Adjoint functors and tree duality

📝 Abstract

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

💡 Analysis

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

📄 Content

arXiv:0805.2978v2 [math.CO] 6 May 2009 Adjoint functors and tree duality Jan Foniok ETH Zurich, Institute for Operations Research R¨amistrasse 101, 8092 Zurich, Switzerland foniok@math.ethz.ch Claude Tardif Royal Military College of Canada PO Box 17000, Stn Forces, Kingston, Ontario Canada, K7K 7B4 Claude.Tardif@rmc.ca 6 May 2009 A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph δH also has tree duality, and we derive a family of tree obstructions for δH from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions. Keywords: constraint satisfaction, tree duality, adjoint functor 2000 Mathematics Subject Classification: 16B50, 68R10, 18A40, 05C15 1 Introduction Our primary motivation is the H-colouring problem (which has become popular under the name Constraint Satisfaction Problem—CSP): for a fixed digraph H (a template) decide whether an input digraph G admits a homomorphism to H. The computational complexity of H-colouring depends on the template H. For some templates the problem 1 tractable bounded treewidth duality tree duality nuf bounded height tree duality δπC fin. duality Figure 1: The structure of tractable templates is known to be NP-complete, for others it is tractable (a polynomial-time algorithm ex- ists). Assuming that P ̸= NP, infinitely many complexity classes lie strictly between P and NP [10], but it has been conjectured that H-colouring belongs to no such intermedi- ate class for any template H [3]. This conjecture has indeed been proved for symmetric templates H [5]. In this paper the focus is on tractable cases. Several conditions are known to imply the existence of a polynomial-time algorithm for H-colouring (definitions follow in the next two paragraphs): it is the case if H has a near-unanimity function (nuf), if H has bounded-treewidth duality, if H has tree duality, if H has finite duality (see [2, 3, 7]). Some of the conditions are depicted in the diagram (Fig. 1). A near-unanimity function is a homomorphism f from Hk to H with k ≥3 such that for all x, y ∈V (H) we have f(x, x, x, . . . , x) = f(y, x, x, . . . , x) = f(x, y, x, . . . , x) = · · · = f(x, x, x, . . . , y) = x. The power Hk is the k-fold product H × H × · · · × H in the category of digraphs and homomorphisms, see [6]. A digraph is a tree (has treewidth k) if its underlying undirected graph is a tree (has treewidth k, respectively). A set F of digraphs is a complete set of obstructions for H if for an arbitrary digraph G there exists a homomorphism from G to H if and only if no F ∈F admits a homomorphism to G. A template has bounded-treewidth duality if it has a complete set of obstructions with treewidth bounded by a constant; it has tree duality if it has a complete set of obstructions consisting of trees; and it has finite duality if it 2 has a finite complete set of obstructions. There is a fairly straightforward way to generate templates with finite duality. For an arbitrary tree T there exists a digraph D(T) such that {T} is a complete set of obstructions for D(T). The digraph D(T) is unique up to homomorphic equivalence∗; it is called the dual of T. Several explicit constructions are known (see [4, 9, 15, 16]). If F is a finite set of oriented trees, then the product D = Q T∈F D(T) is a template with finite duality and F is a complete set of obstructions for D. This construction yields all digraphs with finite duality [15], thus also proving that finite duality implies tree duality. Encouraged by the full description of finite dualities, we aim to provide a construction for some more digraphs with tree duality. To this end we use the arc-graph construction and consider the class δπC of digraphs generated from finite duals by taking iterated arc graphs and finite Cartesian products. We show that all templates in this class have tree duality. We provide an explicit construction of the resulting tree obstructions, which allows us to show that all the digraphs in δπC have in fact bounded-height tree duality, that is, they have a complete set of obstructions consisting of trees of bounded algebraic height (these are tree obstructions that allow a homomorphism to a fixed directed path). In this context we also prove that the problem of existence of a complete set of obstructions consisting of trees with bounded algebraic height is decidable. The arc-graph construction is a special case of a more general phenomenon: it is a right adjoint in the category of digraphs and homomorphisms

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