17 research outputs found
Algebra and the Complexity of Digraph CSPs: a Survey
We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems
A Brightwell-Winkler type characterisation of NU graphs
In 2000, Brightwell and Winkler characterised dismantlable graphs as the
graphs for which the Hom-graph , defined on the set of
homomorphisms from to , is connected for all graphs . This shows that
the reconfiguration version of the -colouring
problem, in which one must decide for a given whether is
connected, is trivial if and only if is dismantlable.
We prove a similar starting point for the reconfiguration version of the
-extension problem. Where is the subgraph of the
Hom-graph induced by the -colourings extending the
-precolouring of , the reconfiguration version
of the -extension problem asks, for a given -precolouring of a graph
, if is connected. We show that the graphs for which
is connected for every choice of are exactly the
graphs. This gives a new characterisation of graphs, a
nice class of graphs that is important in the algebraic approach to the -dichotomy.
We further give bounds on the diameter of for
graphs , and show that shortest path between two vertices of can be found in parameterised polynomial time. We apply our
results to the problem of shortest path reconfiguration, significantly
extending recent results.Comment: 17 pages, 1 figur
Series-Parallel Posets and Polymorphisms
We examine various aspects of the poset retraction problem for series-parallel posets. In particular we show that the poset retraction problem for series-parallel posets that are already solvable in polynomial time are actually also solvable in nondeterministic logarithmic space (assuming P 6= NP). We do this by showing that these series-parallel posets when expanded by constants have bounded path duality. We also give a recipe for constructing members of this special class of series-parallel poset analogous to the construction of all series-parallel posets. Piecing together results from [5],[15],[14] and [12] one can deduce that if a relational structure expanded by constants has bounded path duality then it admits SD-join operations. We directly prove the existence of SD-join operations on members of this class by providing an algorithm which constructs them. Moreover, we obtain a polynomial upper bound to the length of the sequence of these operations. This also proves that for this class of series-parallel posets, having bounded path duality when expanded by constants is equivalent to admitting SD-join operations. This equivalence is not yet known to be true for general relational structures; only the forward direction is proven. However the reverse direction is known to be true for structures that admit NU operations. Zádori has classified in [26] the class of series-parallel posets admitting an NU operation and has shown that every such poset actually admits a 5-ary NU operation. We give a recipe for constructing series-parallel posets of this class analogous to the one mentioned before. Then we show an alternative proof for Zádori's result
Clones over Finite Sets and Minor Conditions
Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra.
In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset.
We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence