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## Separable discrete functions: Recognition and sufficient conditions

Discrete Mathematics. 2019. Vol. 342. No. 5. P. 1275-1292.
Boros E., Cepek O., Gurvich V.

A discrete function of n variables is a mapping g:X1×…×Xn→A, where X1,…,Xn, and A are arbitrary finite sets. Function g is called separable if there exist n functions gi:Xi→A for i=1,…,n, such that for every input x1,…,xn the function g(x1,…,xn) takes one of the values g1(x1),…,gn(xn). Given a discrete function g, it is an interesting problem to ask whether g is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of n variables is known only for n=2. In this paper we will show that a slightly more general recognition problem, when g is not fully but only partially defined, is NP-complete for n≥3. We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for n≥4.

The general recognition problem contains the above mentioned special case for n=2. This case is well-studied in the context of game theory, where (separable) discrete functions of n variables are referred to as (assignable) n-person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of nvariables for any n, thus generalizing the above result known for n=2. Our proof is constructive. Using a graph-based discrete algorithm we show how for a given weakly totally tight (partially defined) discrete function g of nvariables one can construct separating functions g1,…,gn in polynomial time with respect to the size of the input function.