An Algebraic Theory of Complexity for Discrete Optimization

Davis Cohen (1) , Martin Cooper (2, 3) , Paidi Creed (4) , Peter Jeavons (5) , Stanislas Zivny (5)

1 Department of Computer Science
2 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
3 UT3 - Université Toulouse III - Paul Sabatier
4 School of Mathematical Sciences [London]
5 Department of Computer Science [Oxford]

Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimization problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterized. These results provide a new approach to studying the complexity of discrete optimization. We use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the Boolean case.

Mots clés: Galois Connection; Constraint Optimization; Discrete Optimization; Valued Constraint Satisfaction Problems; Weighted Clones; Weighted Polymorphisms

Domaines: Intelligence artificielle [cs.AI]; Apprentissage [cs.LG]; Logique en informatique [cs.LO]; Informatique et langage [cs.CL]

Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny. An Algebraic Theory of Complexity for Discrete Optimization. SIAM Journal on Computing, 2013, vol. 42 (n° 5), pp. 1915-1939. ⟨10.1137/130906398⟩. ⟨hal-01122748⟩