# The complexity and expressive power of valued constraints

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### Abstract

This thesis is a detailed examination of the expressive power of valued constraints and related complexity questions. The valued constraint satisfaction problem (VCSP) is a generalisation of the constraint satisfaction problem which allows a variety of combinatorial optimisation problems to be described. Although most results are stated in this framework, they can be interpreted equivalently in the framework of, for instance, pseudo-Boolean polynomials, Gibbs energy minimisation or Markov Random Fields.

We take a result of Cohen, Cooper and Jeavons that characterises the expressive power of valued constraints in terms of certain algebraic properties, and extend this result by showing a new connection between the expressive power of valued constraints and linear programming. We prove a decidability result for related algebraic properties.

We consider various classes of valued constraints and the associated cost functions with respect to the question of which of these classes can be expressed using only cost functions of bounded arities. We identify the first known example of an infinite chain of classes of constraints with strictly increasing expressive power. We also present a full classification of various classes of constraints with respect to this problem.

We then study submodular constraints and cost functions. Submodular functions play a key role in combinatorial optimisation and are often considered to be a discrete analogue of convex functions. It has previously been an open problem whether all Boolean submodular cost functions can be decomposed into a sum of binary submodular cost functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of cost functions, we answer this question negatively.

These results have several corollaries. First, we characterise precisely which submodular polynomials of arity 4 can be expressed by binary submodular polynomials. Next, we identify a novel class of submodular functions of arbitrary arities that can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the (s,t)-Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.