We define a new general rule-based nonmonotonic framework which allows an external acyclic priority relation between rules to be interpreted in several ways. Several entailment semantics are defined via a constructive digraph, with one being given a declarative fixed-point characterisation as well. Each of these semantics satisfies Principle 1 of [Brewka and Eiter 1999]. The framework encompass...

An (n,N, d)–connector is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length d joining each input to its corresponding output. We consider the problem of construction of sparse (n,N, 2)–connectors (depth 2 connectors) when n N . The probabilistic argument in [1] shows the existence...

Given an edgeor vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NPhard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pai...

A directed network connecting a set A to a set B is a digraph containing an a-b path for each a ∈ A and b ∈ B. Vertices in the directed network not in A ∪ B are Steiner points. We show that in a finitely compact metric space in which geodesics exist, any two finite sets A and B are connected by a shortest directed network. We also bound the number of Steiner points by a function of the sizes of...

Given an edge-weighted undirected graph and two vertices s and t, the next-to-shortest path problem is to find an st-path whose length is minimum among all st-paths of lengths strictly larger than the shortest path length. The problem is shown to be polynomially solvable if all edge weights are positive, while the complexity status for the nonnegative weight case was open. In this paper we show...

The Max Cut problem is an NP-hard problem and has been studied extensively. Alon et al. studied a directed version of the Max Cut problem and observed its connection to the Hall ratio of graphs. They proved, among others, that if an acyclic digraph has m edges and each vertex has indegree or outdegree at most 1, then it has a directed cut of size at least 2m/5. Lehel et al. extended this result...

The shortest path problem in which the (s, t)-paths P of a given digraph G = (V,E) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, k-sum and k-max objectives,...

Given an edge-weighted undirected graph and two vertices s and t, the next-to-shortest path problem is to find an st-path whose length is minimum among all st-paths of lengths strictly larger than the shortest path length. The problem is shown to be polynomially solvable if all edge weights are positive, while the complexity status for the nonnegative weight case was open. In this paper we show...

The paper is organized as a self-contained literate Haskell program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The code, tested under GHC 6.6.1, is available at http://logic.csci.unt.edu/tarau/ research/2008/fSET.zip. We introduce ranking and unranking functions generalizing Ackermann’s encoding to the universe of...

We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority opera...