Digraph related constructions and the complexity of digraph homomorphism problems

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first order reductions. The analysis also yields a complete characterisation of the main polymorphism properties for digraphs whose symmetric closure is a complete graph. The influential Dichotomy Conjecture of Feder and Vardi [19] proposes that every constraint satisfaction problem over a fixed finite template is either solvable in polynomial time or NP-complete. Some well known dichotomies pre-dating the conjecture are special cases: Schaefer’s dichotomy for the complexity of CSPs over two-element templates [31] and the Hell-Nešetřil dichotomy for graph colouring problems [20] (which is equivalent to the CSP dichotomy conjecture in the case of simple graph templates). Since Feder and Vardi’s seminal contribution, the dichotomy has been established for a wide variety of other restricted cases: some of the broadest cases are the dichotomy for three-element templates (Bulatov [11]), for list homomorphism problems (also known as conservative CSPs; Bulatov [12]) and for directed graphs with no sources or sinks (Barto, Kozik, Niven [9]). A key tool in more recent advances, including each of [9, 11, 12], has been the universal-algebraic and combinatorial analysis of “polymorphisms” of CSP templates. Polymorphisms are a generalisation of endomorphisms, and give a CSP template a kind of algebraic structure. When A has polymorphisms satisfying certain equational properties, then CSP(A) is amenable to tractable algorithmic solution; this is already present in Feder and Vardi [19] for example, where a solution by local consistency check algorithm is shown to hold in the presence of polymorphisms witnessing what are known as “near unanimity” equations. In the other direction, the failure of A to have polymorphisms satisfying some families of equations can be used to deduce hardness results for CSP(A); see Larose and Tesson [28, Theorem 4.1] for example. The polymorphism structure of a CSP template A has an extremely tight relationship with the complexity of CSP(A) and appears to relate to the precise structure of algorithms; for example, it is known that solvability by a local consistency check algorithm is equivalent to the presence of specific polymorphism properties (Barto and Kozik [6, 7]). CSPs over directed graphs (henceforth, digraphs) are particularly pertinent to the investigation of CSP complexity. First, digraph homomorphism properties are a The first and third authors were supported by ARC Discovery Project DP1094578. The first author was also supported by ARC Future Fellowship FT120100666 and the second author was supported by ARC Future Fellowship FT100100952.