Jan Kára

The Complexity of Equality Constraint Languages

Abstract We classify the computational complexity of all constraint satisfaction problems where the constraint language is preserved by all permutations of the domain. A constraint language is preserved by all permutations of the domain if and only if all the relations in the language can be defined by boolean combinations of the equality relation. We call the corresponding constraint languages equality constraint languages. For the classification result we apply the universal-algebraic approach to infinite-valued constraint satisfaction, and show that an equality constraint language is tractable if it admits a constant unary polymorphism or an injective binary polymorphism, and is NP-complete otherwise. We also discuss how to determine algorithmically whether a given constraint language is tractable.

The complexity of surjective homomorphism problems-a survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of whose complexity remains open.

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

AbstractThe visibility graph V(P) of a point set P \subseteq R2 has vertex set P, such that two points v,w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P). We characterise the 2- and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main result is a super-polynomial lower bound on the chromatic number (in terms of the clique number).

Clustered Planarity: Small Clusters in Cycles and Eulerian Graphs

We present several polynomial-time algorithms for c-planarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(|C| + n)-time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graphs, namely graphs obtained from a 3-connected planar graph of fixed size k by multiplying and then subdividing edges. An O(3 · k · n)-time algorithm is presented. We further give an O(|C| + n)-time algorithm for general 3-connected planar graphs. Submitted: December 2007 Reviewed: November 2008 Revised: February 2009 Accepted: August 2009 Final: September 2009 Published: November 2009 Article type: Regular paper Communicated by: S.-H. Hong and T. Nishizeki An extended abstract of this paper appeared in proceedings of Graph Drawing 2007. [14] Department of Applied Mathematics is supported by project 1M0021620838 of the Czech Ministry of Education. Institute for Theoretical Computer Science is supported by grant 1M0545 of the Czech Ministry of Education. The 4th author was supported by the grant GAUK 154907. The 5th author and the 6th author were supported by grant 201/05/H014 of the Czech Science Foundation. The 5th author was also partially supported by the ERASMUS program and by the DFG, project NI 369/4 (PIAF), while visiting Friedrich-Schiller-Universitat Jena, Germany (October 2008–

A fast algorithm and datalog inexpressibility for temporal reasoning

We introduce a new tractable temporal constraint language, which strictly contains the Ord-Horn language of Bürkert and Nebel and the class of AND/OR precedence constraints. The algorithm we present for this language decides whether a given set of constraints is consistent in time that is quadratic in the input size. We also prove that (unlike Ord-Horn) the constraint satisfaction problem of this language cannot be solved by Datalog or by establishing local consistency.

Clustered planarity: small clusters in Eulerian graphs

We present several polynomial-time algorithms for c-planarity testing for clustered graphs with clusters of size at most three. The most general result concerns a special class of Eulerian graphs, namely graphs obtained froma fixed-size 3-connected graph bymultiplying and then subdividing edges. We further give algorithms for 3-connected graphs, and for graphs with small faces. The last result applies with no restrictions on the cluster size.

The complexity of equality constraint languages

We apply the algebraic approach to infinite-valued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the constraint types are Boolean combinations of the equality relation, and we call the corresponding constraint languages equality constraint languages. We show that an equality constraint language is tractable if it admits a constant unary or an injective binary polymorphism, and is NP-complete otherwise.

Fixed parameter tractability of independent set in segment intersection graphs

We present a fixed parameter tractable algorithm for the Independent Set problem in 2-DIR graphs and also its generalization to d-DIR graphs. A graph belongs to the class of d-DIR graphs if it is an intersection graph of segments in at most d directions in the plane. Moreover our algorithms are robust in the sense that they do not need the actual representation of the input graph and they answer correctly even if they are given a graph from outside the promised class.

An Algorithm for Cyclic Edge Connectivity of Cubic Graphs

The cyclic edge connectivity is the size of a smallest edge cut in a graph such that at least two of the connected components contain cycles. We present an algorithm running in time O(n 2log2 n) for computing the cyclic edge connectivity of n-vertex cubic graphs.

Optimal Free Binary Decision Diagrams for Computation of EARn

Free binary decision diagrams (FBDDs) are graph-based data structures representing Boolean functions with a constraint (additional to binary decision diagrams) that each variable is tested during the computation at most once. The function EARn is a Boolean function on n×n Boolean matrices; EARn(M) = 1 iff the matrix M contains two equal adjacent rows. We prove that the size of optimal FBDDs computing EARn is 2?(log2 n).