(s):  
TOMER KOTEK
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
External Field ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Study Phase ◽  
Exceptional Set ◽  
Exponential Time ◽  
Low Dimension ◽  
Special Case

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

scholarly journals Fine-Grained Complexity of Temporal Problems

2020 ◽  
Author(s):  
Konrad K. Dabrowski ◽  
Peter Jonsson ◽  
Sebastian Ordyniak ◽  
George Osipov
Keyword(s):  
Time Complexity ◽  
Unit Length ◽  
Solution Space ◽  
Worst Case ◽  
Exponential Time ◽  
Fine Grained ◽  
Novel Algorithms ◽  
Open Question ◽  
Special Case ◽  
Single Exponential

Expressive temporal reasoning formalisms are essential for AI. One family of such formalisms consists of disjunctive extensions of the simple temporal problem (STP). Such extensions are well studied in the literature and they have many important applications. It is known that deciding satisfiability of disjunctive STPs is NP-hard, while the fine-grained complexity of such problems is virtually unexplored. We present novel algorithms that exploit structural properties of the solution space and prove, assuming the Exponential-Time Hypothesis, that their worst-case time complexity is close to optimal. Among other things, we make progress towards resolving a long-open question concerning whether Allen's interval algebra can be solved in single-exponential time, by giving a 2^{O(nloglog(n))} algorithm for the special case of unit-length intervals.

Convex Optimization and Numerical Issues

2019 ◽  
pp. 191-200
Author(s):  
Pierre-Loïc Garoche
Keyword(s):  
Convex Optimization ◽  
Simplex Method ◽  
Time Complexity ◽  
Path Following ◽  
Linear Constraints ◽  
Exponential Time ◽  
Linear Problems ◽  
Effective Use ◽  
Path Following Methods ◽  
Floating Point Implementation

This chapter aims at providing the intuition behind convex optimization algorithms and addresses their effective use with floating-point implementation. It first briefly presents the algorithms, assuming a real semantics. As outlined in Chapter 4, convex conic programming is supported by different methods depending on the cone considered. The most known approach for linear constraints is the simplex method by Dantzig. While having an exponential-time complexity with respect to the number of constraints, the simplex method performs well in general. Another method is the set of interior point methods, initially proposed by Karmarkar and made popular by Nesterov and Nemirovski. They can be characterized as path-following methods in which a sequence of local linear problems are solved, typically by Newton's method. After these algorithms are considered, the chapter discusses approaches to obtain sound results.

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