= Strategic Companies =

<<TableOfContents>>

== Problem Description ==

Strategic companies is a well-known NP^NP^-complete problem that has often been used for systems comparisons, and also in the previous ASP Competitions.

In the Strategic Companies problem, a collection 

  C = c,,1,,, ..., c,,m,,

of companies is given, for some {{{m>=1}}}. Each company produces some goods in a set {{{G}}}, and each company c,,i,, in C is possibly controlled by a set of owner companies O,,i,, (where O,,i,, is a subset of C, for each i=1,...,m). In this context, a set C' of companies (i.e., a subset of C) is a "strategic set" if it is minimal among all the sets satisfying the following conditions:

 * Companies in C' produce all goods in G; 
 * If O,,i,, is a subset of C', the associated company c,,i,, must belong to C'(for each i=1,...,m). <<BR>>

In our setting, instances are subject to two additional restrictions:

 1. each product is produced by at most four companies; and 
 1. each company is controlled by at most four companies. <<BR>>

Under these restrictions the problem is still NP^NP^-complete. <<BR>>
Instances of strategic companies are encoded by means of the predicates {{{produced_by}}} and {{{controlled_by}}}. In particular, there is a fact {{{producedBy(p,c1,c2,c3,c4)}}} when a product p is  produced by companies {{{c1}}}, {{{c2}}}, {{{c3}}} and {{{c4}}}, and a fact {{{controlledBy(c,c1,c2,c3,c4)}}} when a company {{{c}}} is controlled by companies {{{c1}}}, {{{c2}}}, {{{c3}}} and {{{c4}}}.
If a product {{{p}}} is produced by less than four companies (but at least one), the atom {{{produced_by(p,c_a,c_b,c_c,c_d)}}} contains repetitions of companies. 
For instance, {{{producedBy(p,c_a,c_b,c_c,c_c)}}} is used for representing that p is produced solely by c,,a,,, c,,b,, and c,,c,,. Analogously, if a company c is controlled by less than four companies.

The goal of the problem is a follows: given two distinct companies c,,i,,,c,,j,, in C, we ask for checking the existence of a strategic set C' such that c,,i,, and c,,j,, occur in C'.

== Predicates ==

 * '''Input''': {{{produced_by/5, controlled_by/5, strategic_pair/2}}}

 * '''Output''': {{{strategic/1}}}

== Input format ==

An instance of Strategic Companies is modeled by the following facts that will be provided in the input file:

 - A set of facts of the form {{{produced_by(p,c1,c2,c3,c4)}}}, where {{{produced_by(p,c1,c2,c3,c4)}}} means that product P is produced by companies c1, c2, c3 and c4;

 - A set of facts of the form {{{controlled_by(c,c1,c2,c3,c4)}}}, where {{{controlled_by(c,c1,c2,c3,c4)}}} means that company c is jointly controlled by c1, c2, c3 and c4;

 - A ground fact of the form {{{strategic_pair(c1,c2)}}}, specifying the query goal: is there a strategic set containing both c1 and c2?

For instance:
{{{
produced_by(pasta,barilla,amato,dececco,divella). produced_by(tonno,callipo,star,almera,asdomar).
controlled_by(callipo,star,almera,asdomar,barilla). controlled_by(barilla,callipo,almera,dececco,star). 
strategic_pair(callipo,barilla).
}}}


== Output format ==
For {{{strategic_pair(c1,c2)}}} in input, if c1 and c2 jointly belong to a minimal strategic set, the output must contain a number of ground atoms of the form {{{strategic(c_i)}}} (where "strategic(c_i)" stands for "company c_i is strategic") encoding the minimal strategic set which c1 and c2 belong to.

== Example(s) ==

Input: {{{produced_by(pasta,barilla,amato,dececco,divella). produced_by(tonno,callipo,star,almera,asdomar).}}}
{{{controlled_by(callipo,star,almera,asdomar,barilla). controlled_by(barilla,callipo,almera,dececco,star).}}}
{{{strategic_pair(callipo,barilla).}}}

Output: {{{strategic_pair(c1,c2).}}}


== Notes and updates ==

Note that this is a search problem, despite the seemingly presence of a ''query'' in input. With respect to former edition of the competition, the output format requires an entire strategic set to be output, and not just the subpart of the set showing that c1 and c2 belong to it. 

  * Encoding updated on 01/02/2011;
  * Training Instances updated on 31/01/2011;

== Problem Peculiarities ==

'''Type''': search/Beyond-NP
'''Competition''': both M&S and System competition

This problem is challenging both from the modelling point of view (it is required the availability of constructs allowing an "\exist \forall" quantification over solutions, or some appropriate optimization construct), and from the computational point of view.

== Author(s) ==
 * Author:  Mario Alviano 	
   * Affiliation: University of Calabria, Italy
 * Author: Marco Maratea 	
   * Affiliation: University of Genova
 * Author: Francesco Ricca 	
   * Affiliation: University of Calabria, Italy