Simple Subgraphs Research Articles

Parsing Chinese Sentences with Grammatical Relations

We report our work on building linguistic resources and data-driven parsers in the grammatical relation (GR) analysis for Mandarin Chinese. Chinese, as an analytic language, encodes grammatical information in a highly configurational rather than morphological way. Accordingly, it is possible and reasonable to represent almost all grammatical relations as bilexical dependencies. In this work, we propose to represent grammatical information using general directed dependency graphs. Both only-local and rich long-distance dependencies are explicitly represented. To create high-quality annotations, we take advantage of an existing TreeBank, namely, Chinese TreeBank (CTB), which is grounded on the Government and Binding theory. We define a set of linguistic rules to explore CTB’s implicit phrase structural information and build deep dependency graphs. The reliability of this linguistically motivated GR extraction procedure is highlighted by manual evaluation. Based on the converted corpus, data-driven, including graph- and transition-based, models are explored for Chinese GR parsing. For graph-based parsing, a new perspective, graph merging, is proposed for building flexible dependency graphs: constructing complex graphs via constructing simple subgraphs. Two key problems are discussed in this perspective: (1) how to decompose a complex graph into simple subgraphs, and (2) how to combine subgraphs into a coherent complex graph. For transition-based parsing, we introduce a neural parser based on a list-based transition system. We also discuss several other key problems, including dynamic oracle and beam search for neural transition-based parsing. Evaluation gauges how successful GR parsing for Chinese can be by applying data-driven models. The empirical analysis suggests several directions for future study.

Frames, $A$-Paths, and the Erdös--Pósa Property

A key feature of Simonovits' proof of the classic Erdos--Posa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique t...

Improved Symmetry Method for the Mobility of Regular Structures Using Graph Products

AbstractMobility analysis plays a key role in form finding and design of novel kinematically indeterminate structures. For large-scale or complex structures, it demands considerable computations and analyses, and, thus, efficient method is of great interest. Because many structures could be viewed as the product of two or three subgraphs, such structures are called regular structures and usually hold certain symmetries. Combining graph theory with group representation theory, this paper proposes an improved symmetry method for the mobility of kinematically indeterminate pin-jointed structures. The concepts of graph products are described and utilized, to simplify the conventional symmetry-extended mobility rule. Based on the definitions of the Cartesian product, the direct product, and the strong Cartesian product, the authors establish the representations of nodes and members for the graph products, respectively. The proposed method focuses on the simple subgraphs, which generate the entire structure, an...

New Aspects of Regions whose Tilings are Enumerated by Perfect Powers

In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (J. Algebraic Combin., 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (Ph.D. thesis, MIT, 1991). On the square lattice with zigzag paths, we consider two particular families of regions whose numbers of tilings are always a power of 3 or twice a power of 3. The latter result provides a new proof for a conjecture of Matt Blum first proved by Ciucu. We also consider several new lattices obtained by periodically applying two simple subgraph replacement rules to the square lattice. On some of those lattices, we get new families of regions whose numbers of tilings are given by products of several perfect powers. In addition, we prove a simple product formula for the number of tilings of a certain family of regions on a variant of the triangular lattice.

On the Hyperbolicity Constant of Line Graphs

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: X \text{ is }\delta\text{-hyperbolic}\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main aim of this paper is to obtain information about the hyperbolicity constant of the line graph $\mathcal{L}(G)$ in terms of parameters of the graph $G$. In particular, we prove qualitative results as the following: a graph $G$ is hyperbolic if and only if $\mathcal{L}(G)$ is hyperbolic; if $\{G_n\}$ is a T-decomposition of $G$ ($\{G_n\}$ are simple subgraphs of $G$), the line graph $\mathcal{L}(G)$ is hyperbolic if and only if $\sup_n \delta(\mathcal{L}(G_n))$ is finite. Besides, we obtain quantitative results. Two of them are quantitative versions of our qualitative results. We also prove that $g(G)/4 \le \delta(\mathcal{L}(G)) \le c(G)/4+2$, where $g(G)$ is the girth of $G$ and $c(G)$ is its circumference. We show that $\delta(\mathcal{L}(G)) \ge \sup \{L(g):\, g \,\text{ is an isometric cycle in }\,G\,\}/4$. Furthermore, we characterize the graphs $G$ with $\delta(\mathcal{L}(G)) < 1$.

Degree Bounded Network Design with Metric Costs

Given a complete undirected graph, a cost function on edges, and a degree bound $B$, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree $B$ satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a $(2+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when $r_{\max}$ is even, and a 5.5-approximation algorithm when $r_{\max}$ is odd, where $r_{\max}$ is the maximum connectivity requirement. In global vertex-connectivity, there is a $(2+\frac{k-1}{n}+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-vertex-connected subgraph problem when $n\geq2k$, where $n$ is the number of vertices. For spanning tree, there is a $(1+\frac{1}{B-1})$-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

Algorithms on clique separable graphs

We define a family of graphs, called the clique separable graphs, characterized by the fact that they have completely connected cut sets by which we decompose them into parts such that when no further decomposition is possible we have a set of simple subgraphs. For example the chordal graphs and the i-triangulated graphs are clique separable graphs. The purpose of this paper is to describe polynomial time algorithms for the recognition of the clique separable graphs and for finding them a minimum coloring and a maximum clique.