Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

Counting and enumeration complexity with application to multicriteria scheduling

Abstract: In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results.

Reverse Search for Parametric Linear Programming

Abstract: This paper introduces a new enumeration technique for (multi) parametric linear programs (pLPs) based on the reverse-search paradigm. We prove that the proposed algorithm has a computational complexity that is linear in the size of the output (number of so-called critical regions) and a constant space complexity. This is an improvement over the quadratic and linear computational and space complexities of current approaches. Current implementations of the proposed approach become faster than existing methods for large problems. Extensions of this method are proposed that make the computational requirements lower than those of existing approaches in all cases, while allowing for efficient parallelisation and bounded memory usage.

Computational Complexity over Finite Fields

Abstract: We give nonlinear lower bounds for the complexity of evaluation of a polynomial function at many points and of interpolation in the case of a finite groundfield.

Key function of normal basis multipliers in GF(2n)

Abstract: A new definition of the key function in GF(2n) is given. Based on this definition, a method to speed up software implementations of the normal basis multiplication is presented. It is also shown that the normal basis with maximum complexity can be used to design low complexity multipliers. In particular, it is shown that the circuit complexity of a type I optimal normal basis multiplier can be further reduced.