Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr)$ edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

A Generalist Neural Algorithmic Learner

The cornerstone of neural algorithmic reasoning is the ability to solve algorithmic tasks, especially in a way that generalises out of distribution. While recent years have seen a surge in methodological improvements in this area, they mostly focused on building specialist models. Specialist models are capable of learning to neurally execute either only one algorithm or a collection of algorithms with identical control-flow backbone. Here, instead, we focus on constructing a generalist neural algorithmic learner -- a single graph neural network processor capable of learning to execute a wide range of algorithms, such as sorting, searching, dynamic programming, path-finding and geometry. We leverage the CLRS benchmark to empirically show that, much like recent successes in the domain of perception, generalist algorithmic learners can be built by "incorporating" knowledge. That is, it is possible to effectively learn algorithms in a multi-task manner, so long as we can learn to execute them well in a single-task regime. Motivated by this, we present a series of improvements to the input representation, training regime and processor architecture over CLRS, improving average single-task performance by over 20% from prior art. We then conduct a thorough ablation of multi-task learners leveraging these improvements. Our results demonstrate a generalist learner that effectively incorporates knowledge captured by specialist models.Comment: 20 pages, 10 figure

Finding the connected components of the graph using perturbations of the adjacency matrix

The problem of finding the connected components of a graph is considered. The algorithms addressed to solve the problem are used to solve such problems on graphs as problems of finding points of articulation, bridges, maximin bridge, etc. A natural approach to solving this problem is a breadth-first search, the implementations of which are presented in software libraries designed to maximize the use of the capabi\-lities of modern computer architectures. We present an approach using perturbations of adjacency matrix of a graph. We check wether the graph is connected or not by comparing the solutions of the two systems of linear algebraic equations (SLAE): the first SLAE with a perturbed adjacency matrix of the graph and the second SLAE with~unperturbed matrix. This approach makes it possible to use effective numerical implementations of SLAE solution methods to solve connectivity problems on graphs. Iterations of iterative numerical methods for solving such SLAE can be considered as carrying out a graph traversal. Generally speaking, the traversal is not equivalent to the traversal that is carried out with breadth-first search. An algorithm for finding the connected components of a graph using such a traversal is presented. For any instance of the problem, this algorithm has no greater computational complexity than breadth-first search, and for~most individual problems it has less complexity.Comment: 22 pages, 4 figure

Molassembler: Molecular graph construction, modification and conformer generation for inorganic and organic molecules

We present the graph-based molecule software Molassembler for building organic and inorganic molecules. Molassembler provides algorithms for the construction of molecules built from any set of elements from the periodic table. In particular, poly-nuclear transition metal complexes and clusters can be considered. Structural information is encoded as a graph. Stereocenter configurations are interpretable from Cartesian coordinates into an abstract index of permutation for an extensible set of polyhedral shapes. Substituents are distinguished through a ranking algorithm. Graph and stereocenter representations are freely modifiable and chiral state is propagated where possible through incurred ranking changes. Conformers are generated with full stereoisomer control by four spatial dimension Distance Geometry with a refinement error function including dihedral terms. Molecules are comparable by an extended graph isomorphism and their representation is canonicalizeable. Molassembler is written in C++ and provides Python bindings.Comment: 81 pages, 26 figures, 3 table

Towards Better Out-of-Distribution Generalization of Neural Algorithmic Reasoning Tasks

In this paper, we study the OOD generalization of neural algorithmic reasoning tasks, where the goal is to learn an algorithm (e.g., sorting, breadth-first search, and depth-first search) from input-output pairs using deep neural networks. First, we argue that OOD generalization in this setting is significantly different than common OOD settings. For example, some phenomena in OOD generalization of image classifications such as \emph{accuracy on the line} are not observed here, and techniques such as data augmentation methods do not help as assumptions underlying many augmentation techniques are often violated. Second, we analyze the main challenges (e.g., input distribution shift, non-representative data generation, and uninformative validation metrics) of the current leading benchmark, i.e., CLRS \citep{deepmind2021clrs}, which contains 30 algorithmic reasoning tasks. We propose several solutions, including a simple-yet-effective fix to the input distribution shift and improved data generation. Finally, we propose an attention-based 2WL-graph neural network (GNN) processor which complements message-passing GNNs so their combination outperforms the state-of-the-art model by a 3% margin averaged over all algorithms. Our code is available at: \url{https://github.com/smahdavi4/clrs}