Computer Science

In context of the Wolfram Physics Project, a certain class of abstract rewrite systems known as "multiway systems" have played an important role in discrete models of spacetime and quantum mechanics. However, as abstract mathematical entities, these rewrite systems are interesting in their own right. This paper undertakes the effort to establish computational properties of multiway systems. Specifically, we investigate growth rates and growth classes of string-based multiway systems. After introducing the concepts of "growth functions", "growth rates" and "growth classes" to quantify a system's state-space growth over "time" (successive steps of evolution) on different levels of precision, we use them to show that multiway systems can, in a specific sense, grow slower than all computable functions while never exceeding the growth rate of exponential functions. In addition, we start developing a classification scheme for multiway systems based on their growth class. Furthermore, we find that multiway growth functions are not trivially regular but instead "computationally diverse", meaning that they are capable of computing or approximating various commonly encountered mathematical functions. We discuss several implications of these properties as well as their physical relevance. Apart from that, we present and exemplify methods for explicitly constructing multiway systems to yield desired growth functions.

For a bilinear map ∗: R d × R d → R d of nonnegative coefficients and a vector s∈ R d of positive entries, among an exponentially number of ways combining n instances of s using n−1 applications of ∗ for a given n , we are interested in the largest entry over all the resulting vectors. An asymptotic behavior is that the n -th root of this largest entry converges to a growth rate λ when n tends to infinity. In this paper, we prove the existence of this limit by a special structure called linear pattern. We also pose a question on the possibility of a relation between the structure and whether λ is algebraic.

Let G=(V,E) be a plane graph. A face f of G is guarded by an edge vw∈E if at least one vertex from {v,w} is on the boundary of f . For a planar graph class G we ask for the minimal number of edges needed to guard all faces of any n -vertex graph in G . We prove that ⌊n/3⌋ edges are always sufficient for quadrangulations and give a construction where ⌊(n−2)/4⌋ edges are necessary. For 2 -degenerate quadrangulations we improve this to a tight upper bound of ⌊n/4⌋ edges. We further prove that ⌊2n/7⌋ edges are always sufficient for stacked triangulations (that are the 3 -degenerate triangulations) and show that this is best possible up to a small additive constant.

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices S of a graph G is a \emph{geodetic set} if any vertex of G lies in some shortest path between some pair of vertices from S . The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless P=NP , there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter 2 . On the positive side, we give an O( n − − √ 3 logn) -approximation algorithm for the \textsc{MGS} problem on general graphs of order n . We also give a 3 -approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs.

Connectivity is a central notion of graph theory and plays an important role in graph algorithm design and applications. With emerging new applications in networks, a new type of graph connectivity problem has been getting more attention--hedge connectivity. In this paper, we consider the model of hedge graphs without hedge overlaps, where edges are partitioned into subsets called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. This model is more general than the hypergraph, which brings new computational challenges. It has been a long open problem whether this problem is solvable in polynomial time. In this paper, we study the combinatorial properties of hedge graph connectivity without hedge overlaps, based on its extremal conditions as well as hedge contraction operations, which provide new insights into its algorithmic progress.

A new metric parameter for a graph, Helly-gap, is introduced. A graph G is called α -weakly-Helly if any system of pairwise intersecting disks in G has a nonempty common intersection when the radius of each disk is increased by an additive value α . The minimum α for which a graph G is α -weakly-Helly is called the Helly-gap of G and denoted by α(G) . The Helly-gap of a graph G is characterized by distances in the injective hull H(G) , which is a (unique) minimal Helly graph which contains G as an isometric subgraph. This characterization is used as a tool to generalize many eccentricity related results known for Helly graphs ( α(G)=0 ), as well as for chordal graphs ( α(G)≤1 ), distance-hereditary graphs ( α(G)≤1 ) and δ -hyperbolic graphs ( α(G)≤2δ ), to all graphs, parameterized by their Helly-gap α(G) . Several additional graph classes are shown to have a bounded Helly-gap, including AT-free graphs and graphs with bounded tree-length, bounded chordality or bounded α i -metric.

In this paper we are interested in decomposing a dihypergraph H=(V,E) into simpler dihypergraphs, that can be handled more efficiently. We study the properties of dihypergraphs that can be hierarchically decomposed into trivial dihypergraphs, \ie vertex hypergraph. The hierarchical decomposition is represented by a full labelled binary tree called H -tree, in the fashion of hierarchical clustering. We present a polynomial time and space algorithm to achieve such a decomposition by producing its corresponding H -tree. However, there are dihypergraphs that cannot be completely decomposed into trivial components. Therefore, we relax this requirement to more indecomposable dihypergraphs called H-factors, and discuss applications of this decomposition to closure systems and lattices.

The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or k -Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs G=(V,E) of skeleton dimension O( log 2 |V|) it is NP-hard to approximate the k -Center problem within a factor less than 2 .

We prove that every planar graph is the intersection graph of homothetic triangles in the plane.

We describe a new non-constructive technique to show that squares are avoidable by an infinite word even if we force some letters from the alphabet to appear at certain occurrences. We show that as long as forced positions are at distance at least 19 (resp. 3, resp. 2) from each other then we can avoid squares over 3 letters (resp. 4 letters, resp. 6 or more letters). We can also deduce exponential lower bounds on the number of solutions. For our main Theorem to be applicable, we need to check the existence of some languages and we explain how to verify that they exist with a computer. We hope that this technique could be applied to other avoidability questions where the good approach seems to be non-constructive (e.g., the Thue-list coloring number of the infinite path).