It is generally accepted that to unify a pair of substitutions θ1 and θ2 means to find out a pair of substitutions η′ and η′′ such that the compositions θ1η ′ and θ2η ′′ are the same. Actually, unification is the problem of solving linear equations of the form θ1X = θ2Y in the semigroup of substitutions. But some other linear equations on substitutions may be also viewed as less common variants...

Monadic Second-Order Unification (MSOU) is Second-Order Unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete. We also prove that Monadic Second-Order Matching is also NP-complete.

It is shown that the MAX2SAT problem is NP-complete even if every variable appears in at most three clauses. However, if every variable appears in at most two clauses, it is shown that it (and even the general MAXSAT problem) can be solved in linear time. When every variable appears in at most three clauses, we give an exact algorithm for MAXSAT that takes at most O(3 n=2 n) steps where n is th...

We show that the problem of accumulating Jacobian matrices by using a minimal number of floating-point operations is NP-complete by reduction from Ensemble Computation. The proof makes use of the fact that, deviating from the state-of-the-art assumption, algebraic dependences can exist between the local partial derivatives. It follows immediately that the same problem for directional derivative...

Single-player games (often called puzzles) have received considerable attention from the scientific community. Consequently, interesting insights into some puzzles, and into the approaches for solving them, have emerged. However, many puzzles have been neglected, possibly because they are unknown to many people. In this article, we survey NP-Complete puzzles in the hope of motivating further re...

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. However, the unification algorithms for EL developed until recently could not deal with ontologies containing general concept inclusions (GCIs...

Disjoint-Set forests, consisting of Union-Find trees are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression). In this paper we provide such a characterization and show that...

Introduction The vast majority of scientific community believes that P!=NP, with countless supporting arguments. The number of people who believe otherwise probably amounts to as few as those opposing the 2 Law of Thermodynamics. But isn’t nature elegant enough, not to resource to brute-force search? In this article, a novel concept of dimensionality is presented, which may lead to a more effic...

We introduce a new type of movement constraints for a swarm of robots in a grid environment. This type is inspired by Alex Randolphs board game Ricochet Robot and may be used to model robots with very limited abilities for self localization: We assume that once a robot starts to drive in a certain direction, it does not stop its movement until it hits an obstacle wall or another robot. We show ...

NP-complete problems fall into a range of difficulty and it is of considerable interest to identify the general characteristics of these problems that account for this difference. A method for measuring problem difficulty is developed based on the amount of time it takes to find an optimal solution. This reveals that there are layers of difficulty and that any NP-complete problem can be separat...