Study of the NP-completeness of the compact table problem

The problem of compact tables is to maximise the overlap when building a word that is to include permutations of every given words (all the words being the same length). This problem is shown to be NP-complete in the general case, and some specific restrictions are studied. 1. Presentation of the problem 1.1. Informal description and examples In several fields, it can be helpful to give a compact representation of a two dimensional table. We are interested in the case of random output tables, where the table is used to express the possible outcomes in n different initial conditions, where results can be described qualitatively with several results. These results can be represented several times for a given initial condition (to express discrete probabilities). We shall suppose that the number of possible outcomes (with multiple occurrences counted multiple times) is always the same (it could be 100 for a simplified percentage scheme, for example). This common number is called the amplitude of the table (`). For example, one can have the following table (on the left) where each row corresponds to a different set of initial conditions and each column corresponds to a different result. The numbers correspond to the possible outcomes, i.e. in case A, outcome α happens 30% of the time, outcome β happens 40% of the time and outcome γ happens with a probability of 30%. The same table can also be explicitly expanded (on the right). α β γ δ A 30% 40% 30% 0% B 10% 30% 20% 40% C 10% 0% 60% 30% D 20% 20% 40% 20% 1 2 3 4 5 6 7 8 9 10 A α α α β β β β γ γ γ B α β β β γ γ δ δ δ δ C α γ γ γ γ γ γ δ δ δ D α α β β γ γ γ γ δ δ Now, to find the outcome in, e.g. case B, one can computes a random number between 1 and 10 and directly look into the table for the corresponding result in the ’B’ line. 2000 ACM Subject Classification: 68Q17, 68Q45, 91A90.