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

Quantified Constraints in Twenty Seventeen

I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions

Surjective H-Colouring over reflexive digraphs

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen (2014) proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3

Quantified Constraint Satisfaction Problem on semicomplete digraphs

We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over semicomplete digraphs. We obtain a complexity-theoretic trichotomy: QCSP(H) is either in P, is NP-complete, or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicomplete digraphs enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right

On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width

The relational width of a finite structure, if bounded, is always (1,1) or (2,3). In this paper we study the relational width of first-order expansions of finitely bounded homogeneous binary cores where binary cores are structures with equality and some anti-reflexive binary relations such that for any two different elements a, b in the domain there is exactly one binary relation R with (a, b) in R. Our main result is that first-order expansions of liberal finitely bounded homogeneous binary cores with bounded strict width have relational width (2, MaxBound) where MaxBound is the size of the largest forbidden substructure, but is not less than 3, and liberal stands for structures that do not forbid certain finite structures of small size. This result is built on a new approach and concerns a broad class of structures including reducts of homogeneous digraphs for which the CSP complexity classification has not yet been obtained.Comment: A long version of an extended abstract that appeared in LICS 202

Gyenge többségi függvények = Weak near-unanimity operations

Az OTKA pályázatom benyújtása (2008. február 11) óta 8 cikkem jelent nemzetközi folyóiratokban, de ezek közül csak kettő cikk született a kutatási programban megadott témában a beszámolási időszak alatt, ezért a többi 6 megjelent cikket nem tüntettem fel a jelentésben. Ezen cikkeken kívül a kutatási tervnek megfelelően további 3 kézirat van publikálásra benyújtva, illetve egy kézirat született, amely nincsen még benyújtva. A kutatási időszakban összesen hat nemzetközi konferencián adtam elő, ezek közül négyen meghívott előadóként. Az [1] cikkben egy irányított fákból álló speciális gráfosztályra, az úgynevezett speciális triádokra bizonyítjuk a homomorfizmus problémára vonatkozó dichotómia sejtés. A [2] cikkben algoritmust adunk arra, hogy láncok direkt szorzatában az azonos elemszámú ideálok melyikében maximális az elemek magasságának összege. A [3] kéziratban a korlátos szélességű és kevés részhatvánnyal rendelkező algebrákra vonatkozó kényszerkielégíthetőségi problémát megoldó algoritmusokat ötvöztem. A [4] kéziratban bebizonyítottuk a Valeriote sejtést reflexív irányított gráfokra. Az [5] kéziratban a Valeriote sejtés több ekvivalens megfogalmazását adtuk meg. A [6] kéziratban a CSP probléma egy teljesen új redukcióját vezettük be, amely segítségével újabb algebraosztályokra bizonyítható a dichotómia sejtés. | Since the submission of my OTKA grant proposal (2008/08/11) I had eight articles appeared in international journals, but only two of them were on a topic listed in the project proposal. Therefore, I did not mention the other 6 in this report. Beside these articles I have 3 submitted manuscripts and one manuscript not jet submitted. During the three years of this research grant I have given 6 talks on international conferences, four of them were invited plenary talks. In [1] we have proved the constraint satisfaction dichotomy conjecture for a special class of directed trees, for the class of special triads. In [2] we give an algorithm to determine the order ideal of a direct product of chains in which the number of elements equals a fixed integer and the sum of heights of elements is maximal. In [3] I have combined the algorithms solving the constraint satisfaction problem for bounded width algebras and for algebras of few subpowers. In [4] we have proved the Valeriote conjecture for reflexive directed graphs. In [5] we have given equivalent formulations of the Valeriote conjecture. in [6] we have introduced a completely new reduction of CSP problems that allowed the proof of the dichotomy conjecture for various new classes of algebras