Complexity Of Algorithmic Problems Research Articles

Critical properties of bipartite permutation graphs

AbstractThe class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique‐ and rank‐width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique‐ and rank‐width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub‐depth, and with respect to other notions, such as well‐quasi‐ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types.

Linear Time Computable Problems and Logical Descriptions

It is a general problem to investigate the trade off between the complexity of algorithmic problems, the structure of the input objects and the expressive power of problem description languages. The article concentrates on linear time algorithms and on first order logic (FO) as problem description language. One of the main results is a proof that each FO-problem can be solved in linear time for arbitrary relational structures of universally bounded degree.

Maximizing a lower bound on the computational complexity

Abstract We solve a combinatorial problem that arises in determining by a method due to Engeler lower bounds for the computational complexity of algorithmic problems. Denote by Gd the class of permutation groups G of degree d that are iterated wreath products of symmetric groups, i.e. G = Sdh−1∗⋯∗Sd0 with Πh−1i=0 di = d for some natural number h and some sequence (d0,…,dh−1) of natural numbers greater than 1. The problem is to characterize those G = Sdh−1∗⋯∗Sd0 in Gd on which k(G):=log|G|/max0≤i≤h−1log(di!) assumes its maximum. Our solution consists of two necessary conditions for this, namely that d0≤d1≤⋯≤dh and that dh is the largest prime divisor of d. Consequently, if d is a prime power, say d = ph with p prime, then a necessary and sufficient condition is that di = p, 0 ≤ i ≤ h − 1.