Very Complex Systems, Very Hard Problems

The Very Particular Structure of the Very Hard Instances

We show that the algorithms which behave well on average may have difficulty only for highly structured, non-random inputs, except in a finite number of cases. The formal framework is provided by the theory of Kolmogorov complexity. An experimental verification is done for graph S-colorability with Brhlaz’s algorithm.

How Do You Like Your Equilibrium Selection Problems? Hard, or Very Hard?

The PPAD-completeness of Nash equilibrium computation is taken as evidence that the problem is computationally hard in the worst case. This evidence is necessarily rather weak, in the sense that PPAD is only know to lie ”between P and NP”, and there is not a strong prospect of showing it to be as hard as NP. Of course, the problem of finding an equilibrium that has certain sought-after properti...

Looking Forward – Looking Backward A Very Simple Mathematical Model for Very Complex Social Systems

Multiparty communication complexity and very hard functions

A boolean function f(x1, . . . , xn) with xi ∈ {0, 1}m for each i is hard if its nondeterministic multiparty communication complexity (introduced in [in: Proceedings of the 30th IEEE FOCS, 1989, p. 428–433]), C(f), is at least nm. Note that C(f) nm for each f(x1, . . . , xn) with xi ∈ {0, 1}m for each i. A boolean function is very hard if it is hard and its complementary function is also hard. ...

Multiparty Communication Complexity: Very Hard Functions

We consider the multiparty communication model deened in DF89] using the formalism from Hr97]. First, we correct an inaccuracy in the proof of the fundamental result of DR95] providing a lower bound on the nondeter-ministic communication complexity of a function. Then we construct several very hard functions, i.e., functions such that those as well as their complements have the worst possible n...