Strategic Companies

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 = c₁, ..., 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).
  

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

1. each product is produced by at most four companies; and

2. each company is controlled by at most four companies.
   

Under these restrictions the problem is still NP^NP-complete.
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.

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
• Original Authors: Marco Maratea, Francesco Ricca
  • Affiliation: University of Genova and University of Calabria, Italy