Jan Foniok

Pivoting in Linear Complementarity: Two Polynomial-Time Cases

We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty’s least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP.

Counting unique-sink orientations

Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarize old and show new lower and upper bounds on the sizes of some such classes. Furthermore, we provide a characterization of K-matrices in terms of their corresponding USOs.

Generalised dualities and finite maximal antichains

We fully characterise the situations where the existence of a homomorphism from a digraph G to at least one of a finite set of directed graphs is determined by a finite number of forbidden subgraphs. We prove that these situations, called generalised dualities, are characterised by the non-existence of a homomorphism to G from a finite set of forests. Furthermore, we characterise all finite maximal antichains in the partial order of directed graphs ordered by the existence of homomorphism. We show that these antichains correspond exactly to the generalised dualities. This solves a problem posed in [1]. Finally, we show that it is NP-hard to decide whether a finite set of digraphs forms a maximal antichain.

Digraph functors which admit both left and right adjoints

For our purposes, two functors ? and ? are said to be adjoint if for any digraphs G and? H , there exists a homomorphism of? ? ( G ) to? H if and only if there exists a homomorphism of? G to? ? ( H ) . We investigate the right adjoints characterised by Pultr (1970). We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where these necessary conditions are satisfied, and the right adjoint indeed exists. Finally, we discuss a connection between these right adjoints and homomorphism dualities.

Combinatorial Characterizations of K-matrices

Abstract e present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies the original proof substantially by exploiting the duality of oriented matroids. As an application, we show that any simple principal pivot method applied to the linear complementarity problems with K-matrices converges very quickly, by a purely combinatorial argument.

Dualities and Dual Pairs in Heyting Algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.

On Finite Maximal Antichains in the Homomorphism Order

Abstract We show that for structures with at most two relations all finite maximal antichains in the homomorphism order correspond to finite homomorphism dualities. We also show that most finite maximal antichains in this order split.

Infinitely many minimal classes of graphs of unbounded clique-width

The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width.

Deciding the Bell Number for Hereditary Graph Properties

The paper [J. Balogh, B. Bollobas, D. Weinreich, J. Combin. Theory Ser. B, 95 (2005), pp. 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number

Deciding the Bell number for hereditary graph properties

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